Hello, I really enjoy reading this tech forum - this is my first post and I don't quite know where to start.... I'm learning to tune and I've got this old book describing to tune to the cirle of fifths. Now after 30, 40 attempts I just can't get it right. I tune all fifths slightly narrow, probably about half a wave per second, but I always end up with the last fifth (d to a) beating too fast - even if I almost exagerate the fifths so they definitely beat too fast? I was thinking maybe I'm losing something in the octaves (by making them a tiny bit too narrow) but I just can't imagine such a small inaccuracy in the octave would cause this. I then work all the way backwards,trying to re-adjust it and judging by the amount I have to nudge the pins back, I'm out by quite a lot, I just don't understand where the mistake lies.
I'm also trying to tune to a different scale, a C scale with the F being slightly sharp. I probably haven't studied the theory enough, but with the ET being equal (ie the end result should always be the same) does it mean the F which is sharp in the C scale is in fact tuned to the same frequency in both scales, in the C scale as well as in the large temperament of the circle of fifths? As I narrow the fifth with the F as the higher note in a fifth it feels as if I'm flattening the F?
I don't want to get stuck on the circle of fifths - I just hope somebody can explain to me what I might be doing wrong!

Welcome Aboard!

What you are doing wrong is not using the 3rds and 6ths as they become available. This will give you a better indication if the fifths and fourths are too wide or narrow. But it is when you tune the 9th note that you find out if you really have it right. Nine notes gives you a major 3rd above and below the starting note.

Can you hire someone to teach you to tune? That is what I did.

Hi Ron,
I've been learning to tune/voice/regulate and repair for the past year. There are several really good methods for getting yourself into some type of standard approach to your tuning.

My suggestion is to experiment with a couple of well-established methods to help you keep your focus if you are going to be self-taught, like I am.

Bill Bremmer's ET via Marpurg is a most excellent method and my first choice. Kent Swafford's Every Which Way method is most excellent as well - I used that method quite frequently. I can't get to the links from this location, but you can find them on the web or one of the other fine techs here can assist.

Jeff's suggestion is timely - having some mentoring sessions with an experienced tech will prove very helpful. I wish I'd done that when I first started out.

Glen

Many thanks for the replies, will look into it.

Ron,

You have identified a very often and sometimes heated discussion both here and elsewhere. Tuning through the cycle of fifths is a very natural way to construct a temperament. You have 12 notes (plus one that creates an octave), so it is rather logical to choose a sequence of fourths and fifths that would lead through an entire chromatic octave. A fourth, is of course, an upside down fifth. Nearly any temperament sequence of any type, equal and non-equal has, over the centuries, used fourths and fifths.

The problem with using only fourths and fifths is that one can never know just how much to temper any particular interval. The piano's natural inharmonicity complicates tuning greatly. An octave that sounds "pure" (beatless) to the ear is still stretched of slightly wider than it would be theoretically. The optimum amount that a central, "temperament" octave should be stretched is the subject of much debate.

Regardless of what anyone may personally believe about how much the temperament octave should be stretched, however, simply tuning it so that no beat is heard is not sufficient. It could still be slightly narrow or range from a 2:1 type to a 4:2 type and still sound apparently the same when only the octave is played.

The exact width of the octave will directly affect how much each of the other intervals within it must be tempered for the end result of an Equal Temperament (ET) to be established. It is virtually impossible to achieve a true ET simply by tuning an octave that sounds "pure" and listening only to fourths and fifths. A very experienced tuner may get close but the usual result is what you have experienced: towards or at the end of the sequence, you find a dilemma. There is no way to reconcile that last few fourths and fifths without "backing up" through the sequence.

If you do that, then it means you have changed what you previously thought was correct. What about those intervals you left untouched? From what I read that you did, you created a temperament that could not be ET at all, even if the final result was that all fourths and fifths sounded reasonable and acceptable.

Unfortunately, there are many publications, including one correspondence course that many people buy because it is cheaper than others, seems easy to read and follow but frankly leaves out the most critical information. Other publications may have more detailed information but people often ignore that seemingly more complicated information and try to tune a temperament using only fourths and fifths. This has lead to many a tuner, often a professional who makes the same kind of errors every time for a lifetime and never once tunes a true ET but firmly believes that he/she has done so.

The essence of aural tuning is the perception and control of beats, both rapid and slow. One can never tune a true ET without being able to perceive and control both Slowly Beating Intervals (SBI) and Rapidly Beating Intervals (RBI).

Coincidentally, another tuning novice, Martin Store has recently gone through much the same problem as you are experiencing. Please find and read the thread:

https://www.pianoworld.com/forum/ubb...emperaments%20-%20which.html#Post1653259

In a very short time, Martin has been able to produce a clinically perfect ET using the suggestions that I and others have provided. He has posted recordings of his results that you can use for your own benefit.

Someone else mentioned the material I have published and I thank that person for that. I have published many articles to try to help novice tuners or those who first learned to tune using and Electronic Tuning Device (ETD) learn how to tune aurally. You will find it to be of great personal satisfaction and gratification if you do as well.

I would suggest you look on my website:

http://www.billbremmer.com/articles/

I suggest that you begin with the Midrange Piano Tuning article but only pages 1-4 would be necessary to study at this point. Then go to the "ET via Marpurg" article and study it carefully. You will have to learn how to tune a sequence of RBI's first and that may take some time and seem quite difficult but once you master that, you can complete the rest of the sequence using only fourths and fifths. The sequence works amazingly well because it avoids all of the kinds of errors that people typically make when using a traditional fourths and fifths sequence.

Good luck and please let us know of your progress. If you have a way to record and post your results, we would all like to hear them and provide feedback. You may well want to know if your first steps at tuning the RBI's sound correct or not. Please be aware that there is also a complete series of videos that cover this subject:

http://www.billbremmer.com/videos/

There is also a sound file that displays just how these initial RBI's should sound. It should be very helpful to you:

http://www.billbremmer.com/videos/ET_CM3s.mp3

All of this material is free to view, download and share.

That's where testing other intervals keep you on track. By the time I tune F#3 to B3, I have the F#3-A3 minor third to check. If it's beating too fast or bordering on dissonant, then I know I'm running 4ths-5ths too pure. That happens early in the temperament (three intervals), which lets me go back and rework early in the game.

With practice, it is indeed possible to get a smooth, clean temperament with smooth, progressively faster M3rds in the temperament octave by running 4ths and 5ths. I've done it for almost 30 years now, and it is the method I used when I passed the Guild tuning exam with a score in the high 90's. (Disclaimer: I'm no longer in the Guild).

Sometimes I think the whole process is made way too difficult by overanalyzing. A proficient tuner should be able to set a good, solid temperament in minutes.

There is just something inherently nice about the (progression of) fifths! (where as the forths are not so melodic). Thanks Bill I will look into your material.

So, I'll be the party pooper...

While it is "quaint" - or perhaps "noble" to pursue aural tuning, modern electronic tuning devices are much more able to construct either equal temperament or any number of tonal temperaments that can then be spread to the rest of the piano by traditional means.

If you are learning aural tuning for your own edification/satisfaction, ignore this post. If you are trying to become a good or even great tuner.... I'd suggest another path.

Ron Koval
chicagoland

Hi Ron
I'm not sure what you mean? Not to focus on the circle of fifths? (I won't, I just used it for my first attempts) Which path would you suggest?
Thanks, Ron.

I agree with you, Ron. While I believe that every tuner should possess good critical aural skills, the goal should be to produce a wonderful tuning.

Every year I bring my financial info to my accountant who does my income taxes for me. It makes no difference to me whether he uses a calculator and computer software or if he does long math with a pencil an paper; in the end I want accurate results and a job well done.

Ron, I have a few suggestions. First, see if perhaps there is a particular note range you have the most problems with, and see if perhaps you might have a slight hearing loss there. I had much the same problems you have described and finally figured that I was tuning my g-g# too narrow. I adjusted that intellectaully and and found that it worked. It wasn't until some years later that a hearing test showed a slight loss there, and a musician/audio tech friend of mine mentioned that this seemed to be the area of common loss for baritones... it corresponded to the area of a baritone's natural break in his voice. My daughter has a weakness in the Eb-E range... her vocal break... hmmm. Secondly, try the circle DOWN from the A till just about the C, then start back UP sweetening it up as you go, and get back to the C from the other side. See if that changes things. Chances are, you will see the difference to be greatly reduced. Using 4-5ths going down makes one tune a bit flat, and going up makes it a bit sharp. BTW, the reversing of the direction of the circle of fifths will help to pinpoint that area of your range that is maybe a bit off. Try it. It works for me and others whom I have talked to...
There are many methods of tuning, and many scales, I use this method, others don't. There is place and customers enough for us all, and this one works for me.

Consider tuning as three skill sets:
1. tuning unisons
2. tuning the temperament (tonal or atonal)
3. transferring the temperament to the bass and treble

1 and 3 are pretty "simple" to do by ear, harder for the machines.
2 is the hardest to learn by ear, and simple for machines.

So. Get an EDT.... now. Use it to set the A3-A5 two octaves - We can help you verify that the A3 -A4 -A5 single and double octaves are in a "good" place before proceeding.
(We can help with how to set one up, depending on what you choose. Tunelab is often the first choice of those beginning because of the price and the ability to try it out before buying.)

Then spread that temperament out to the bass and treble. Again, as you have questions come back here and people can give specific suggestions.

Ron Koval

Haha, that sounded amusing - as though your tuning may have turned rogue against the establishment.

Haha! Actually it's more benign than that. In 1995 I went through a move and thought it was pretty certain I'd be leaving the trade, so I didn't bother to renew my membership. I've thought of getting in again, but I don't feel the need to go through the hassle of having to take the tests again, so I'm content as a former RPT.

Yes, please. I've been practicing aural tuning and learning to use TuneLab at the same time. I used TuneLab to record each note of a Steinway that had been tuned aurally by a trusted tuner/technician. I then had TuneLab calculate a tuning curve for the same piano using inharmonicity measurements. TuneLab's octave stretching was quite a bit more conservative than the tuner's. So I'd be very interested to hear how you would check the temperament octave stretch before proceeding.

I'd add pitch raising and setting the pins and strings (principally good hammer technique) to that list. And lets not forget about customer relationship, business, and time management skills among others.

Aural tuning can teach much about octave sizes, dealing with inharmonicity, interval tests, and adds to "knowing what I am doing" as well as "why am I doing it."

IMHO. I would not suggest to buying an ETD until a beginning tuner can tune at least using the sanderson-baldassin two octave temperament by ear, only because of the lessons that are learned and skills gained.

I can't say I completely agree. I think that any technician should be able to deliver a solid tuning by ear, just in case there is an emergency situation and one is required to tune without a device. I personally tune with a machine, but feel I can deliver a satisfactory tuning without one, though I feel I can do a better job, faster with one.

Just my thoughts,

Ah yes, the "what if something happens to the ETD defense"... What happens if your tuning lever breaks, if your mutes burst into flames, or your tuning fork drops down a sewer? That's right, you use a backup.

I'm all for technicians choosing to enhance their skillset by learning aural tuning. I've followed up on too many "experienced" aural techs that leave 10 cent errors in the temperament to ignore the value of modern electronic tuning devices for consistent results.

Ron Koval

I'm not saying that the tuning would be perfect, or that aural tuning is "always" the gold standard, but there have been times when I have left my ETD 80 miles away at home, and was left with only my tuning fork to tune. I was very grateful that I could tune and pitch raise that spinet by ear, and not have to drive back to get my ETD.

I am willing to admit that I am a better tuner with an ETD, but I can still tune without one, just as I can navigate (poorly) without a GPS, but wouldn't want to leave home without one. Aural tuning is not that difficult to learn, and even if it isn't learned initially, should eventually be attainable to a reasonable degree by a quality technician.

Here's one approach:

Consider how you tune two strings of a unison - one string is tuned and then the second is brought slightly above, below, above and then back to the cleanest spot. Your ears and hands work together to "know" where the cleanest spot by moving through it and then stopping there.

You can tune an octave the same way. First tune A4 to tunelab. Next tune A3 to tunelab. Now while playing the octave, slightly move A3 up and down in pitch to determine the cleanest spot (without watching tunelab if needed) Compare that with the tunelab calculation. Then do the same with A5. Next play the A3-A5 double octave as a "go/no go" check. Is the double octave good?

That's where you want tunelab to place the A3-A4-A5 ladder. (You may adjust tunelab at this point to make it agree.)

After you are good with that, add D3 and E5 to the mix to compare the octave, the double octave and the octave fifth for equal "cleanness".

Machines don't need to be limited to the A-A temperament. Many instruments do better staying away from the lowest half octave of plain wire string to set the temperament octave or two octave. I often use a C4-C6 double octave to start.

Ron Koval

Would you be able to point me to a (web) source for this temperament?

After 2 months, I find learning to set the pins the hardest. I feel fairly confident that eventually I will know how to set a scale, but as yet I don't feel I've improved in setting the pins properly.
I've begun to make notes of my unison tuning. If I do all three-stringed notes on the piano and come back to the piano a few hours later, maybe 10 of them need a little adjusting. When I check again next day, I still need to adjust maybe 5, and often there are a couple of new ones which didn't need to be touched at the first check! This could go on for days, there's always a couple of notes I'm not happy with, and it's not because the pins are loose.

@Mark: the temperament is included in the appendix section of the accu-tuner IV manual:

Accu-Tuner IV Manual (pdf, 750 KB)

Octaves this, ETDs that, beat rates, too! But ya know what? People continue to tune an excellent ET using the circle of fifths. And no matter what the temperament sequence, you don’t know if you have it until the ninth note.

But there is a real difference in using the circle of fifths: the fourths and fifths sound better! And remember that the circle of fifths is why there are 12 notes. And when it comes to the inevitable compromises that are required on some pianos, I’ll take better sounding fourths and fifths over progressive 3rds and 6ths any day of the week.

Let’s not throw the baby out with the bath!

Heaven help me, I agree with Jeff! :P

Uh, if you like I could edit my post

Does anybody happen to have a description of the circle of fifths tuning sequence and which tests are available along the way?

I am going to directly challenge this. I have seen the beatrate curves for RBIs on a Baldwin Studio as tuned by a Verituner. They were picture perfect. But I know how the SBIs would sound: HORRIBLE!.

It reminds me very much of the 70’s when I learned to tune. Strob-o-tuners were becoming popular because they were supposed to be more accurate. They didn’t know better, I guess. Oh, but now we do and the answer is modern ETDs? I do not buy it. Sure, if you cannot set a temperament aurally, you might as well use a machine. But that does not mean that a machine does a better job than someone that can tune aurally. Want to prove if you can really set an accurate aural temperament? Use the circle of fifths!

And another thing, what is this nonsense about “tonal” vs “atonal” temperaments? ET is omnitonal, not atonal. UT is … heck I don’t know what it is! It just sounds out of tune!

I agree with you on this also Jeff. My teacher in college twice demonstrated (on two different pianos)to my entire class that it is entirely possible to set a perfect E.T. using only 4ths and 5ths. It took him about 5 minutes and some backtracking and reconciliation of some intervals. It was explained to us that different pianos, as for size and scaling, presented extremely small variations to the amount of tempering and that with experience you come to recognize this and adapt. The fact that we as students did not even deal with other checks and intervals (other than the octave) for the first 3 months relayed the importance of SBI's to us in setting the temperament.

Later, as we progressed, we learned the other intervals and checks that are at our disposal, but the importance of the 5ths and 4ths was never ignored nor relegated to the back seat of a RBI driven temperament. It was also very important for us to reasonably recognize the (theoretically correct) F3-A3 beat rate. Although the application of it is not set in stone for all types and sizes of pianos in general, the deviancy from it is not that great either, that one should not bother to learn it.

Thousands of tuners have learned to tune in much the same way. There are other methods that take short cuts or completely avoid learning some of these rudimentary things. It also annoys me to no end when I come across an aurally set temperament with a beautifully cascading set of M3rds and I hear a beatless perfect fifth beside another one thats beating faster (to the correct side)than what the fourths are.

An interesting thing that I came across years later when incorporating an ETD in the setting of a temperament is that when i did my aural testing of the temperament, I have rarely had to adjust anything in regards to RBI's, but on more than a few occasions, I would tweak the SBI's to my satisfaction and do a quick check on the RBI's before moving on.

Here is a link to a book online. It is considered to be the standard way of tuning by 4ths and 5ths:

http://www.archive.org/stream/modernpianotunin00whit#page/n3/mode/2up

And here is a recent Topic on the subject with some variations:

https://www.pianoworld.com/forum/ubbthreads.php/topics/1612251/Re:%204ths%20and%205ths,%20I%20love%20'em.html

The thing to remember is that every interval can be formed with 4ths and 5ths. By comparing the beatrates of these other intervals, errors in the 4ths and 5ths can be detected and corrected. With practice, the errors become smaller and less frequent. Then the errors in a piano’s scaling show up instead.

When you get past a certain level of tuning, it doesn't matter much where you start aurally. I speak a lot in favor of CM3's, and I always start by placing them, but then refine the temperament guided by 4ths/5ths.

I think pretty much everybody agree that both SBI's and RBI's are needed. Just as Jeff and Emmery, I give priority to the SBI's, and I believe most tuners do.

Please feel free to challenge this, it's just from the top of my mind.

If you will follow along, the recommendation I made suggested using a temperament location higher than an aural setting. This avoids the problems that you mention with the Baldwin studios because the temperament is then spread out with a hybrid method that leans heavily on the old aural tradition of using octaves/doubles/and octave fifth to place those tenor/bass notes. RBI's are ignored with this method.

As to tonal/atonal - look it up in the dictionary. A tuning without a built in tonal center is by definition atonal. You can call it omnitonal, but I don't think that is really how it functions. Tonal tunings on the other hand are just that - tunings that have a tonal center.

Ron Koval

Pat:

I certainly do not challenge how you tune. I do challenge how tuning sequences are explained. There is a strong insinuation that a firm ladder of CM3s is the only correct way to start a temperament and that these should not be touched after being set.

But what I find impossible to decide on, when I attempt to start with a ladder of CM3s and something needs to be adjusted, is if it the error is the F, the C# or both? It becomes ambiguous where the error is. There is a solution, but only for well scaled pianos. It is using ET via Marpurg with additional RBI checks immediately after tuning the temporarily tuned SBIs.

Ron:

Sure!

Here are definitions from the Merriam-Webster website:

Definition of TONALITY
1: tonal quality
2a: key 7 b : the organization of all the tones and harmonies of a piece of music in relation to a tonic
3: the arrangement or interrelation of the tones of a work of visual art


Definition of ATONAL
: marked by avoidance of traditional musical tonality; especially : organized without reference to key or tonal center and using the tones of the chromatic scale impartially


This has to do with written music NOT tuning theory!

But your suggestion of using an ETD to set the temperament only in a well scaled part of the piano is a good one. I wonder how a non-aural tuner is to know where this is.

Yes it has to do with written music - AND how a tuning can be constructed based on that theory of writing music, either tonal or atonal.

Usually about a half octave to an octave above the stringing break the scale of a piano becomes more consistent. On little spinets that means sometimes my two octave temperament may be from A4-A6!

Ron Koval

I have never felt the need for using CM3's in a temperament sequence. I understand that many people do, but my main complaint of it is that once you get above 8-10 bps on an interval, your aural perception of beat rates diminishes exponentially and the term "souring" is more applicable than "beat rate". In a practical sense of what you are doing it may not be as important but two well seasoned tuners can argue untill the cows come home about whether two sets of intervals comparitively are this or that when you get above this threshold.

I wouldn't entirely agree about most tuners refining the SBI's in this day and age. ETD users often ignore checking these intervals and let them sit wherever the machine leaves them. If the SBI's are not incorporated into the temperament sequence as a primary function aurally, you also run the risk with aural tuners that they skip refining some of them and move on, out of haste and speed.

I agree, and that's why I throw in the CM3's pretty fast, and spend more time on the refinement. To me, the CM3's get me close, fast, and the ladder of CM3's also helps me to find a good octave width.




Yes, and that goes to priority once more. But you may be right, maybe prioritizing SBI's isn't as common as I'd want it to be.

Jeff: I use something very close to ET via Marpurg for starters, then I go into comparing every note of the temperament against it's 4th and 5th. Then I start moving things around until I'm happy, kind of a molding method... The potterY temperament

I rarely have to move F3, and basically never A3.

I do not know if this is the best way to come out landing on your feet, but it works for me

Welcome to the wonderful world of tuning! For starters, congratulations on attempting to learn to tune aurally. While I perceive that RonTuner was being facetious calling aural tuning "quaint" and "noble" I actually agree in a literal way - it is quaint and noble, which is what makes it fun and rewarding. When you master it, you will feel part of an elite club of piano technicians that have taken the time and energy to learn the most basic skill of our craft. I have yet to meet someone who regrets having taken the time to learn to tune by ear. I have met many technicians who wish they could do it.

When I decided to pursue this as a career, I took a look at who the top piano technicians were in my region in order to know what my goals should be. Every one of them are excellent ear tuners. Some regularly use an ETD in their work, others don’t. But they all feel that aural tuning is an important foundational skill. Also, not having aural tuning skills will close certain doors to you. Many university jobs will not be available to you, and if you ever wanted to work for Steinway & Sons as a technician or contract tuner, you will be shown the door if you are dependent on an ETD.

One thing that comes to mind is that we don't know what kind of piano you are working on. If it is a console or spinet, achieving a "textbook" tuning will be an impossibility. I suggest not practicing on any upright under 45" or grand under 5'4". Otherwise the wacky tension changes that occur around the temperament of small pianos will really mess with your beat rates.


One of the problems beginners run up against is expecting too much of any temperament sequence. There is the hope that if you find the "right" sequence your temperament struggles will be over. Put that idea behind you. If you end up with a great temperament after one pass through your sequence consider yourself very lucky, and don't expect it to happen often.

The sequence, in most cases, is only roughing in the temperament. It's a starting place, not an end result. It's like the rough sandpaper you use before the fine sandpaper. So what you need now is a collection of temperament troubleshooting techniques that can refine your results.

When I’m refining a temperament I listen to parallel intervals up and down: 3rds, 4ths, 5ths, and 6ths. I pick out the most out of place interval and then try to figure out where the problem is. Both notes are put on trial and the jury is a series of aural checks like contiguous 3rds and 4ths and inside 3rd/outside 6th test etc. If you think you may need to raise a note up slightly, you need to look how that change will affect all the other intervals associated with that note. Developing a good repertoire of checks is essential.

I have tuned using both the 3rd 6th type of approach, and have also used the old Braid White method. Either can be a great starting point. My approach now is a sort of hybrid of the two approaches. As time goes on, you will learn what works best for you.

One thing I highly recommend is developing relationships with skilled mentors. Find out who the very best technicians are in your region and seek them out. Be respectful of their time and be willing to pay them. In some cases, becoming a member of PTG can be a big help. This depends largely on your local demographics and the health of the local organization.

Patrick, I know an elderly fellow who is a PTG member that exclusively tunes all his temperaments in the F2-F3 range. He feels that his M3rds come out more refined in the end. I really seen no issues with his temperament and how it succesively gets expanded into the tuning so I think there is a high amount of leniency on ones' approach to where and how its done. Perhaps the differences in approach can also be attributed to varying strengths and weaknesses of each individuals capabilities and limitations. When I first started tuning I had great difficulties with 4ths for some reason, yet most of my classmates had complained about 5ths. I would love to know exactly how and why there are these differences. I find it to be one of the more attracive traits of this trade, to be able to find what best works for yourself in achieving a decent tuning.

That's helpful. I hadn't heard about the lower half octave of the plain wire string before. My understanding of TuneLab was that once it has a piano's inharmonicity measurements, it constructs a tuning curve that lets the piano be tuned in any sequence - even note by note all the way up the scale. I wouldn't have thought that it would be more reliable for some strings than others.

So if I want to stretch an octave more than TuneLab does, say by making A5 sharper than the given tuning curve, does TuneLab incorporate that custom offset when calculating its recommended frequency for A6, A7 etc. or does it just let you record that you're using a different frequency for the single note A5? That is, will the entire right half of the tuning curve be more stretched based on my tuning of A5?

You will have to adjust the curve to get it to cross the point of the custom offset by use of the arrows up and down once you have the curve displayed.

The custom offset should show red. Use the menu to lock the custom offset note so it doesn't shift as the rest of the curve moves. You can set a couple of notes to really customize the tuning curve to better match the instrument.

Ron Koval

You mean checking the chromatic m3rds or m6ths, right? I remember that you wrote about this a few years ago. It was a good test to get under the belt.

If you hate reading stuff online here's a free pdf of the same.

Kees

As an aside, here's an intersting quote from White's book about ET>

Kees

I didn't realize that the method White is describing in his book is also attributed to the Circle of Fifths.
The tuning sequence I thought was the one and only "Circle" of Fifths is as follows:

A3 to fork,
A2 to A3 (octave down),
E3 to A2 (fifth up),
E2 to E3 (octave down),
B2 to E2 (fifth up),
F-sharp3 to B2 (fifth up),
F-sharp2 to F-sharp3(octave down),
C-sharp3 to F-sharp2 (fifth up),
G-sharp3 to C-sharp3 (fifth up),
G-sharp2 to G-sharp3 (octave down),
D-sharp3 to G-sharp2 (fifth up),
D-sharp2 to D-sharp3 (octave down),
A-sharp2 to D-sharp2 (fifth up),
F3 to A-sharp2 (fifth up),
F2 to F3 (octave down),
C3 to F2 (fifth up),
G3 to C3 (fifth up),
G2 to G3 (octave down),
and finally
D3 to G2 (fifth up).

Or just get your own copy:

http://www.amazon.com/gp/offer-listing/9996267636/

One of the first books I bought after Reblitz.

Why pay $10 to some random company unrelated to the author if it's free?

Kees

Close: minor 3rds and Major 6ths. Probably just a typo.

But what I am saying is: that is the only way I can make them work. And I don't know if it would work across a break.

Have you tried setting a temperament by only tuning 4ths and 5ths and listening to 3rds and 6ths? Even if you don't enjoy it, you may learn a thing or two.

Kees:

Thanks for posting the quote from Dr. White. Reading it reminds me of just how objective he was, not some villian that destroyed music like some portray him.

And why pay $10 for a book if it can be read for free online? Well, I like books! I can enjoy them while sitting in a relaxing chair and holding the book at a comfortable position.

Free if you don't mind sitting in front of your computer to read it. I suppose my iPad might suffice, but it seems to have been taken over by the rest of the family I'm as much a computer nerd as anyone, yet there are many reasons I prefer real books to screens, no matter how portable.

I just opload the pdf to the nearest copy shop and have it nicely bound. That's more expensive than buying the book but I have the illusion that I got it for free.

Kees

I haven't tried that order exclusively, but you're right - I should, and will.

Emmery, I completely agree.
Ryan, the same goes for your post above.

I could have signed both of these. But you beat me to it.

Pat:

Thanks!

Jeff: I apologize if you took offence, it didn't sound bad to my ears. That happens sometimes due to English not being my native language. I don't really hear nuances of the words that I can't list here...

This said, during my years in NY I never understood why cop movies on public TV channels replaced every hint of a curse with a *beep*, although those words was heard everywhere on the streets, from people of all possible sex/age/belief/ethnicity aso. That is, and still is, double standard to me.

Where DO you draw the line? Are you representing the majority? I am just curious, and I'm still trying to learn your customs.

I doubt if I represent ANY majority.

I don’t know what they do on TV anymore. It is up in the attic.

I once heard that Catholics tend to avoid taking “The Lord’s Name in Vain” but use sexual explicit terms instead. Protestants tend to do the opposite. Having gone to sea for 24 years most things do not bother me, although the setting can be inappropriate, such as this Forum…. A true Curse does get under my skin, though, if directed at a person, even jokingly. But it can be humorous when directed at an inanimate object. Is there really a place of eternal punishment for false beats?

I remember my arrival in NY 1994. At the airport, a cab forced our airport bus driver to hit the brakes. Out jumped both drivers, starting to yell obscenities at each other.

They kept on doing so for about a minute, gesticulating wildly, then they went back to their vehicles and got back to work.

Where I live, the first few worlds would have been enough to start a fight where only one would remain standing.

You could look at it this way: Neither were standing, they were both groveling in the mud.

How true.

I also remember when I once made an impression of a friend of mine, loosely quoting the way he speaks, and he instantly yelled "I would NEVER take that word in my mouth!". In my ears it was an innocent curse in line with his everyday talk, but somehow there was a clearly defined line drawn that I never saw.

Now, Jeff, how can we tie all this back to tuning along the circle of 5ths?

Well wait a minute. The music being played is what is atonal or tonal, regardless of temperament. A piano is an atonal instrument. Only when someone sits down and plays something in the key of whatever does it become tonal. If someone sits down and plays a 12-tone Schoenberg piece, then it's atonal. This is a function totally outside the temperament or the tuning, so I'm not quite getting how a temperament is being called tonal or atonal.

I think calling them equal and unequal is clearer than calling them atonal and tonal.

Kees

Ah, ok! Thanks, Kees.

Back to topic:
Any comments on my circle of fifths tuning sequence listed above? Does anybody still tune after this? The questions I had in my first post in this thread related specifically to this method, tuning octaves and ascending fifths only.
Thanks!

Why would you be setting that temperament down that low? A2 is in the bass section. I start at A4 and then:

A3 to A4
E4 to A3
B3 to E4
F#3 to B3
C#4 to F#3
G#3 to C#4
D#4 go G#3
A#3 to D#4
F4 to A#3
C4 to F4
G3 to C4
D4 go G3

Along the way I'm checking M3rds and M6ths.

Here is link to White's tuning scheme. You have to add 1 to all note subscripts to conform to present day American notation.

Kees

Sorry, I made a mistake with the annotation, I got it wrong by an octave. This is the correct annotation: (hopefully)

A4 to fork,
A3 to A4 (octave down),
E4 to A3 (fifth up),
E3 to E4 (octave down),
B3 to E3 (fifth up),
F-sharp4 to B3 (fifth up),
F-sharp3 to F-sharp4(octave down),
C-sharp4 to F-sharp3 (fifth up),
G-sharp4 to C-sharp4 (fifth up),
G-sharp3 to G-sharp4 (octave down),
D-sharp4 to G-sharp3 (fifth up),
D-sharp3 to D-sharp4 (octave down),
A-sharp3 to D-sharp3 (fifth up),
F4 to A-sharp3 (fifth up),
F3 to F4 (octave down),
C4 to F3 (fifth up),
G4 to C4 (fifth up),
G3 to G4 (octave down),
and finally
D4 to G3 (fifth up).

Ron:

I can see a potential problem with including all these octaves. Let's say someone has a tendency to tune octaves haphazardly. Some a little wide and some a little narrow. The circle may be completed, but the resulting fourths will not be even, which of course means it will not be ET.

This problem can be lessened by listening to the resulting fourths when tuning octaves. And any time a fourth or fifth is in question an octave can be tuned to check the complimentary SBI. I especially do this when tuning an SBI across a break. By why include all these octaves when the circle can be made with alternating 4ths and 5ths? I suppose it could be a coping strategy if someone has real difficulties with 4ths. There is a time and a place for everything.

But lets also look at which RBIs come available when. The first is the A3-F#4 Major sixth which would beat about 10 bps: a bit fast to start off with. The next is the A3-C#4 Major third at about 9 bps: this one isn’t so bad and along with it is E3-C#4 Major sixth at about 7.5 bps: that is a nice speed, but if this is a smaller piano, E3 is probably on the bass bridge and there can be significant jumps in beat speed.

The next type of important check that will become available happens when half the circle of fifth has been tuned. It is the M6 outside – M3 inside check. When D#4 is tuned, the B3-D#4 M3 should beat about 10 bps, the same as the first RBI which is A3-F#4. Again, this is a bit fast, but useable.

Now let’s jump ahead to the all important ninth note: F. When F3 is tuned it forms a Major third below A3. (A3 can easily be checked with the beginning note of A4.) This F3-A3 major third has previously tuned chromatic M3s above and below it. If everything tuned to this point is a bit wide it will beat too slow when compared to the other M3s. If everything is tuned a bit too narrow, it will beat too fast. If there is an error, then the previously tuned notes must be checked and retuned as need be. Since octaves were included, this adds many more places for errors and many more notes that need to be retuned. Also, since the sequence has gone only in one direction from A, corrections cannot be made from only one end towards the middle. It can often happen that not all intervals are in error, but if the sequence is only in one direction in the circle of fifths, it may be necessary to retune all the notes. But if D and perhaps G had been tuned at the beginning, correcting any errors that only became obvious when the ninth note is tuned (which would then be A#3 or perhaps D#3) would probably be easier.

Well, this illustrates some of the things to consider when choosing a 4ths and 5ths sequence. Much can depend on the size of pianos that are tuned and the particular tuner’s abilities. Try different sequences and see what works for you. It may not be a 4ths and 5ths sequence that you prefer, anyway.

Jeff, thank you for going through this sequence for me. Would you mind explaining the "M6 outside, M3 inside" test that you mention a bit further? How do you quickly find the corresponding M3s and M6s?
Thanks.

You are very welcome!

The M6 outside, M3 inside test is very easy to find if you know your dominant 7 chords. Consider the G7 chord. It is made from G, B, D and F. If the chord is arranged so that the seventh is on the bottom (F, G, B and D) then it is in the inversion needed for the test. For example, the F3-D4 M6 should beat the same as G3-B3 M3.

If you do not know your dominant 7 chords then take any M6, and compare it to the M3 that is a whole step (two semitones) higher. Or take any M3 and compare it to the M6 that is a whole step lower. If you are not sure of the sharps and flats, use the key signature for the note a fourth above the bottom of the M3.

Hi Bill. Sorry to hear about your accident, I hope your situation is improving. The above link to the beat rates is indeed very useful as is your entire website.

Also to others: I wonder whether on can find all typical bps rates on the internet as sound files, it would help to memorize them and to have them for comparison, next to the piano!

Interesting that the Sanderson manual, for all of the tuning sequences, emphasizes getting the M3's correct first, and then setting the M5's and M4's.

EDIT: Not trying to be contrary here. When did the more recent emphasis on setting the M5's and M4's start? Was there a specific writer who shifted the focus to the SBI's, or has there just been a gradual movement in that direction?

Jake:

I have wondered how tuning with a ladder of Major thirds (CM3s) became popular! It used to be that tuning with 4ths and 5ths was the norm.

Best I can figure it happened around the time the PTG starting testing with ETDs. And it seems to go with the accuracy that is required by the exam. A deductable error is the same amount of error as would be easily recognizable in the beatrate progression of CM3s.

Hm...One other thing I notice about the Sanderson tuning sequences is that they start on C to form the temperament. I think Braid White does the same. (?)

I'm learning using a C fork at 523.3 Hz and setting the temperament from F3 to F4 (keys 33-45).

how do you count the beat at 4th or 5th?

i was told by a steinway tuner that they tune by using 3th 6th and 10th

iam so amaze and curious...

I feel like people are wondering how shoes were ever tied before velcro.

Jake,

First of all, let's get out abbreviations correct: it is P4 & P5 (not M4 and M5).

Jeff,

You are exactly right about that. The re-introduction of the contiguous thirds (CM3) idea came into popularity beginning about 1980 with the simultaneous introduction of the the first Sanderson tuning device and electronic scoring of the PTG tuning exam.

Dr. Sanderson, in consultation with Bill Garlick RPT who was at that time an instructor at the North Bennett Street school and later at Steinway, identified the (now) well known 4:5 beat ratio of CM3's.

You may be interested to know that Owen Jorgensen had always used the classic 4ths & 5ths type sequence from Braide White's writings. It had become habitual for him, so he saw no need to learn another method. His daily existence included tuning the school of music pianos, including large numbers of Baldwin Hamilton Studio pianos.

The latter frustrated Dr. Sanderson with its irregular scale that simply didn't work very well with a calculated electronic tuning. The CM3's afforded the best solution for Sanderson to aurally verify and correct a calculated tuning.

Jorgensen said that while he immediately saw the value in the use of the CM3's, he never put them into practice because basically, he knew what he was doing with a 4ths and 5ths sequence. The "value" as he put it, was that it served to divide the temperament octave into three very precise, equal parts.

Other tuning experts such as Rick Baldassin RPT and Jim Coleman RPT, jumped on the CM3 bandwagon and began to teach it. Thus, we now have the "Baldassin-Sanderson" temperament sequence to which Coleman's name has also at times been applied.

I had never used the CM3's until well after I became an RPT but learned about them at PTG seminars. By the time I went to my first Steinway training seminar with Bill Garlick in 1986, CM3's were the foundation of my temperament sequence. By that time, I knew that if I fist had the temperament octave equally divided, I could hardly go wrong with the remaining 4ths and 5ths. F3-A3-C#4-F4 were always key pivot points. This worked well for me and Garlick saw that I could execute a flawless temperament every time with great efficiency.

Late in the 1980's, I published my own ET sequence which is very much like Randy Potter's of the present day. After the initial CM3's, one has a choice of tuning either an RBI or SBI thereafter and immediately checking the results.

In 2003, I set out to re-think the entire process in order to help people who wanted to pass the PTG exam. That lead to the "Mid-Range Piano Tuning" article. A few years after that, I hit upon the idea that the two octaves, F3-F4 and A3-A4 were key to perfecting the CM3's. One needed the final F4-A4 M3 as an "outside judge" of the three CM3's below it.

I still suggest that after the initial CM3's, John Travis' sequence of "Up a third, Up a third, Down a 5th" leads to the most available checks and balances as the rest of the sequence is constructed. However, one can really use whatever sequence had become familiar.

You may be interested to know that when I was asked recently to tune a 1/4 Comma Meantone temperament and not really knowing from memory any published method, I quickly decided to tune it from C3-C4. I used the CM3's, C3-E3, E3-G#3 and G#3-C4 to set the parameters of the scale. C3-E3 and E3-G#3 are pure and that leaves G#3-C4 very wide and dissonant. From C3 and E3, I could construct 4ths & 5ths which balanced proportionately (each tempered far more than in ET) but lead to another pure M3. The 1/4 Comma Meantone was indeed easier to construct that way than ET.

I cannot imagine tuning a Baldwin/Hamilton studio with evenly progressive CM3s. But then some people don't even like scrapple. I don't understand that either.

And here lies one of my problems with the CM3 approach. I have very real problems discerning a beat that is this fast (at least on my piano). So I can never really say whether the F4-A4 M3 is "right" or "wrong", compared to the other three. Even C#4-F4 is sometimes difficult for me to discern.

On the other hand: to disregard beat rates and simply grade the CM3s in terms of "progressive sourness" isn't very effective or accurate for me either.

I haven't tuned many pianos, and I'm not trying to bash the CM3 approach either. I'm just saying that as a relative beginner, I can't place the uppermost M3 with great confidence, so unfortunately for me, it feels like a guessing game.

If you want to use thirds as a check, you should tune from E3 to E4. There is no reason to restrict yourself to one particular octave when setting the temperament. For that matter, there is no reason to restrict yourself to an octave.

Excellent point, BDB. One octave is a good starting point, but a two octave temperament will give you a lot more information to go on. I will very often make corrections in my starting temperament octave as I move up towards the treble. Ideally, in the end, the entire piano is a temperament.

I feel the same. I've had the opportunity to try it on a grand once and it works just as advertised. The F4A4 beats are fast but clearly audible. But on the less than perfect uprights I have the same problem. If I can hear the C#4F4 beats clearly I consider myself lucky. The crucial F4A4 check is not available it seems. On those pianos I do better by just timing F3A3 to 7bps with my watch, and make A3C#4 and C#4F4 progressive.

Not that I would ever tune equally except for the exam (some day I'll take it) of course.

Kees

To all of the above: you know you have it wrong when you can either hear the F4-A4 beat easily or it is so fast that you can't hear it at all. If you can hear it easily, it is too slow and that means F3 is too sharp. If it is so fast that you can't hear any beat, it means that F3 is too fast. It is as simple as that and very clearly and explicitly said in my ET via Marpurg article.

If you are in doubt, compare again C#4-F4 and then F4-A4.

Yes Bill, I understand but my dilemma on crummy uprights is that if everything is right I still can't hear the F4A4 beats. It's not that they are too fast, I just don't hear them.

Maybe I'll just have to focus my brain on them.

Kees

That doesn't seem very accurate, Bill.

F3-G4 is enough for me to refine the temperament. It gives some critical octave checks for the first SBIs and a few more RBI and inside/outside checks later on. It is pretty rare for me to need to adjust any of these notes later on, but sometimes I think I don't get things as stable as I'd like when getting used to a piano.

Very good point about continuing the temperament to the entire piano. 4ths, 5ths and 12ths are available if you know how to use them. But I wonder if those that set the temperament, or even just begin the temperament, with RBIs learn to use them to their full potential.

It's extremely accurate. There is only one place for each of the five notes to be for them all to be correct and that is why it is virtually infallible.

If the difference between "can either hear the F4-A4 beat easily" and "it is so fast that you can't hear it at all." is more than 1 cent, then I doubt if all the RBIs can be made to beat progressively.

Below some error analysis of the method.

Kees

You can move your CM3 ladder down to a slower beating area where the beat rates are easier to compare, to establish bearings. Then those are helpful when tuning up into the F4-A4 range.

Jim, you tune(d) C#3-F3-A3-C#4, if I remember correctly?

Yup. But recently I've been doing Eric Nikiforoff's 1-2-3, which switches immediately to complementary and opposing 4/5ths after the CM3 ladder. It seems to be more fun than how I was doing it before. I'm not sure why. Perhaps it's the symmetry of the sequence.

But back to the topic of this thread -- I'm not sure 1-2-3 is as good for the beginner, as it was touted in Eric's PTG Journal article, since you need a feel for 4/5ths. When you're just starting out, you mess with octaves and unisons, and then there's this chasm you have to leap to get to a temperament, which involves learning the sound and feel of all these intervals. When you're focused on M3rds and octaves, you can get a lot closer to ET, faster. Then work on refinement. Perhaps that's another contributor to the switch in emphasis by those doing the teaching. And of course the effect of ETDs.

On the other hand, perhaps that's why ET via Marpurg works, because you don't have to be as good at gauging the 4/5th beat rates, you go for equal beating. But I didn't come at it from that way, and once you "know," it's hard to go back and be in the frame of mind of "oh yeah, that would've been easier for me to learn."

Well, that is, when I'm not tuning EBVT, and I'm not using Tunelab to set temperament. I try to do a little bit of everything just to keep myself entertained

If the CM3s beat progressively as they should, there cannot be more than about 1/2 cent error at the most. The difference between being able to hear F4-A4 easily and for it to be too fast to be discernible is about 1 cent. Get it to where you can just barely hear it and you have it right.

I suggest for anyone who has trouble hearing the F4-A4 M3 to do what I said for novices who are trying to hear the F3-A3 M3. Tune A4 to the fork, then move F4 sharp and flat until you can hear the beats. Surely, you will be able to hear the beats when that interval beats slowly.

Now, flatten F4 until the beats increase to the point where they are too fast to hear. Sharpen F4 again until you can just barely perceive the very rapid beat. F4 should then be about right. Tune F3 from F4 as the same type of octave you have tuned for A3-A4, then fill in C#4. The entire sequence should come out either correctly or within a range that you can make small adjustments for it to be exactly right.

Yes, you can tune the CM3s lower down first, such as from C3-E4, C#3-F4, D3-F#4, etc., but the only way to be sure that A4 ends up exactly at 440 is to start with A4 at A-440 and work from there. The F4-A4 interval needs to end up correctly at some point. If you start somewhere else and don't even check it, chances are that it will be incorrect.

Thanks to Kees and Jim M. for your posts.

Unfortunately, Jeff, we have been through all of this before. Gadzar provided some excellent analysis in the past as well. It seems that no matter what anyone says about this, you still come up with the "I doubt..." and "I still don't think..." remarks. Have you ever really followed the directions even one time yet? Or did you read the directions and then make up your own and find they did not work and then proclaim that the CM3s are inaccurate?

Braide-White wrote nothing of this but the CM3s have been taught by PTG now for thirty years.

Excellent table! It shows clearly that one can easily detect a 1 cent error in any note of the CM3s by hearing at the progression of the beat rates in the CM3s set.

In every row of this table a significant jump is found in the beat rates of the CM3s that tells us exactly how to correct the tuning of which notes.

Hi!
I studied the chart of the ladder of M3's and one question came up. Since the thirds are sensitive to adjustment, and if you adjust the common tone of two contigous thirds you can find the exact spot for the notes very accurattely. My question is: If I tune F3-A3-C#4-F4-A4 correctly, what happens if i tune D4 pure to A4, and F#3 pure to C#4, what if i tune A#3 so that F#3-A#3 and A#3-D4 are equalbeating, would this be the right spot for A#3 ?

Lets see.

D4 pure to A4 = 440/1,5
F#3 pure to C#4 = 2^(-8/12)*440/1,5

Beatrate of F#3-A#3 is 4*A#3-5*F#3
Beatrate of A#3-D4 is 4*D4-5*A#3

The thirds beat equally, their beatrates are the same, so 4*A#3-5*F#3=4*D4-5*A#3

Which gets down to 9*A#3=4*D4+5*F#3
A#3=(4*D4+5*F#3)/9

Popping in D4 and F#3 from above gives us A#3=233,0306041
Real A##=233,0818808

Difference in cents=log(A#3/A#3real;2)*1200 which is -0,3809 cents.

A#3 would be 0.3909 cents flat, theoretically. I hope I didn't make a math error somewhere.


On tuning the contiguous thirds, assuming that A3 is spot on at 220hz, when F3-A3 beats at 6 bps I got that F3 is 1,8419 cents sharp. But then it's hard to get the others beat progressively. For example, when F3-A3 is 6bps, and you make A3-C#3 9 bps, C#4-F4 is 12,15 bps and F4A4 is 12 bps, but if you make A3C#3 10 bps, C#4-F4 is 10,9 bps and F4-A4 is 12 bps. In the first one, an upper third would be beating faster than a lower one, in the second the beatrates doesn't get progressively faster. I haven't tried it too much so I cant say how noticeable the flaw in the second beat rate progression would be, but I guess it would be.

Let's say you make F3-A3 6,5 bps, then A3-C#4 8,9 bps, which gives C#-4F4 of 11,475 bps and F4A-4 13 bps. Errors for F3 and F4 is 0,85 cents and for C#4 0,26 cents. Is it possible to notice the error in such a beat rate progression?

I put together a small excel file where you put in F3-A3 and A3-C#3 beatrates and it gives you the other beatrates and how many cents each note is off. It uses perfect octaves

http://www.speedyshare.com/files/28124849/Contiguous_thirds_beatrates_calc.xls

interesting, how would things change if i tuned D4 pure from A3 instead of A4? the A3-A4 octave is 4:2.
How would things change if F#3 is temporarly tuned pure from B3 if B3 is at its correct position.I'm looking for a way to find B3s correct position by tuning it between other interwalls that are equalbeating. I don't know how to calculate it myself... got any tips on links for theoretical calculations?

// martin

Bill:

If you remember, you recently posted a question about how many cents a note can be off before a ladder of CM3s is no longer progressive. It is about 3 cents:

http://www.pianoworld.com/forum/ubbthreads.php/topics/1553498/ET%20via%20Marpurg%20Revisited.html

I am going to spend some time and find the place in Dr. White’s book where he considers how an octave can be divided by M3s (and maybe m3s?) and his reason for not using it to set a temperament.

But certainly it is fine to use ladders of CM3s if the major goal is to have progressive M3s. But if the goal is to have ALL RBIs progressive, a ladder of CM3s may not be sufficient. Perhaps it is for those that can really hear a 4:5 beat ratio. Not everyone can, especially when it is over 10bps. I certainly cannot.

I didn't take into account inharmonicity, but in that case the A3 would come a little down, D4 also with A#3, I guess making it a little more flat than it is theoretically, but inharmonicity would complicate everything further, because it changes from note to note and we use different partials of the same notes, which are sharp by different amounts.

Anyway, I got that A#3 would be 1,34 cents sharp if you tune F#3 pure to a correct B3 and make the thirds equal beating.

You can get the A#3 pretty close using the method you suggested, if you then tune F#3-A#3 as a chromatic third after F3-A3, same with A#3-D4, and then start it over again for B3, you would probably get it pretty close.

Basically you are doing the up a third up a third down a fifth sequence, but instead of estimating a tempered fifth to get the first note of the first third, you do it pure, then do another fifth pure for the second note of the second third and move the note common to the two thirds so that they are equal beating. Not a bad idea I think and I guess you could use all the other tests used in the up M3, up M3, down P5 sequence as well and I don't think you will be any worse off.

For calculating the difference of two notes in cents:

http://en.wikipedia.org/wiki/Cent_%28music%29#Use

For calculating the theoretical frequency:

2^((+-semitones from A4)/12)*440

G4 is 2^(-2/12)*440

To calculate the beatrate, you have to know at which partials the frequencies are close and then calculate the difference of those partials. For a M3 it is the 5th partial of the lower note and the 4th of the higher, for P5 it's 3rd partial and 2nd, for p4 its 4th and 3rd, for m3 it's 6th and 5th.

For example:

Beatrate of A3-C4 m3 is 6*A3-5*C4, of A3-C#4 is 4*C#4-5*A3

For a pure interval, multiply or divide by the just intonation ratios, 3/2 for P5, 4/3 for P4, 5/4 for M3, 6/5 for m3.

A pure m3 down from A5 is A5/(6/5)

Doel:

Thanks for posting the error analysis table. You did not say what you think it shows.

What I see is, generally, a one cent error will produce a 1 bps change in beat rate. Since chromatically progressive RBIs have a difference of about 1/2bps in the temperament, I see this as a problem. There can be difficulties later in the sequence in trying to get ALL the RBIs to beat progressively.

All:

Here it is from page 72 of Dr. White’s book:

http://www.archive.org/stream/modernpianotunin00whit#page/72/mode/2up

”We have reached the central position in the science of tuning. What has gone before has been enough to show that one cannot obtain a series of pure diatonic scales, in the quantity required for the performance of music, with a key-board comprising only twelve keys to the octave. The particular method adopted in Chapter I for the purpose of showing the truth of this assertion might of course be matched by a dozen others; without altering the facts in the least. For example, I might have pointed out that an ascending series of perfectly tuned perfect Fifths, although nominally equal to seven Octaves, yet actually exceeds them. I might have shown that three major Thirds should be equal to an Octave, if tuned pure one above the other; but that in fact they fall considerable short thereof. There are many other possible illustrations; but I have already shown, in the simplest manner, that some form of compromise is needed if pianos are to be tuned so as to make the performance of music in all tonalities tolerable despite the defective and inadequate 12-to-the-octave key-board.” (Bold added for emphasis.)

And from page 85:

http://www.archive.org/stream/modernpianotunin00whit#page/84/mode/2up

”Experience shows that it is easiest to tune by Octaves, Fifths and Fourths; by Fifths and Fourths for the octave of tones, usually F2-F3 chosen for the “bearings” or foundation work and by Octaves up and down thereafter. The other intervals involved are best used for testing the correctness of the work as it proceeds.” (Bold added for emphasis.)

Since Dr. White knew an octave can be divided by M3s and “Experience shows that it is easiest to tune by Octaves, Fifths and Fourths” I think he knew about the possibility of starting with a ladder of CM3s and chose another method based on experience. Just because he did not write about a sequence does not mean he did not know of it. Even today there are those that know of tuning with CM3s and choose not to, for the same reasons.

You must mean 0.3 cents.

Here is the link, again. Look for yourself:

http://www.pianoworld.com/forum/ubbthreads.php/topics/1553498/ET%20via%20Marpurg%20Revisited.html

Hello.

Perhaps this article about 4ths and 5ths makes more sense in this thread:

By A.L.Silver (1955): "An unequal temperament is described in which the fifths and fourths of the tuning chain have the same beat rate."

http://gfax.ch/literature/Equal_Beating_Chromatic_Scale--Silver.pdf

Regards, a.c.

Your math looks right to me, but you had a typo. It is 0.3809 cents flat like you also posted.

When it comes to looking at the relationship of beatrates, you can pretty much ignore iH. The beatrates, and especially the beatrate relationships in the temperament, are largely self-correcting. What iH giveth in one place, iH taketh away in another. Until you cross a jump in scaling...

UnrightTooner, but the nearly 3 cent error is when F3-A3 and A3-C#4 beat almost the same and then there is a jump of almost twice in beatrate with C#4-F4 and F4-A4 beating almost the same. I guess you could call it progressive, but I think such a mistake could be noticed easily. When I try various combinations in the excel file, I can't get an error of more than 1 cent of any note without making the beatrates rather uneven.

I cannot say how big of a mistake in the beatrates could be noticed. In the example I posted before, with beatrates of the contiguous M3s of 6,5 8,9 11,475 and 13, the biggest error is in the F's: 0,85 cents. I'm guessing it isn't that easy to notice a flaw in that progression and it might get accepted by the tuner.

Increasing F3 by 0,8 cents for example increases the F3-C4 P5 beat rate by about 0,24 bps. Double that an octave higher. Is it possible to lose 0,24-0,48 bps somewhere a little at a time and make up for it somewhere else when tuning a series of 12 fifths? That would mean an error of 0,8 cents. But on the other hand when tuning contiguous thirds and the F is already 0,85 cents off, you'd probably make other small errors tuning the other notes as well.

Yes we have been through all of this before. I have already told you what I have done. You act as if it is impossible for CM3s to be progressive and still have errors - even across a break. Of course what is an error to one tuner may not be to another…

But here is something I haven’t mentioned before. When I was first tuning, back in the 70’s, I noticed a check that became available when tuning C#4 using the classic sequence by Dr. White. I noticed that if F4 was also tuned, C#4 could be checked by listening to the M3 higher and lower. I thought I was really onto something. My excitement was short lived. I soon found out that yes, of course, any note must be placed so that the M3 below and above are progressive. This is the idea behind CM3s. But it does not help much when chromatic M3s AND M6s are the goal. Then I understood the power of 4th and 5th tuning. EVERTHING has to be right or errors will show up. And besides, anyone that is tuning aurally should be able to hear the difference between 7, 9 and 11 bps anyway. That’s about all CM3s do for you.

Yes, I agree that a beatrate sequence that has two beatrates nearly the same would not sound progressive. 3 cents is a maximum number. But look at Doel’s table (Thanks again, Doel!) Every scenario shows progressive beatrates. But what would sound evenly progressive is another story. And looking at a table is different than actually listening to beatrates. Some claim to be able to hear the difference between a 4:5 and a 5:6 beatrate ratio. I cannot.

I am not sure where you meant to go with your last paragraph. You certainly can fudge any tuning sequence to get things to work fairly well if there is a bit of an error here and there. And if the piano has an unfriendly pinblock you pretty much have to. So, how much is too much of an error to “sweep under the rug?” I’ll just quote what I said before that was not taken favorably:

Very interesting!

Have you tried this tuning? How does it sound?

It is easy to measure if you have an ETD.

Just tune by ear the set of CM3s, and then check with the ETD the error made.

With some practice you will notice that you can tune F3 with an accuraty of +/- 0.3 cents.

And this is almost the greatest accuraty a good aural tuner can achieve.

If you try the same experiment for tuning a fifth using usual tests, i.e.: m3-M3, M6-M10, you will find a lesser accurate result.

That is why top tuners like Jim Coleman, George Defebaugh, Rick Baldassin, and many others, who learned to tune with the "Classical 5th/4th method" switched to the RBIs based sequences.

Take also for example Virgil Smith, who used a sequence based on M3s and m3s, to establish the correct tempering of the 4ths in his temperament.

The question here is why do all these top tuners tune RBIs to find out the correct tempering of SBIs? (Opposed to what Braid White's method does: tuning SBIs using RBIs as checks).

I believe this is because an octave is hardly divided into 12 equally tempered 5ths, compared to the easier job of dividing it into 3 equally tempered M3s.

I can not understand how a tuner would be able to tune a 5th with the m3-M3 test, which involves comparing a m3 to its contiguous M3, which have beat rates in the ratio of nearly 8:7, while this same tuner is not able to tune two contiguous M3s in a ratio of approximately 5:4.(see the edit note)


It is not a matter of accuraty in the ratio. It is a matter of eveness in the progression of the CM3s. That's why the set of CM3s tuning is more accurate than the tuning of a single 5th using contiguous m3-M3, in the later you are estimating a single ratio. In the CM3s you are striving for an even progression, which gives you more accuraty.


Edit: I guess the difference in tuning a m3-M3 compared to the M3-M3 is the fact that you can hear at the 5th and you can not hear at the augmented 5th. But that means that you are not really tuning a 8:7 ratio between the m3 and the M3, but rather tuning a 5th directly by its colour or tempering and using the check only to ensure that the 5th is narrow and not wide, that is the m3 beats indeed faster than the M3.

Yes Gadzar, I find Silver's model interesting too, especially when I think it was elaborated in 1955. I've not heard it on a piano but the author says it is easyer to tune than ET. And I believe him, since he could only refer to our first ET model, and he must have gone through what many of us have experienced.

But perhaps, even today, for many tuners it would be more confortable a regular approximation than a messy ET attempt (as the author suggests) and perhaps that simple beat-rule is also time-saving. It would be nice to know what the first EBS 12th (D3-A4) sounds like.

Regards, a.c.

It could be because they tune well scaled pianos. If I tuned well scaled pianos I would probably use ET via Marpurg.



The primary reason that I do not use CM3s is because they give me poorer result than 4ths and 5ths. Do you have a problem with people preferring what you do not?




Why use a single or even three or four intervals to obtain high accuracy? Until the 9th note is tuned, you cannot be sure any intervals are really correct.

Alfredo,

What would be the effect of the beat rate of 5ths doubling every octave?

This will have a "stairs" shaped beat rate curve.

Is it really easy to tune equal beating 5ths? We have so long heared equal tempered 5ths, that EB can sound "bizarre" and "wrong" to our modern ears.

Unright,

Thank you, but I am not interested in talking with you as you strive to "control" people by making them getting upset.

For all who may be interested in what I am talking about, here is the thread:

How to [i]"control"[/i] people on the internet

Or this other one:

How to [i]"get fun"[/i] at the expense of others

Bye.

Here's the original table as powerpoint, which includes a column with idealized audio. http://www.cs.ubc.ca/~kvdoel/tmp/cm3ppt.zip

Kees

Kees,

Excellent demonstration!

Of course there are not piano sounds but the uneveness in the progressions can clearly be heard!

(I can not get the presentation to work, but I heard directly the .wav files)

I practiced for about 3 weeks a while ago when I thought I might take the RPT exam soon. My errors went down from 2 cents to at most 1 cent on one note. It's encouraging to hear that when I would have kept going I could get it that accurate. (Assuming you don't have a special secret diet that allows you to achieve this.)

Do you even run into bad piano's where you just can't hear the higher M3's?

Kees

Folks:

I decided to find something out for myself. Yesterday on a customer’s piano with a friendly pinblock, I tuned Dr. White's sequence (circle of fifths) without any RBI checks or retuning previous intervals. The F-F octave ended up as a pure 4:2. The only note that needed adjusting for proper RBI progression was D#4. It must have been poor pin setting towards the end of the sequence because neither G#3 nor A#3 needed adjusting.

So now what? I guess I have been using the RBI checks as a crutch.

If you think I am lying, that’s OK. I would think the same thing.

Realizing that I can do this bothers me in ways that would be complicated to explain. I slept poorly last night. I think I will take a break from posting for a few days and consider the implications to Pride and Compassion. I am not sure the best way for me to post about tuning now.

The tuner teaching me, does exactly that and does it well, based on 35 years of practical experience, which makes it so difficult to learn from!

With all due regards Gadzar, his statement was a mere observation of human nature in some people, not a proclimation of his intentions. I've read many of Jeff's postings and don't really think he's trying to control anyone into becoming a Patty Hurst-like desciple of Braide White. Why all the paranoia?

Kees,

When I began to tune the CM3, I used to follow the instructions given by Randy Potter in his course of piano technology. But they weren't clear enough to me at how to tune F3. A sort of try and error procedure.

Then I've found the article of Bill Bremmer about tuning the midrange. I read it carefully and I followed his instructions step by step. And I discovered the correct way of setting the CM3s.

First of all, he explains exactly how to tune F3, which was not obvious in other documents I've read before. And secondly, the tuning of C#4 is a key step for the success of the CM3 setting.

You can start puting F3 where you hear a nice tempered M3. And it is the piano who tells you if F3 is sharp or flat. The way you tune C#4 is key in order to know how F3 must be retuned.

Before reading the Bremmer's article, I used to tune F3 as a 7 bps M3, then F4, and then I tuned C#4 by puting it in the middle between A3 and F4, striving to temper equally both M3s A3-C#4 and C#4-F4. That is an error!

If you do this, things smooth out, and you have a hard time to verify the even progression of M3s, as you already have smoothed the progression between A3-C#4 and C#4-F4.

So it is hard to say if F3 should be corrected and in what direction. F4-A4 is so fast that it helps only a little. And you finish with an acuraty of about 1 to 2 cents.

The point is to tune C#4 hearing the beat rate of A3-C#4 and tuning a 5:4 ratio from F3-A3. Do not hear at C#4-F4 when you are tuning C#4. Only tune a 5:4 ratio between F3-A3 and A3-C#4. Do not cheat, tune the 4:5 ratio!

How?

The speed of F3-A3 is at the limit of what I can reproduce with an alternating movement of my tonge between my teeth. And I can switch from a 4 ticks to the beat to 5 ticks to the beat. So it is easy to find the correct ratio. You can start practicing at a lower speed first, switching from 4 to 5 beats, and then at tempo (at about 7 bps). If your tonge is not fast enough you can play it mentally. Anyway there will always be a way for you to find the correct ratio. Everyone has his own means. Claping hands every 4 beats and every 5 beats, etc...

The trick is to find the correct rythm 4 to 5.

Once you have tuned C#4 in this way you can now check what the beat rate is for C#4-F4. If it is slow you have to raise F3 and if it is fast then you must lower F3.

When you have it right you check the whole set from F3 to A4, and make any needed adjustments, but chances are you have it already right.

The key here is to not put C#4 in the middle of A3 and F4, but try to tune the 5:4 ratio between F3-A3 and A3-C#4.

When I understood this, then my accuraty went to half a cent or even better.

And yes of course, there are pianos where F4-A4 is hard to hear.

My Petrof, for example, has a weakness in F3-A3 which is hard to hear, while F4-A4 is clearly discerned.

Even voicing can affect what you hear from M3s as the relative strength of partials is changed.

One more point.

In piano tuning there are no magic numbers!

The 5:4 ratio is not a magic number (of 1.25), but an approximation. It will change from piano to piano depending on it's scaling.

Once you have tuned C#4 to have a 4:5 ratio between the beat rates of F3-A3 and A3-C#4 you check C#4-F4, and you retune F3, F4 and C#4 until it is too in the 4:5 ratio.

Now, you finally have to check F4-A4. Why? Because the correct ratio of the CM3s for this piano can be other than 4:5. You can find that F4-A4 is faster or slower than you have expected, and then you have to do some adjustments in the tuning of F3.

The theoretical value of the CM3s ratio is near 1.26 but the iH of the piano changes this ratio all along the scale. However it can be considered constant in the span of the temperament octave, with a little error.

So tuning with a 4:5 ratio will put you very near, and you'll have to do only some little adjustments to have it right.

Thanks Gadzar, indeed I never tried to set the 5:4 beat ratio accurately.
I can do the tongue movement up to 12bps (a left over from the days I was playing baroque concerto's on recorder). I'm going to work on it that way.

Kees

Please let me know if your accuraty gets any better.

Another interesting thing I noticed, when F3-A3 is tuned right and you tune A3-C#4 for the same beatrate as F3-A3, C#4-F4 and F4-A4 beat almost the same, with C#4-F4 beating 0,6 bps slower than F4-A4. The beatrates would be 6,9 6,9 13,25 and 13,85. If you tune F3-A3 less than 6,9 bps and make A3-C#4 the same, C#4-F4 will beat faster than F4-A4. For example if F3-A3 is 6,5, the others are 6,5 14,475 and 13, but if you tune F3-A3 7,5 bps, the others are 7,5 11,625 and 15.

So you can check the accuracy of F3 by tuning the first two contiguous thirds equal beating and if F3 is right the other two should beat roughly the same, with the first of the two beating 0,6 bps slower.

That´s very interesting, this is in fact a way of testing F3 for doing a first approximation of near 1 cent!

For F3 tuned at -0.5 cents we have:

F3-A3: 7.2 bps
A3-C#4: 7.2 bps
C#4-F4: 12.9 bps
F4-A4: 13.9 bps

For F3 tuned at + 0.5cents we have:

F3-A3: 6.7 bps
A3-C#4: 6.7 bps
C#4-F4: 13.6 bps
F4-A4: 13.9 bps

But for F3 tuned at + 1 cent we have inverted CM3s:

F3-A3: 7.4 bps
A3-C#4: 7.4 bps
C#4-F4: 13.88 bps
F4-A4: 13.86 bps

In order to have inverted CM3s the error in F3 must be greater than +1 cent.

I think you made an error somewhere, or took into account inharmonicity or some other variables, because the F4-A4 should beat twice as fast as F3-A3 if we forget inharmonicity.

Here are the numbers I get by inserting the beatrates for the first two thirds:

F3-A3: 7,2 bps
A3-C#4: 7,2 bps
C#4-F4: 12,5 bps
F4-A4: 14,4 bps

And

F3-A3: 6,7 bps
A3-C#4: 6,7 bps
C#4-F4: 13,9 bps
F4-A4: 13,4 bps

Beatrates when F3 is correct:

F3-A3: 6,9 bps
A3-C#4: 6,9 bps
C#4-F4: 13,3 bps
F4-A4: 13,9 bps

I haven't tried this in practice and cannot say how hard it is to determine whether one third is beating 0,5 bps faster or if they would sound the same at almost 14 bps, but it seems if the inversion of the CM3s beating that fast can be heard you could set the F3 with 0,5 cent accuracy using this without having to guess anything.

You are right, I made an error.

In fact it is as you say: For F3 tuned at + 0.5 cents CM3s have already inverted with C#4-F4 = 14.0 bps and F4-A4 = 13.4 bps.

So we can detect a deviation of less than + 0.5 cents in the tuning of F3.

I've not tried this technique neither but I can anticipate it's easy to discern if one of the CM3s is faster and if they are inverted or equal beating.

Thank you Rafael,

This had always been my sense of it although I was unable to prove it mathematically. If properly executed with all five notes from F3-A4, the greatest possible error would be within 1/2 cent.

Incidentally, the first piano I tuned on Monday was a Steinway model A. I tried my latest suggestion. I fist tune A4, then F4 to it. There is a very small margin between a beat that is easily heard (too slow) and one that is too fast to discern. I placed F4 at the spot where it can just barely be heard clearly.

I then tune F3 to F4 and filled in C#4. I don't really think so much in terms of 4:5 but merely what I call a "small difference" (not the smallest difference that can be discerned but a small difference). The sequence revealed a perfectly even progression on the very first attempt.

Chapeau!

I guess you already have the ear of how a M3 should sound on a S&S. I seldom have the opportunity to tune such a nice piano.

When I set the CM3s, usually it takes me from 2 to 3 tunings of F3: my initial estimation and then a second tuning after hearing the progression, sometimes I have to retouch F3 if F4-A4 doesn't fit the rest.

I've never tuned F4 to A4. I'll give it a try.

Gadzar,

I while back, Patrick brought up an interesting point. While the beat rate of all Major thirds in ET is different, they are all of the same size and therefore the same quality sound. Perhaps the idea of tuning the F4-A4 M3 first would not work well for all people but it is worth giving it a try.

Some have said that they cannot hear the beats in that interval at all. Indeed, at some point, the coincident partials are too high and faint to be heard. In my writings, I have always said that the F4-A4 M3 is at or near the limit of discernibility. Normally, we would not listen to chromatic M3's this high in the scale.

However, for the A4 to be truly and exactly at pitch, we must work from that note. Therefore, the F4-A4 M3 must be correct. If one likes 4ths & 5ths, one can also tune D4 and E4 between A3 and A4 and make the resultant 4ths and 5ths be correctly proportionate.

To tune F4 as the first note after A4, sharpen or flatten F4 until you can hear a gentle beat, then flatten F4 until the sound becomes too fast, "sour" and indiscernible. Sharpen F4 again until you hear that moderate sound, not easily heard and barely discernible. Now, this may be no more accurate than guessing at the approximately 7 beats per second that the F3-A3 M3 should have but if the initial estimate is inaccurate, the rest of the sequence will show that.

Certainly, if one has F3-A3-C#4-F4-A4 with a reasonably even progression, no two exactly alike, no large differences, the two pairs of octaves, F3-A4 and A3-A4 are proven similar in width and D4 & E4 are tuned as proportionate 4ths & 5ths, one has fully half of the notes of the F3-F4 temperament octave already tuned plus one more outside note. How could anyone go very far astray with a framework like that?

Theoretically there can be an up to 3 cent error without reversing the beatrate progression of the contiguous thirds. But with a little practice I don't think I'd make an error of over 1 cent without noticing. With more practice it could become even more accurate.

The 0,5 cent error was for the other method of setting F3 by making F3-A3 and A3-C#4 beat the same and then the other two thirds should be almost the same, with C#4-F4 being 0,6 bps slower than F4-A4.

Ron:

I just skimmed through this entire Topic. It is obvious that you are not interested in CM3 sequences. Sorry that this Topic got hijacked and I apologize for my part in it.

I realize I have been “on the fence” about what 4th and 5th tuning is really all about. I had fallen for the trap of trying to beat someone at their own game. 4th and 5th tuning is not about making the RBIs progressive. It is about tuning properly tempered SBIs. RBIs can only go so far into helping this happen.

It would be like trying to be a good person by obeying all the rules rather than by developing good character. Obeying some rules shows poor character. And trying to explain what good character is, by putting forth a bunch of rules, is self-defeating. Likewise, if SBIs are properly tempered, the progression of RBIs is not an issue. It happens naturally. Just like a person with good character does not steal, unless it is to take something away that is meant to harm and probably many other reasons. (See what I mean about rules?) And RBIs should not be progressive when crossing a jump in scaling if the goal is to have SBIs properly tempered.

So putting aside RBIs, for now, how do you properly temper SBIs? It is pretty simple, really. In the temperament octave the 4ths should beat about 1 bps and the 5ths about 1/2bps. And by saying ”about” I do not mean that on some pianos it will be 1.2 bps for the 4ths and on others 2/5bps for the 5ths. I mean any 4th or 5th on any piano will be about these speeds in the temperament. Why? A number of reasons:

First, Inharmonicity is largely self-correcting when considering beatspeeds in the temperament, unless there is a jump in scaling. And a couple of tenths of bps on a SBI is very little change in pitch. It may not even be tunable depending on the pinblock, and won’t make a bit of difference in RBIs that are already properly progressive. RBIs are tempered 7 times as much as SBIs.

So what about RBIs and their checks? Let me quote Dr White again:

From page 85:

http://www.archive.org/stream/modernpianotunin00whit#page/84/mode/2up

”Experience shows that it is easiest to tune by Octaves, Fifths and Fourths; by Fifths and Fourths for the octave of tones, usually F2-F3 chosen for the “bearings” or foundation work and by Octaves up and down thereafter. The other intervals involved are best used for testing the correctness of the work as it proceeds.” (Bold added for emphasis.)

Of course we would now say F3-F4, the notation has changed. ”The other intervals involved are best used for testing the correctness of the work as it proceeds.” I now understand what this statement really means. Without getting into detail of the tests themselves, which Dr. White does later, let me explain the right and wrong way to use the RBIs as checks.

As the circle of fifths is constructed more and more RBIs become available. Each one gives an indication of how equal the temperament is. The first one only gives beatrate information, which is actually reliable despite iH. And if there is a question, simply go back and listen to the SBIs. If the 4ths are about 1bps and the 5ths are about ½ bps, continue on! That is the right way. The wrong way would be to adjust the 4ths and 5ths to obtain a theoretical beat speed for an RBI. If we do that, then we are letting the tail wag the dog.

The next RBI will give us two different beatrates to compare. Then later some beatrates will be the same. Then some beatrates will fall between two others. As the circle is formed it may be noticed, for example, that two beatrates that should be the same are not quite the same. It is then an easy thing to check the SBIs that formed those RBIs. If any seem a bit slower or faster than others, and bringing them in line with the others will make the RBIs beat closer together, then do so. If not, it is probably a due to a jump in scaling. If it is not a jump in scaling then perhaps you do not have a good feel for what a properly tempered SBI sounds like. This takes practice. But it is easy to recognize jumps in scaling.

A jump in scaling can be expected from one bridge to another, when going from wound to unwound strings on the same bridge, and when the number of strings per unison changes on the same bridge. Usually, major thirds will beat faster than expected when straddling a break and slower than expected above a break. Also if a break is much lower than it should be for the size of the piano, the lowest M3s above the break will beat slower than expected.

But there are exceptions. On the Kimball console, the M3s that straddle the wound/unwound break beat slower, not faster than expected when the SBIs are properly tempered. Kohler Campbell consoles have some surprises, too.

So then how to determine the correct beatspeeds for intervals that straddle or are near a jump in scaling? By tuning properly tempered SBIs to construct them. Don’t let the tail wag the dog!

OK, more about actually tuning properly tempered SBIs. It is challenging to play an SBI and say if it is properly tempered, although it is easy to play a series of them and tell which ones are different. Here is my suggestion for actually tuning SBIs. Start a bit below pitch so that the beat is easily heard. If both faster and slower beats are heard, lock onto the slower one. The faster one is from a set of higher partials and will lead you astray. Bring the pitch up with a smooth pull until it beats the proper speed and get this sound in your head! It is not just a beat speed it is also a tonal color. We are tempering intervals not just setting beatrates. It is this color difference that makes incorrectly tempered SBIs stick out more than the beatspeed. Now you are going to have to do whatever is necessary to actually leave this interval where it belongs. Usually some overshoot and then easing the pitch back down. But let’s not get into hammer technique. That’s a whole other subject.

Don’t know if this will help you, Ron. I hope it does. It helped me. I am now off the fence. I won’t fall for trying to beat someone at their own game in regards to 4th and 5th tuning. I am off the fence now. (Still have a couple splinters in interesting places, though )

Jeff: I tried a bit of 4th and 5th tuning.

A3 from fork (ETD actually). Then D4, G3 C4 F3. Now I check F3A3. It was too slow. So I adjust F3 so F3A3 beats at 7bps and go back and fix G3 C4 and D4 to all beat proportional.

Now why not set F3 right away at 7bps from A3 and then tweak G3 C4 and D4 to fit?

Kees

I computed the error from blindly tuning 5ths 1/2 bps and 4ths 1 bps, using White's sequence. "Error" as compared to taking the F3F4 octave that would result from such a scheme and then dividing that (stretched) octave in 12 equal parts. All with zero inharmonicity. I get the following errors:

C 0.0
C# 0.9
D -0.2
D# 0.9
E -0.0
F 1.2
F# 2.0
G 0.8
G# 1.8
A 0.9
A# 2.1
B 1.3

Doing the same with a typical IH model doesn't change this much.

Not sure what to conclude. I guess what I'm after is some instruction on what to do if the RBI tests (e.g. steps 5 and 8 in his sequence) fail the test. Not that I mind figuring out what to do myself, but the RBI tests should come with some instruction on what to do if the test fails.

Kees

UnrightTooner, I wonder how big is the jump in scaling, does the IH double?. How much does the expected beatrate change and if you make the M3s beat correct, is there an obvious flaw in the fifths?

Because then you are not

Because then you are not creating a M3 by tuning 4ths and 5ths, you are fitting 4ths and 5ths into a M3. (The tail is wagging the dog.) But how do you know that the M3 was beating too slow? If it was way, way slow then your SBIs were way off. If it was only a little slow, that may be the correct speed for that M3.

The first M3 is no place to work backward. If the first couple of RBIs show a problem it is best to start over. Including the complimentary SBIs and listening to the resulting octave will help things get started right.

I was not very satisfied with how I wrote about this. Sorry that my thoughts did not come across very well. Let me try again.

But first, there seems to be a problem with your error list. I checked the error for F and G and get only -0.3 cents. Not that it matters. If you take a look at the theoretic or even the actual SBI beatrates, they will all be about 1 bps for 4ths and ½ bps for 5ths. And any SBI that is tuned to these beatrates will be acceptable. But I am not saying that all of them can be tuned to this beatrate. For that matter I don’t think it would be possible to do so. It is beyond human capability to tune consistently within 0.1 bps. That is about 0.2 cents for these intervals.

You can form any RBI with 3 or 4 SBIs. If the SBIs are about at these beatrates, the RBI will also be at about its correct beatrate. Temper the SBIs correctly and the RBIs will also be tempered correctly. When I did my parlor trick experiment the other day, I was surprised at the resulting beatrates. F3 was wound and as expected F3-A3 beat faster than F#3-A#3. This is what I meant by a proper progression. But it was much more than I would have tuned it if I was using RBI checks all along the way. I had been letting the tail wag the dog and didn’t realize it. It was disturbing…

OK, let’s look specifically at the tests for tuning D4 and A3. If F3-D4 is way off then an SBI may have been tempered to the wrong side of just intonation. If it seems a little fast or slow, then the 4ths and 5ths should be checked. If they are tempered correctly, then that is the beatrate for the F3-D4. Otherwise the tail is wagging the dog.

Ron mentioned a little while ago that to avoid problems with scaling, the temperament (with an ETD!) should be set at least half an octave away from the break. I guess he should know, though I doubted it at the time. So unless the piano is a full sized upright or a parlor grand any beatrates involving F3 can be off a bit. That is one reason why I have been using a sequence where the first RBI is A#3-G4. I may change what I am doing now, though…

Anyway, the test for A4 is an interesting one. Since the only RBIs are F3-D4 and F3-A3, and F3-D4 has been determined to be correct (or you shouldn’t tune A4 yet) then the test is comparing the beatrate of these two RBIs. The difference should be about 1 bps regardless of scaling and regardless of the actual beatrates. This is the M3-M6 test for a 4:3 P4. You are directly testing the tempering of the SBI that was tuned last: A3-D4.

I have also wondered why Dr. White did not give more detailed instructions in what to do with the tests and what they actually mean. And also about including F3 at the beginning but not having a better check for C#4, like with F4. That is why I include A#3 at the beginning, so there is a good check with F#3. Now I think that Dr. White meant all these tests to merely assist in learning to tune properly tempered SBIs. When this is done, then there are just minor adjustments to be made as you proceed, like adjusting a bunched up curtain on a rod. And, as he mentions, F3 is tossed in to get the ball rolling at the beginning. F4 is when the circle is completed, not A#3. Then it makes sense that the ninth note in the circle (starting with C4, not F3) is G#3 giving a M3 that is tuned between two others and includes the starting tone (C4).

I hope you have more questions!

Here is a link to some graphs that include iH curves: http://www.goptools.com/gallery.htm

Notice that the scale for iH is logarithmic. It doubles about every 8 semitones. It is not like a piano has an iH number for all its strings. Problems occur when the curve bends or jumps. Then the combined effects of iH are no longer self-correcting in regards to beatrates.

Typically when F3 is wound and A3 is not, it will beat 1 or 2 bps faster than theoterical. And yes, if you make the M3s beat at their theoretical speed the SBIs sound horrible, at least to me.

UnrightTooner, are the 5ths and 4ths progressive in the temperament region? Because if you also have to worry about making them progressive, it is much easier to make mistakes. Maybe the F3 should have been made 1 or 2 bps faster and the difference should have been divided between the fifths? How certain can you be the fifths were correct, especially if they should be made progressive? F is four fifths away from A.

I used to worry about this. It is too small an amount to be able to tune. Here is how it works. If SBIs double every octave, then the 12th root of two can be used to determine what the tempering error is if they are not progressive. Like if a particular SBI beats 1 bps and is tempered 2 cents from just intonation, and then if it is tuned 4 cents from just intonation it will beat 2 bps, the same as the SBI an octave higher. So how much would an SBI be in error if it beat the same speed as the one a semitone away? Only about 1/8 of a cent! And using the same method on an M3, it takes seven times as much error to make an M3 unprogressive: about 7/8 cent. The SBIs do not need to be progressive for the RBIs to be progressive. But when we are talking about SBIs that are contiguous, like F#3-B3 and B3-E4 then yes they are tuneably progressive. I have pointed out the method, you can do the math.

But SBIs less than double each octave, so the error is a little less. And 5ths are tempered less than 2 cents, because of iH, which reduces their error even more. But none of this is very noticeable.

And actually F is two 4ths and two 5ths away from A. I mention this now anticipating later questions about the inside/outside test.

If they should be progressive, but too little to be tunable, doesn't that allow too much room for error when you are doing one after another? There is the human error, maybe 0,1 bps or less, also the fact that they should be progressive by a little amount which in not tunable and when you do four of those in a row using the last one for the next one, aren't mistakes too easy to make?

I calculated the error for F3 by going from A4 to D4, to G3, to C4, to F3 and by making every 5th beat 0,5 bps and 4th 1 bps. The F3 would be 1,499 cents flat, making the F3-A3 M3 beat faster. I guess it would pass, because it isn't way off and there is nothing to compare it to. If you add in the human error and consider that these are only the first 5 notes, the error could get bigger at some point, but would probably correct itself so the last P4 or P5 wouldn't sound wrong, because if you take the average beatrate the last one should be roughly right, because at some point you tune the interval flat, at others sharp.

In the end I think it comes down to if it is easier and more accurate to set three notes with an absolute error of 1 cent and work from there or do a fifths and fourths from the beginning, very possibly going flat more than 1 cent somewhere and making up for it somewhere else.

What is the inside/outside test?

Could you check your math? I get 1/2 cents wide, or about 1/4 bps error on the F3-A4 M3. But I am not just talking about math. I am talking about aural tuning, too.

I think 1 cent error is too much for a polished temperament. Something won't sound right somewhere. But how would YOU set three notes with an absolute error of 1 cent across a jump in iH, and know that the 4ths and 5ths will be properly tempered?

The inside outside test is when you compare the beating of F3-D4 with G3-B3. They should beat the same. Well aurally, not necessarily theoretical.

That sounds more right, but the other guy got the same numbers as I. You realize you are comparing against a stretched octave? (Academic, I agree it matters not.)

I think the tail is still wagging the dog. The goal is really to have all semitones to be locally equal. Tuning with 4ths and 5ths is a technical trick to achieve this, there is nothing particularly musically important about those intervals. I could argue in your example it's worse to let F major be harsher than F# major and it would be better to accept some irregularity in the 4th/5ths.


I suspect he didn't know what to do. It's probably like the famous prime number factorization problem; there is no method to compute the prime factors of a very large number, but once you have the factors it's easy to check they are right (by just multiplying them).

Kees

Here's how I calculated it:

A4=440hz
D4=(2*440+0,5)/3=293,5
D4-A4 beatrate=3*293,5-2*440=0,5
G3=(2*293,5+0,5)/3=195,833..
G3-D4 beatrate=3*195,833..-2*293,5=0,5
C4=(4*195,833..+1)/3=261,44..
G3-C4 beatrate=3*261,44..-4*195,833..=1
F3=(2*261,44..+0,5)/3=174,462963
F3-C4 beatrate=3*174,462963-2*261,44..=0,5

F3 should be 440*2^(-16/12)

Difference in cents = log(174,462963/(440*2^(-16/12)))to base 2 * 1200=-1,499 cents

Remember, there are different perspectives on which intervals to compromise, the RBI's or SBI's. It seems that you feel that the SBI's are what make a piano sound "in-tune", while others will say that it is the RBI's. I remember an article that Owen Jorgensen published shortly before he died, in which he recommended tuning progressive minor thirds below the break, and leaving the atrocious sounding octave, as he felt musicians were more sensitive to changes in the RBI of the m3 than the quality of the octaves, fourths and fifths.

The idea behind CM3's is the have evenly tempered RBI's, not SBI's.

Kees:

I really enjoy your hard nosed analyses!

Yes, I realize now that the difference is stretched octaves. Not sure if you stretched them in the order it was tuned, though. And I agree, it matters not.

But if there is nothing musically important about 4ths and 5ths then there certainly could not be anything musically important about locally equal semitones. Who could notice?

Yes, you could argue that it is worse to let F major be harsher than F# major. And I would counter with it is better to have harmonious SBIs, which I believe is the goal, not locally equal semitones (…uh, at which partial? )

Edit: Hoooo! But wait! If the goal is locally equally semitones, the closer the partials are to the fundamental (lower partials) the better. That means SBIs would give more locally equal semitones than RBIs.

We are talking about the F3-A3 M3. You started on A4.

Yes, I do believe it is better to have good sounding SBIs. It matters most when there is a jump in scaling. But I think it matters on well scaled pianos, too. You can have pretty darn progressive RBIs and still have so-so SBIs. But when SBIs are tuned, with the parameter also being progressive RBIs (on well scaled pianos) then you have the best of both worlds.

Jeff: Remember the old discussion we had on how ET can actually be defined precisely/mathematically given a realistic but smooth IH curve? I thought I had that figured out: it's the tuning where all intervals change as smoothly as possible in size (in cents).

The assumption however was a smooth IH curve. Now how can we define ET when there is a discontinuity in the IH curve? I haven't come up with an answer yet but am thinking of how to tune a hypothetical piano which has 12000 keys per octave equally, across the break, for a superior being that can hear intervals of 0.1 cents. Once we figure that out we can keep 1/1000 of the keys an have a good ET across the break.

Looking forward to hearing your approach which will undoubtedly be quite different.

Kees

You are right, if I start at A3 the F3 will be 0,5 cents flat. But if I add in human error in the right direction, 0,1 bps per interval, I can make the F3 an additional cent flat. If you made the F3-A3 interval sound correct and divided the difference evenly to the fifhts and fourths, maybe that is the right way they should sound? After all, they aren't all supposed to be half or 1 bps, the error is small but it adds up.

If you were to divide the error in the F3-A3 to the fifths and fourths, would you notice an error in them? And if there is no noticeable error, ie all the intervals sound correct, isn't it the better way to set the temperament?

I will give you a direction I have thought about, and maybe you can make it work.

Rather than a smooth curve at the first partial, as we were playing with before, what about a smooth curve at the 3rd or the 4th or the 3-1/3rd or the root mean square or weighted average or something. But perhaps there would be a more direct approach by concentrating on the beatrates of the SBIs. Some algorithm should certainly be possible. And while I am typing this I remembered the simplest way of all: pure 12ths. But Mr. Stopper has already done that. - sigh - Still, a 12th beatrate curve is another option.

The best way to set a temperament is the way that works best for the particular tuner on a particular piano

Because of iH, the error will be less than 0.5 with the ½ and 1 bps SBIs. 5ths do beat a little slower than theoretical. But you are mistaken to look at tuning with 4ths and 5ths as a textbook thing. It is an experience thing. Very small differences in tempering can be heard; smaller than the pin can be set. (I think tuning 4ths and 5ths directly as much as possible, such as when expanding the temperament, will assist in learning to set the pin.) What often ends up happening is leaving the pin in a stable condition as close to where the SBI is correctly tempered as possible. Then if down the road an RBI relationship can be improved, the SBIs are listened to and the ones that are a little out of line with the others are adjusted.

I had thought as it seems you do. I had thought, as others have posted, “Who knows how the SBIs should sound for any given piano? And how could they be tuned that accurately? All that can be done is to take your best guess and fix the problems as the crop up.” What this leads to is poor tuning of the individual SBIs, just roughing it in, not even setting the pins very well because it will probably have to be changed later, and then trying to tune to notes that aren’t even staying where they were put! And then when a break in scaling is encountered, trying to get the RBIs to beat theoretically or at least progressive. To do this the SBIs are mushed around and somehow the circle is sloppily closed in frustration.

I remember something the old timers said when I was first tuning. They said the theoretical beatrates are only approximate. They can be different, sometimes very different, especially in spinets. I wanted to ask which ones would be faster and which ones would be slower and how would you know how much? I never did ask, partly because I did not want to seem ignorant, and partly because I didn’t think they really knew, otherwise they would have said how.

Now I know how. Tune properly tempered SBIs. Make minor adjustments for RBIs relationships except where scaling problem show themselves, then let the chips fall where they will. As long as each 5th beats about ½ bps and each 4th beats about 1 bps the resulting beatspeed of the RBIs will be acceptable.

Jeff, you are too predictable: I predicted you'd come up with something unpredictable. Indeed there will be a jump in at least all but one of the partials (including 1) at the break, so this will have to be weighted in some way. Interesting to think about.

How do you tune the notes around the break? Do you tune "normally" above the break, then tune the notes below the break, or do you also adjust notes above the break to help the transition to be smooth?

I think pure 12ths is not the answer as we want something that reduces to standard ET when inharmonicity goes to zero.

Kees

Doel:

I tune properly tempered SBIs<Period> These produce progressive RBIs above, below and while straddling the break, but not when going from one to another. To do this, very minor corrections are sometimes made to the SBIs, but no major ones like would be needed to make RBIs progressive when transiting (good word for it I think) a break. What happens is an RBI won't be in progression, so then I listen to the SBIs. Sure enough I find one or two that really aren't quite tempered right. I slap my hand and say Bad Tuner! You are sloughing off by tuning improperly tempered SBIs. You can do better than that! And when I do really good, there are no corrections.

I was more interested in tuning calculations in order to understand how it can be done rather than to actually do it. And when I wrote my program I also wanted to have any automatic tuning calculations result in theoretical ET with no iH. What I came up with was pretty poor, automatic-tuning-wise, but answered most of the questions I had.

But consider what it means to have a tuning scheme that produces zero stretch with zero iH, but max stretch with high iH. That really is not what is desired. It is best to get as much stretch as appropriate in low iH pianos so that the high treble does not sound flat. 12ths are a great tool for this. Not that they need be pure, but working with just the 1st and 3rd partials should simplify things and create wider octave types on lower iH pianos.

Would it be difficult for you to use your tuning calculation tools to see just what happens to the RBI and SBI beatrates when 12ths, pure or otherwise, are tuned across a jump in scaling? I could never quite trust my program's results on this. I used the original Young's equation for partial frequencies and I also have reason to suspect that the partial frequencies that I was kindly given were "cooked" by Verituner.

Do I sense a new Topic coming up?

Thanks to all contributors of this thread. There's so much interesting theory and guidance to read, sometimes I have to remind myself it's more important to practice tuning than to spend time reading about it!

I hear ya

Yep, but remember to always practice stability. Nothing else works well without that.

You mean stability, as in setting the pins? Absolutely the hardest skill to master.

Yep!

So, you guys that did learn to tune with fourths and fifhts, how did you practise and how long did it take until you could set a working temperament?

I took lessons from a blind floor tuner. He had me read parts of Dr. White's book. At the second or third lesson he demonstrated how the temperament was set. Then I set a working temperament on the first try. Getting a good temperament took a couple more tries during practice. At the fifth lesson he said that there was nothing more he could teach me.

BUT, that did not prepare me for when I tried to tune a temperament across a break, like is common with spinets. I thought at the time that I wasn’t tempering the 4ths and 5ths correctly. It was a real blow to my self-confidence. I thought the warning about beatrates on spinets meant that the RBIs would beat slower or faster, but still be progressive. It was some things that I read on this great Forum that clued me into what really happens.

I can´t believe that you did that so fast, I think i tried tuning by fourths and fifths maybe 50 times and didn´t get it even once... I have been trying bills sequences now for a while and i think i´m starting to get deacent results now...

Touché!

Bill: what sequence to you use yourself? and how did you learn to tune in first place?

You asked, so I told you. Rather than tell you to go pound sand, I will consider how frustrated you must be and reply to your Topic about tight pinblocks. It really is OK. I understand. If the strings don’t stay at pitch, no temperament sequence will work very well. It is like trying to chain rolling logs on a moving truck.

Then I guess I won't bother telling about some really amazing things I have done, like herding bear.

This is another point I was making in another thread some time ago, the tuning pin system we have today doesn't have enough accuracy, because as you say, the speaking portion's tension changes, we can hear the difference in frequency and tempering, but the pin cannot be set that accurately.

You say that when a break in scaling occurs, the thirds will cause the fifths to be mushed around to fit. But the fifths and fourths are supposed to be mushed a little from ½ and 1 bps, they aren't supposed to beat exactly at that rates, plus the mushing effect of human error, you said humans can't tune consistently within 0,1 bps. Instead of doing the fourths and fifths in a row using one tuned note for another and then finding that F3-A3 beats faster than F#3-A#3, if you would make the thirds correct and let the fifths fall where they will, would you say the fifths and fourths are tempered wrong when let's say 1 cent is divided between four fourths and fifths, keeping in mind that they aren't all supposed to beat ½ or 1 bps? And if the end result is all intervals apparently sounding correct, isn't that the purpose?

today i actually set my first temperament within the range of 1 cent (partial2) by tuning fourths and fifhts. The key was to use the following formula:

contiguous 5ths ratio 2:3
5th and 4th top common note 2:3

5th and 4th bottom common note 3:4
contiguous 4ths ratio 3:4

D4 from A3 about 1 bps

G3 from D4 so that G3-D4 beats 2 times in the same time as A3-D4 beats 3 times.

C4 from G3 so that C4-G3 is a little faster than A3-D4 and G3-D4 beats 3 times in the same time as G3-C4 beats 4 times.

F3 from C4 so that F3-C4 beats 2 times when G3-C4 beats 3 times.
At this time i could check with the M3-M6 test (F3-A3 = 7bps F3-D4 8bps)
Then i tuned:

A#3 from F3 about 0,7 bps, checked so that F3-A#3 beats 4 times compared to F3-C4
At this time i could also check that F3-A3 , F3-D4 and A#3-D4 all are accelerating.

D#4 from, A#3. a bit faster than A3-D4.

G#3 from D#4 a little faster than G3-D4 and G#3-D#4 should beat 3 times compared to A#3-D#4. G#3-C4 should beat about 1 bps slower than A#3-D4.

C#4 from G#3. A3-C#4 should fit inbetween A#3-D4 and G#3-C4. C#4-G#3 should also fit in between G3-C4 and A3-D4.

F#3 from C#4 a little bit slower than G3-D4 but faster than F3-C4. F#3-A#3 should be 0,5 bps faster than F3-A3.
F#3-C#4 should beat 2 times compared to G#3-C#4 3 times.
again the M3-M6 test.

B3 from F#3. a little faster than F3-A#3. F#3-B3 should beat 4 times compared to F#3-C#4. at this time G3-B3 should fit inbetween F3-A3 and A3-C#4 and progress evenly.

The last note is E4 which is tuned from A3 and B3. B3-E4 should beat 3 times when E4-A3 beats 2. C4-E4 should also beat a little faster than B3-D#4. E4-B3 beats 4 times compared to F#3-B3.

This was the first time i tried this sequense by comparing the interwalls against each other. The " fourths about 1 bps wide and fifths about 0,5 cents narrow" didn´t work for me. I noticed how much more accurattely i could tune when i have to compare different intervalls against each other. especially if the note beeing tuned is a common note. for instance C4-F3 to G3-C4 If they are equal beating C4 is too low. (this didn´t happen in this sequence).

One thing i did notice was that i didn´t get very much stretch with this method. how could i add a little bit stretch?

I don't know if i was just lucky today, but it seemed pretty easy. as i wrote earlier i have made over 50 attempts to tuned by fourths and fifths but now when i found out about the beating ratios beteween the different intervalls it got a lot easier.

As a jazzpiano player i find it relatively easy to play 3 beats over 2 or 4 over 3. i use that kind of polyrythmic ideas in my playing.

Feel free to critisize...

Martin

I tried a hybrid method today which worked out great for me. I tune F3 to A3 at 7 bps, making sure of the beatrate with a metronome. This is something I can do very accurately. Then I tune the 4ths and 5ths making up F3A3, first C4 to F3 then D4 to A3 then fit in G3 to make G3C4 and G3D4 be "correct". I can do this very accurately too. At this point I have F3 G3 A3 C4 and D4 solidly in place and just tune by 4ths and 5ths checking any RBI's that become available and if tests failed I listen to the 4ths and 5ths again and usually discover a problem.

With the contiguous M3 method I have problems placing C#4 as the A3C#4 and C#4F4 beats are sometimes very hard to hear.

I guess there'r many ways to skin a cat.

Kees

Do bother, it sounds very interesting.

I have played ney on top of Haleakalā mountain and got the almost extinct native ney-ney bird to reply to me.

Kees

In my opinion, this is the biggest mistake anyone can make. It is the reason why the rest f the temperament often does not work out, especially on short or irregularly scaled pianos.

I often read that with these, 4ths & 5ths sound good but the RBI's are irregular or not progressive (or digressive) across breaks, etc.

When I tutor someone, I tell them not to try to "count" or time beats with any measuring device whatsoever. The actual rate of the F3-A3 M3 depends on the inharmonicity and the size octave chosen. The contiguous thirds approach with two pairs of octaves will find that rate. If you find the F4-A4 M3 impossible to discern, the reason may well be that it is far too fast. If the F4-A4 M3 is too fast, then the F3-A3 M3 is too fast.

In theory you are right of course, but according to my calculations even on short pianos the error remains well under 1 cent and on nice grands it's more something like 0.2 cent.

Kees

Whatif you use a hybrid version where the F3-A3 is set by tuning the first set of contigous thirds?, then you would have a frame for tuning the rest of the notes, C4-G3-D4 between F3-A3 and F#-B3-E4 between A3 and C#4 and finally G#3-D#4-A#3 between C#4 and F4...

Martin

Kees,

I am not even thinking about theory! I am a seat-of-the-pants type guy. When I regulate, I never measure anything! I let the dimensions find themselves. So, why impose an arbitrary beat rate of which you can never be certain? Let the beat rate find itself!

If you have not had success with my descriptions, try investigating what Jack Stebbins RPT (long term primary instructor at the North Bennett Street School and three decade PTG tuning examiner) says in his material on "Let the piano tell you!"

There seems to be something very intelligent coming out of Finland these days! If you can get the series of contiguous thirds from A3 to A4 and verify that both octaves, F3-F4 and A3-A4 are at least similar in size, everything else will work, no matter how you go about it. Tune all the 4ths & 5ths you want after that until kingdom come and you will be able to correct yourself, now matter how you go about it.

I refuse to believe that people "cannot hear" the F4-A4 M3! If you say that you cannot, then you must do as I said, practice moving F4 against A4 until you can. Surely, you will be able to hear when you have a slow beat. Flatten F4 when you do hear it as you hear the speed increase to beyond the point of discernibility. You will hear it if you try.

There must always be an interaction and cross check between SBIs and RBIs in ET. Leaning too heavily on 4ths & 5ths will almost always result in Reverse Well, especially if you try to count 7 beats per second for the F3-A3 M3. That is a quick and easy recipe for Reverse Well.

Leaning too much on RBIs easily results in a Marpurg type quasi ET. The RBIs sound good but the 4ths & 5ths are all over the place.

Equal Temperament means exactly what it says: all intervals are tempered equally.

Well I find just about any recipe to set the ET temperament works for me, so I just like to play around with different methods.

What I like about setting the beat rate of F3A3 and taking it from there is that I can, if appropriate, set it to a low value (like 4bps), tune the 4ths and 5ths that span it proportionally, and then deal with the rest as I please to make up any unequal temperament I like.

Kees

Martin:

Congratulations on reaching your goal of setting a temperament with 4ths and 5ths! Getting SBIs to be progressive is an accomplishment beyond simply tuning the temperament with SBIs. You must be doing better with your hammer technique.

You may not understand just what I mean by “about ½ bps” for the 5ths. I would say that all your fifths beat about ½ bps. And while we are on the subject, you mentioned that you did not get much stretch. If the temperament octave was 4:2 that really is satisfactory. If it was not, well, try making the 5ths beat a hair slower, like closer to ½ bps If the SBIs are tuned to theoretical beatrates, the octave will be narrow due to iH. The 4ths will be OK, but the fifths will be a bit too narrow.

You may think that going from 2/3 bps to ½ bps (about ½ cent) would cause a total stretch of 6 cents in an octave when going around the circle of fifths, but it does not anymore than a pure fifths tuning results in an octave that is 24 cents wide. ½ cent in a fifth is divided by 7 semitones which results in less than 1 cent for an entire octave.

Since listening to the beatrate ratios is working for you, consider trying slightly different beatrate ratios. Instead of the 3:2 and 4:3 ratios when comparing the 4ths and 5ths, try 2:1 and 3:2. This will probably only be necessary until you learn to hear the tempering of the SBIs directly instead of counting beats.

Kees:

I agree that theoretical beatrates are very useable on well scaled pianos. The combined effects of iH are largely self-correcting. And the idea that the octave size makes much of a difference just doesn’t cut the mustard in my book. If you are changing the octave width enough to make a difference, then you are beyond the point of octave sounding even close to pure. An M3 would have to be changed about 1 cent before it beat the same as the next chromatic M3. Over the course of a set of CM3s, this would be about 3 cents in an octave, almost enough for a pure fifth tuning!

But when the scaling is not good, well I won’t repeat my litany again right now.

” You say that when a break in scaling occurs, the thirds will cause the fifths to be mushed around to fit.”

Yes, if you use RBIs to set the temperament. That is why I use SBIs.

You really should use the Forum’s quote feature so what is said is kept in context.

I herded Alaskan Brown Bear with a NOAA Hydrographic Survey Launch:

According to my calculation on a poorly scaled piano the difference in F3A3 beatrate between 4:2 and 6:3 octaves is something like 0.1 bps. On a well scaled piano much less.


Again according to my calculations on a small piano the beatrate of F3A3, when based on a 4:2/6:3 compromise octave gets as low as 6.7 bps. That is only 0.2 bps less than the ideal number (6.9) and corresponds to an error of 0.4 cent. Surely on crummy pianos you can live with that. On well scaled concert grands the error is less than 0.1 cents if you go for 6.9 bps with a metronome.

Of course I could have made an error in my calculations. It would be nice to know what is the bps of F3A3 on spinets in your practical experience, if you know?

Kees

too bad there's no "like" button on here, I like the boat c/o Jeff.

Personally I just balance it out, hear it, go with it, ... every piano is different and to get the most pleasing and aesthetically perfect sound from it requires personal presence and attention.
SM

Maybe slightly off topic, but I though brown bears were land animals.

Kees

I have seen them swimming across Shelikof Straight, like 30 miles.

One of the biggest jumps in scaling I have come across was the Wurlitzer Petite Grand. I have to make F3-A3 and A3-C#4 beat at the same speed for the SBIs to sound best. On a Kimball Spinet (not a Whitney, different scaling...) F3-A3 beats the same as G3-B3. But I often wonder just what I am hearing in the 5ths. There is often much trash. And there would be a big difference between the 3:2 and 6:4 partial matches. It could very well be that I am tuning for the least obnoxious sound. I may not actually be tuning "about 1/2 bps" for the fifths.

Here is a scaling graph of a Wurlitzer Petite Grand. It isn't the same scaling as the one I encountered. That one had bichords up to G3 or so. But notice what happens around the break:

Jeff:

Yes if the break occurs in the temperament octave all bets are off. If not, I claim F3A3 stays pretty darn close to 6.9 on any instrument.

Why would you not move the temperament octave up to be above the break on that Wurlitzer Petite Grand?

Naively I would assume that the highest M3 that crosses the break will beat slower than you would expect compared to its upper neighbor, as the lowest note of this M3 will have a higher IH constant than expected, so the 5th partial will be closer to the 4th partial of the upper note. Does this conform to reality?

Kees

I agree, but there are always exceptions.



Then the RBIs beat too fast for me to be sure that they are progressive. As a compromise, I have modified Dr. White’s sequence so that the first RBI is A#3-G4, about 10.5bps. Then when E4 is tuned, I have an inside/outside check (A#3-G4, C4-E4) to make sure things are on track before there is an RBI progression that transits a break, such as F3-A3, G3-B3.



Ugh! In my haste this morning (more bears to brand…) I chose a poor example for a challenging scale. It is definitely different than the Wurly I tuned. This one shows a more typical example where the M3s straddling a break causes them to beat faster than those above the break. Tuning the SBIs and then listening to the resulting RBIs will tell the story.

Steinway S: