After a number of different Topics and some experimentation I am considering tuning pure twelfths over the entire keyboard, even across the breaks. Because of the P5-P8 test for pure twelfths, this should be practical.

Other than Mr. Stopper and Mr. Swafford I don’t know of anyone that deliberately tunes with pure twelfths. Can anyone share their experiences? Both aural and ETD responders are welcome.

I haven't in the past, but will probably try it out. Particularly after some of the news coming out of the Kansas convention where Bernhard Stopper's presentation of his aural sequence seems to have generated a lot of interest.

Let me know how it works out. This may turn me into an A-forker.

Yup, tried it many times - I think Ric Brekne posted a method years ago using tunelab.

Often stretches the middle too much for my ears in the mid-section of smaller or more regular pianos.

using only the 3:1 or 6:1 is still relying on single partial matches between two notes that may or may not have a regular partial pattern...

I much prefer to balance between the 12th, the double octave and then bring in the single octave to make sure it doesn't obviously beat.

Easy using the Verituner.

Ron Koval

Well, what prevents you from trying it and reporting the results?

From my theoretical perspective I can contribute that P12 tuning is electronically not distinguishable from a reasonable 6:3 bass smoothly to 4:1 in the treble tuning on reasonably large pianos. This is the default in tunelab and will be acceptable to customers without specific requirements. I have simulated P12 schemes on my sister in law's Wurlitzer spinet on my recent holiday, but sort of concluded that instrument can only be tuned aurally.

If you're interested I can put up some data on that piano. It actually sounded quite nice after I tuned it aurally, but what I did to get that effect is unclear. I measured everything and plan to look into it.

It's ironic that ETD's are all you need to tune concert grands close to perfection, but for those nasty neglected spinets that are tuned only every decade for a bargain price you need an aural tuner, which seems to be an endangered species. There are apparently only 2 aural tuners in my local RPT chapter.

Kees

Thanks, Ron:

I remember reading some hot and heavy volleys between Brekne and Stopper on the subject, mostly about precedence. But since I don’t use an ETD, the method itself was not of use to me.

I do know what you mean about the middle getting stretched too much sometimes. The M6s in the 4th octave can be edgy. But on the whole I think there is much in favor of pure 12ths.

Of course when there are wild partials you need to make adjustments for anything that is obnoxious. But what I now realize is that dealing with a typical jump in iH can be done coherently. With pure 12ths, the 3:2 fifth and the 2:1 octave will be consistently tempered. What more could you ask for?

I may post some results when I have some AND if there is genuine earnest interest. (Once burnt, twice shy.) Mr. Stopper’s description of how to tune pure twelfths does not work for me. I do not hear what he does. Whatever I may come up with, others may not hear what I might. This sort of thing can be fodder for those that wish to be abusive. It would be irresponsible for me to be an instrument in someone else’s misbehavior without real benefit to others.

I realize that pure twelfths are often tuned during other tuning schemes. But deliberately tuning them may be a different thing. I am thinking that it naturally requires a more exacting ET than other methods. The less tempered an interval is, the more accurate it must be to be consistent.

Sometimes I think the best way to tune the bass on a spinet is to tune it so that none of the octave partials line up. Then it is not so obvious when some can be made to line up without others being obnoxious, and some can hardly be made not to be obnoxious at all. Twelfths may fit this requirement.

If you want to put the graphs of the Wurly spinet on a new Topic you can bet I’ll enjoy studying them

This may be one of the most important and oft forgotten statements ever made on tuning... What one finds easy and intuitive, another finds baffling!

There are many paths...

That's one of the key reasons I suggest hybrid tuning or getting comfortable with modern electronic tuning devices instead of the blanket suggestion to learn aural before anything else. The longer I've been tuning, the more I find that a really musical tuning lives in the unison and how whatever temperament is spread out to the ends of the piano.

Ron Koval

Ron: I agree on the intuitive/baffling perception!

I was in Bernhard's class in KC, and he really doesn't go for 3:1's, 6:2's or 9:3's per se. He listens for beat canceling (= masking). That alone would make it quite different from a 3:1 setting on an ETD.

Then, in the outer ranges, he goes for musical justness. Very interesting to see him tune, because it was far less theoretical than I would have imagined. When I spoke with him later, he indeed explained that he went for it aurally, then established the theory later. Much like any long-lasting discovery through the years

Is Bernhard publishing his aural approach? I read somewhere on the PTG forums that Kent Swafford was (?)

So... did anyone there happen to have a ppc version of onlypure running to see if his aural approach matched the machine calculation??

Ron Koval

Thanks for posting, Pat:

I am going to try many things in my experimentation. Like octaves, I think I can tune more consistent 12ths by tuning 4ths and 5ths instead. But I also wonder about the aurally pure 12th vs. a proof tested 12th, vs. a 12th with an octave/fifth or fifth/octave included.

Did it seem to you that the temperament was exceptionally equal? I wonder if that is part of the purity that people hear.

Instead of the graphs just a result: no matter how you tune the whole piano with SBI's (which can be P4 P5 P8 P12 P15), there is always a large jump in M3 beatrate across the break. It varies from a 2 bps jump (up when going down if IH drops below the break) for the Wurlitzer, to 1 bps for a large Heintzman upright. (Those are the only two pianos I have IH data for all notes.)

Conversely, if you tune the lower midrange paying attention to contiguous M3's there will be unprogressive SBI's across the break.

I don't know what your customers prefer but mine prefer the M3's to behave well (which is also unequal so probably not comparable).

Kees

Not sure what you mean by the "burnt" stuff. I think you refer to a couple of folks who argued that what you posted was unimportant because they did not understand it.

For my part I have genuine interest in your posts about tuning fundamentals and you have opened my eyes to many issues which I was not aware of and are not covered in any tuning literature. I have and am still learning a lot from your contributions.

Thank you Jeff!

Now that I'm in the acknowledgements mode; to get to the level of understanding Jeff I studied the materials on Bill Bremmer's website which was all I needed to understand piano tuning. Thank you Bill!

Kees

Doel:

You are welcome and I very much appreciate your mathematical confirmation of what I understand intuitively and experience in practice. There is no one else.

My customers like how I tune, otherwise they would not continue to be my customers! Your 2 bps jump in the M3’s on the Wurlitzer spinet is a bit more than what I experience. But I am thinking that I let the RBIs influence me too much sometimes.

Do you remember how the least overall jumps in beatrates (I think I may start calling it what it really is: jumps in tempering) across a break in the graphs that you produced for another Topic were for pure twelfths? I am now trying to figure out the best way for me to tune pure twelfths. I think it will probably be by playing a stacked P5 and P8 simultaneously. Whether tuning either the upper or lower note, the interval being tuned will be ... tempered as much as the other one when the twelfth is pure. This can be determined by the experience of “beat cancelling” that occurs when the P5 (3:2) and P8 (2:1) are beating at the same rate.

But tuning the temperament itself is another challenge since this cannot be done with twelfths alone. The basic sequence is up a fifth, up a fifth, down an octave. The octave is problematic for me and I think the answer will be to create the octave by tuning down a fourth from the second note in the basic sequence. Then the tempering of the second fifth can be checked by testing the 2:1 octave, unless the fourth is influenced by a break. Then the fourth may need to be altered instead. The RBI checks are created the same as any 4th and 5th sequence, but with even more available. In particular there are many more M6s which makes the listening to m3s completely unnecessary and, just as important, more inside/outside tests. Of course there will be jumps in … tempering across scaling breaks except for the fifths and octaves.

Another interesting thing is the choice of the first twelfth. D3-A4 seems a natural for how the first RBIs fall when going two steps CCW in the circle before starting back at A and gong CW. The first RBI will be G3-E4. I can live with that. G3 is rarely wound. As I said before, this may make me into an A-forker!

If you feel like it, you might want to try 8:4 octaves to see what the M3's do then. I expect they will be much smoother because the test for the 8:4 octave is the m6-M3 test.

Jim: Bernhard is, as far as I understood, reluctant to posting them in a public forum. He will, however, give them to anyone who PM:s him.



Ron: No, i don't think anybody ran the OnlyPure software side-by-side. Bernhard himself, though, says he gets the same result aurally and by his ETD, and I have no reason to believe otherwise.


Hi Jeff,

Yes, the temperament seemed very equal indeed - this with a tuning sequence that didn't use any RBI's at all. That was actually one of the most impressing things about Bernhard's tuning!

Indeed a little less, but still a large effect on the spinet.
Somewhat irrelevant I think as 8:4 octaves in the midrange are not an option I think.

Kees

Debugging today I found I made an error. With 8:4 octaves the M3's are progressive across the break even on the Wurly, exactly as you predicted. The P5's however jump from 1/2 bps narrow below to 1.5 bps wide after the break.

Using a 6:3 in the bass to 4:1 in the treble scheme the M3 bps jumps from G3 to G#3 from 8 to 7 (more like what you observed I think), the P5 from 1 bps narrow to 0.3 bps narrow (anti-progressive).

Using a P12 scheme M3 jumps from 9 to 6 (very bad) and P5 from 0.3 to 0.8 (progressive).

Kees
PS (added in edit): A scheme that produces nice tuning curves and strictly progressive P5's is tuning by equal beating P4/P5's from the lower common note. Not sure how you could set such a temperament aurally. This seems to produce nice P12's too in the midrange at least (under 0.3 bps).
PPS (added in edit 2): On larger pianos the P4/P5 scheme produces too narrow octaves. Adjusting the scheme to have P4 beat twice as fast at P5's from a common lower note works well in that case.

Doel:

Thanks for crunching the numbers!

Yes, tuning 8:4 octaves is “off the table” due to the tempering of other intervals. And since that is what is needed for progressive M3s, for such a piano if tuned "coherently", then progressive M3s are “off the table” along with strict CM3s. This is the true value of this sort of investigation. To find out what cannot be done.

The 6:3 bass 4:1 treble is Dr. White’s prescription (take a fifth in the morning and another at night…). But his reasoning was for useful tests rather than dealing with iH. I wonder if there has been a general study of things that have been done right, but for the wrong reason.

And I don’t see where tuning equal beating 4ths and 5ths with a common bottom note would be that difficult. Maybe that is what often happens, anyway. The fifth is a very complex sound, especially on spinets.

And thanks for the heads up on what is expected to happen to M3s on a Wurly spinet tuned to pure 12ths. But I am not sure that a 3 bps jump would be that bad. I guess I’ll have to find out someday. Interesting that you came up with a 2 bps jump when tuning aurally.

Thanks for the reply. Could you clarify a bit, though? When you say that the sequence didn't use any RBIs at all, do you mean just that no RBIs were tuned, or that no RBIs were even listened to as checks? And were there RBI checks after the temperament was set to prove that it was ET, or did it just seem to be very equal? There was an ugly discussion about this sort of thing not long ago.

I must have been unclear, that's not what happened. I set the temperament octave using my ETD to a specific unequal temperament and the result was completely messed up M3 rates, which are what's important for this temperament. I tried to set ET too with the ETD with the same result; about a 2 bps jump in M3 beatrate in the wrong direction.

Then I tuned the whole thing aurally in unequal temperament getting the M3's right and it sounded fine. Of course now the P5 and P4's were not according to spec anymore but they are more or less random in most UT's anyways.

Kees

I'm not Pat, but I attended the first class Bernhard taught, (a second one was added at the convention), and Bernhard did not use any RBI checks as he tuned. He only listened to the three note combinations described in class, unisons as he went.

When he was done, he played all the major thirds chromatically. They seemed a little rough from where I was sitting, and I simply blamed that on the cold air blowing on the tenor break of the piano, but when I actually got to sit down at the piano, acoustics as they were, the RBI did indeed sound quite smooth and equal. Nothing jumped out at me when I played chromatic octaves, 12ths, and double octaves across the scale. I then went to quickly grab the temperament worksheet he handed out, but someone was at the piano before I could get back, and most people were interested simply in playing on the piano, which is very important, but I can't tell anything about a tuning except for unisons when I listen to playing, so that was as far as I got in analyzing the tuning.

After that I was thoroughly impressed with his work.

Does that answer some questions?

Regards,

No RBI's used for tuning whatsoever. For verifying, Bernhard ran a few chromatic M3's. They sounded smooth.

I now understand, that I don't understand. I am in the dark when it comes to UTs. It's OK.

Thanks, Thomas and Pat:

So there was a temperament worksheet handout! If it is not copyrighted, could it be posted here? Or do you think it best for me to contact Mr. Stopper directly?

Jeff,

Although I'm neither Thomas nor Pat, I would suggest that you do the latter. Having done so myself, I think that you, in particular, may find his hand-out interesting. I am currently trying to wrap my mind (and hopefully, in the near future, my ears too) around it, because it intrigues me.

Jeff and all,

I wrote my last post in a rush, but found it to be true even as I read it today Bernhard didn't use any RBI's for tuning, but ran them for confirmation. They sounded good, and proved his method.

Jeff: Bernhard hesitates posting his sequence in this forum, but would without doubt provide it if you contacted him personally. As you don't use PM's at pianoworld, you'd have to get in touch with him using his official email, info@piano-stopper.de.

I don't think it's OK. In my opinion you can't understand equal temperament if you don't understand unequal temperaments which it evolved from, and is a particular instance of. The historical temperaments all try to balance the badness of the M3, as this is musically most significant. When music reached the stage where all M3's are equally used it started making sense to make the all "the same". The 5ths were never an issue, except for nonsense tunings like Kirnberger I.

Just my personal opinion, not intended to start any kind of argument. I am aware that your position is also defensible.

Kees

Doel:

I think we can continue to have discussions and not arguments.

I have no idea what to think when I read about the “badness of M3s” when I also read that M3s weren’t used for tuning until modern times. I would have to study the subject much more than I have the interest for it to really understand UTs.

But could you elaborate on: ”In my opinion you can't understand equal temperament if you don't understand unequal temperaments which it evolved from, and is a particular instance of.” Maybe you mean in a musical rather than in a theoretical sense.

Reading something doesn't make it true. One of J. S. Bach's son wrote an article somewhere around 1770 or so on tuning keyboards which basically says use P4/P5 for tuning and M3's for checks. I think I have a facsimile of the article somewhere.


Yes that's what I meant, now that I think of it. Of course I overstated it a bit.

Since we don't know what ET is, we came to the matter of "taste", namely should we focus on P5 smoothness of M3 smoothness (simplifying a bit by leaving out other intervals). Until very recently perhaps the focus of UT's including "attempted ETs" has always been to use a particular distribution of unequal 5ths be the means to achieve the goal of a particular pattern of M3's.

So if you start favouring smooth P5's over smooth M3's you are going against 400 years of tradition. Fine with me, I hate tradition for tradition sake anyways, but a nice historical perspective I think.

Kees

Like form follows function?

There is no circle of thirds, just a triangle of thirds. Then you have to either go back to the circle of fifths or simulate it with the square of sixths.

I don't understand what you wrote.

Kees

I don't tune perfectly pure 12ths, I make them equal beating with double octaves. They sound very nearly pure but technically, they are not. However, when tuning the high treble from F6 to the top, I often tune pure double octave-fifths.

The equal beating double octave and octave-fifths (the PTG Journal's preferred nomenclature) provide for excellent "beat masking" (as Bernhard Stopper calls it). I also think of beat masking as beat cancellation or beat suppression. When that exact compromise is achieved, (take the example of the F3-F5 double octave), playing F3-A#3-F4 and F5 at the same time will yield an uncanny stillness even though none of the intervals are actually beatless. This is true for ET, Quasi ET and any mild Well Temperament or mild Meantone or mild Modified Meantone.

It makes the whole Midrange and well into the Bass and Treble sound very clean and in tune with itself, regardless of temperament. I have done this for at least 25 years.

The Sostenuto pedal is a great tool for implementing this idea but if the piano does not have one, the damper pedal can also be used the same way the Sostenuto pedal is deployed (play the notes first, then press the pedal) to adjust the note being tuned. The only difference between using the Sostenuto and the damper pedal is some collateral noise but the intended goal of "beat masking" can still clearly be heard.

Bill, I find that the collateral noise using the damper pedal is too distracting for me. I have thought about a tool to simultaneously play double octave, octave fifth and tuning note similar to Bernard's tool but with three pads. However, the different configuration of sharps and whites may be a problem for the design. Alternatively I have thought of a tool that will play a forth but with enough weight to keep it down, then use it to play and sustain the double octave and octave fifth and then quickly move your hand up to play the tuning note. What are your thoughts on such tools to ease your method of mindless treble tuning?

I second that post, having almost identical thoughts.

Kees

Bill:

The double octave is one of the intervals that I check but do not directly tune. I think I will leave more comments on this sort of thing for another, current Topic.

Generally, I expect any decent tuning will have 12ths and 15ths that are virtually pure. When I started studying this in depth I was surprised how Plain Jane octaves will produce pure 12ths and wide 15ths when a moderate amount of iH is present.

I have tried your mindless octaves, but for me, it is not quite enough stretch in the upper treble when listening melodically.

A four sided tool might work. Or a 15th spanner and then use fingers for additional notes.

A temperament can be created using 4ths and 5ths, like Dr White’s method. And a temperament can be formed with 3rds and 6ths like Defebaugh’s method. Especially in the case of jumps in iH, the two methods will not have the same result.

I am thinking that the same could be true for UTs, and the development of them, and even for the expansion of the temperament.

There are nuances that can only be heard during the time that an interval is being tuned that cannot be heard when the interval is created by tuning other intervals. For instance, an M3 and an M6 can be tuned from a common note and produce a P4. This P4 might be a bit different than it would be if tuned directly. Or a 15th can be tuned by expanding the temperament with octaves. And this 15th might be a bit different than it would be if tuned directly.

In the case of UTs, I am only guessing at some things from what I read in your posts. I can imagine that during the development of UTs the sound of some intervals, as they were being tuned, affected how the UTs developed. Like maybe some 4ths and 5ths were tuned a certain way in the hopes of the 3rds and 6ths turning out as desired. But then it was decided that it was really preferred to have the 4ths and 5ths sound a specific way instead.

It was at the end of 2009 that, elaborating on this subject, I too thought that a double-spanner could have been used for an easy tuning.

Thread: about mindless octaves [Re: Inlanding] alfredo capurso
Full Member

..."About practical tuning and techniques, how do you think it would work using a double-spanner?

I mean, on top of a 12th spanner, say at 180° degrees, we could have a 15th spanner.

Working it out on paper only, it seems to me that we may need to tune only three octaves, and the rest could be done with this double-spanner.

Do you think it could be worth giving it a try?"...

Today I'm sure you'll be able to find a practical solution for putting this tool together. Indeed, at that time, I worked on a fairly banal sequence that could cover (I cannot remember if) three or four octaves by adding the aural tuning of three octave intervals. All the rest could be done with 12ths and 15ths.

In any case, one thing the tuner will always have to consider is the amount of stretch, which will depend on many well known factors. Thus, in my opinion, it could well be put in the chapter of easy tuning, perhaps useful in case of pitch raise (?), or for beginners that can deal with three intervals only.

My 0.002 cents.

Regards, a.c.
.

Too clumsy for me. I don't mind the collateral noise when I have to use the damper pedal instead of the sostenuto. However, if such a tool would work for you, go for it!

Precisely, virtually pure. I cannot usually hear any beat between them, so a strictly aural tuner may actually think they are both pure. However, an ETD will show that technically, the double octave is slightly wide and the octave-fifth is still slightly narrow.

I also agree with your comment on the high treble. As you will note that I said, beginning at or about F6, I begin tuning a pure double octave-fifth. That does the trick nicely.

Bill, I am confused about the words "double octave fifth". Does that mean go up a double octave plus a fifth (a 19th?, or tuning the mindless 'double octave versus the 12th' (which is an extended fifth). Can you please clarify, maybe using note names as an example? From middle C, or A440, what is a double octave fifth?

Thanks in advance,
Erich

I would think that using an ETD set to tune pure 12ths, would make a stretch that is closer to an aural tuning. I don't know if this would sound good on smaller grands, especially Yamaha's.

It means tuning the note being tuned with the note which is a double octave plus a fifth below. Example: Tune F6 pure with A#3. That is also called a 6:1 octave. The first partial of the note being tuned matches the 6th partial of the note found a double octave-fifth below it. "Octave, octave-fifth, double octave, double octave-fifth and triple octave" are now the preferred nomenclature of the PTG Journal, so that is why I use those terms, even though other terms are synonymous and just as valid.

I try to conform to the PTG Journal standards, mostly because if I use any of my previously written material in a Journal article, I won't have so much editing to do. It never hurts to be able to think of things in more ways than one.

Bill:

Have you tried pure 12ths in the bass?

Thanks Bill for the clarification. If I may ask two more questions, I know that a compromise between 4:2 and 6:3 octave is optimal for midrange and 6:3 is optimal for the bass. What is good for treble, and then high treble? Is it 4:1, then 6:1 when you get up to a certain point (F6) as you posted earlier?
And that brings me to my second question. You said that the test for 6:1 is a pure double octave plus 5th, but what is the test for 4:1? (I would guess pure double octave, but maybe I'm off in my reasoning).
I know you can't just stick these numbers on any size piano, which is part of this thread's discussion. But it would help me to understand what partial ratios I'm going for outside the midrange. Am I correct in assuming that 6:1 will be more stretch than 4:1 and thus sound brighter in the high treble? Ok, that's 3 questions - sorry.

Thanks for your insight. Jeff, you are a whiz with the numbers - feel free to post an answer to my questions. Do pure 12ths in the bass give you 6:3 octaves, or even more stretch in the bass? Any info from anybody will be much appreciated.

-Erich

That I can answer theoretically. For upright and worse: Less stretch if you tune 3:1 12ths compared to 6:3, about the same if you tune 6:2 12ths.

For larger grand pianos the answer is different.

Kees

What he said!

And I think this is the most important feature of 12ths, pure or not. They cause wider octave types where the iH is low (when you want more stretch than what is naturally provided by iH) and narrower octave types where iH is high (so that octaves do not become too wide and cause beating).

Glad you brought up the 19ths. I hadn't thought about them for a while, and I have not tuned them. I imagine it would be much easier with an ETD than aurally with an m3-M17 test.

So I checked some previously calculated tuning curves and found that 19ths would cause much, much more stretch than 12ths. This is not to criticize how it sounds, because I don't know how it would sound.

It shouldn't be hard to check this sometime while you are tuning to see if the 12ths actually are wide with pure 19ths in the high treble. Some squirrely things can happen up there due to iH.

I do the same as I do in the treble although I usually don't use an ETD to see that the double octave is wide and the octave-fifth is narrow. They bothsound virtually pure to me. The lowest Bass, A0-B1 (more or less), I tune the inverse of the way I do the high treble, pure 1:6.

Either that, or after all the rest of the piano is tuned, I tune the lowest single strings by widening the octave until it is obviously too wide, then step on the damper pedal and sharpen the string being tuned when it seems to blend best with all of the coincident partials above. That tends to give the low Bass a deeper sound and makes the piano sound larger than it really is.

A 4:1 is a bit narrow. That means double octaves are pure. I tune a compromise between double octave and octave fifth. That way, octaves, octave-fifths, double octaves, double octave fifths and triple octaves all blend nicely. None but perhaps the double octave-fifth and triple octave are actually pure but they all sound virtually pure.

If you read what BDB has said bout this, it is apparently what he does too.

There is no "test" for 4:1 or 6:1 octaves. You either tune them pure, wide or narrow.

I've had extremely good results with tuning between an octave and a fifth and double octave. It helps provide a consistent method for creating an evenness to expanding the temperament. As well, it gives a quietness and purity that, until the time I started using it - it was hit or miss. I do check them against an octave to ensure I've not stretched too far. It is a quick and very reliable method for expanding and checking how the temperament is extended up and down the register. The first time I used it, I was surprised at how much time I saved and how truly even the octaves sounded.

Glen

Glen:

Have you ever tried pure twelfths?

Folks:

I was thinking about the difference between expanding the temperament with one interval and expanding it with two.

For instance, the temperament octave can be expanded upwards by making the 4th beat a little faster than the 5th. The average beating of the 4th and 5th will be determined by the width of the M2 that exists between the lower notes. And what is being done is really making a compromise based on that M2. Of course the octave and other intervals may be checked for anything out of kilter, but by nature this is expanding the temperament by compromise.

Or the same temperament can be expanded by listening and tuning the 5th by itself and then checking with the 4th. If the fifth is good but the 4th is not, then it points out an error in the M2 that exists between the lower notes. Here there is no compromise. The fifth is tuned and then other checks are used.

I see the use of a single interval for expanding the temperament as demanding higher accuracy in the temperament.

Jeff,
I did try pure twelfths at one point during my learning journey, but found that I had to make other adjustments when expanding out from the temperament because setting one interval as pure was at the expense of other tempered intervals. Since 5ths were tempered when I set the bearings, then making them pure as I could outside of that range made the piano sound a bit strange, not uniform as I expected from my best attempt at ET. I am sure there is a way to do it, but I did not take any more time trying to make twelfths pure.

At one point when I was learning more about stretching octaves, I noticed the 5ths getting wider, to pure, then to ever so slightly wide, but it was a consequence of the stretch. So, at some point I am sure the 12ths were pure.

I did like how attempting to make the 12ths pure helped me in expanding the bass octaves, however. At one point I tried stretching the octaves in the bass, but ended up over-stretching to the point they would roll too much.

But once I tried balancing the double octave, octave and a fifth and the octave down the register, it helped me to control how much stretch, if any I would put in the bass. I became much less concerned with whether it was 6:3 or 8:4 type in the bass, once I started balancing the double octave, 12th, and octave. I still use m3-m6 as a check and the inside outside M3-M6, and a sanity check using 10ths to ensure beats are as progressive as possible. Some pianos are more forgiving in that regard than others. I just try to make sure to minimize the amount of roll in the lower bass.

Hope this helps - I know it is a long answer to a short question, Jeff.

Glen

Thanks everyone (Bill, Jeff, Kees) for the responses to my questions. Man, just when I think I have all the aural tuning techniques in my bag of tricks, I realize that there are much more out there. There are so many ways to skin a cat (poor cat doesn't have any skin left).

Now that all these techniques for expanding the temperament and stretching the bass/treble have been presented and discussed, I would really like to hear some of these methods in action. Can anyone point to a piano recording where they either did the tuning themselves, or know how that particular artist likes their tunings to be done, or knows the tuner and his/her methods using one of the previously discussed methods?
Links to YouTube clips or even short demo clips on iTunes or Amazon that emphasize bass or treble stretch would be great.

It never ceases to amaze me the amount of knowledge I've gained from this forum. So much more to learn - I guess that's the beauty of it!

Thanks again to all.

-Erich

Theoretically the test would be M3 below bottom note for 4:1 and m3 above bottom note for 6:1. I guess you mean their beat rates are too fast to be useful as a test?

Kees

Thanks, Glen:

I was looking for the experiences of others with pure 12ths, and your long answer to a short question did just that.

Yes, I know how an ordinary temperament octave on a mid-sized piano may not be enough stretch for 12ths and imposing them will skew the temperament. I have studied this and found that for spinets 4:2 octaves will produce pure 12ths and as pianos get larger, the octave must be wider, but never as wide as 6:3 in the temperament. However, it will need to be 6:3 or even greater in the bass.

I am still working out how to implement a pure 12th tuning. It will probably be along the lines of a D3-A4 temperament on pianos with D3 being unwound and a F3-G4 modified White sequence for other pianos by making an educated guess on the octave size.

Also, the 12ths have to become pure before the 5ths become pure. Interestingly a plain old 4:2 octave if actually tuned to the upper treble will produce wide 5ths. I think that is too much stretch. However they may be used, 12ths are a good moderator for stretch.

Erich:

There was a soundclip from Mr. Stopper that had a recording of a Bach piece (if I remember right) on a piano tuned by him. It was incredibly beautiful, but I can't seem to find it.

Duh! Here is a link to Mr. Stopper's site with the recordings: http://www.piano-stopper.de/html/sound_examples.html

Jeff, thanks for the link. I really liked listening to Ravel's Miroirs, in particular the movement "Une barque sur l'ocean". There is a purity and stillness to the sound, especially when a lot of 5ths are being played in the music. Also, I can't find anything too over-the-top with the treble and bass stretch on this particular piano. It sounds like a Steinway concert grand, so I'm assuming it can handle the wide octaves due to the pure 12ths.

I am assuming that right outside the temperament area, the octaves would roll pretty prominently. More so than they would using m3-M6 or Bill's mindless octaves? Is this generally acceptable to most musicians to have that much movement in the octaves? Regardless, I would think that this tuning would be a favorite with string players, guitarists, orchestras(piano concerto where the piano needs to cut through) and even voice, because of the pure and still 5ths sound.

I wonder how this method will sound on a spinet though. I play and teach on everything from a grand to a spinet, so the next time the spinet is due for its tuning, I will give the pure 12ths a try. I'll post my findings of putting the 12th method on this humble piano. But if it is anything like what the grand sounds like, I think I'll be pleased.

Thanks for the link,
-Erich

Erich:

I'd be very interested in knowing how it turns out for you. What I expect you will find on a spinet is abaut a 2 bps jump in the M3s across the break along with slight beating of double octaves across the break. And I expect that the octaves will sound virtually pure. With pure 12ths the lower 5th and the upper octave (2:1) will beat at the same rate, like 1/2 bps. How the 4:2 and 6:3 octaves turn out depend on the iH.

Thanks Jeff, I guess I was thinking incorrectly about the octaves being too wide right out of the gate from the temperament. I suppose the initial temperament octave width will keep the subsequent octaves in check. Let me run a pseudo-scenario by you all and see if we are on the right page about the 12th method:

First, set F3-F4 temperament octave (I am currently using Bill's Marpurg ET shortcut). Next fill in F#4,G4,G#4,till I get to the A440 I originally tuned to the pitch source. Then I tune downward from F3.. E3,D#3. When I reach the D3 right below the temperament area, I now have my first 12th that I can tune pure with A4. Then I keep going down C#3 to G#4, C3 to G4, and so on. Typically I stop short of the low bass to fill in the treble above the A4. So now I likewise tune upwards from A4... A#4 to D#3,B4 to E3, and so on. Then I do the low bass and the high treble last. Nothing but pure 12ths for the whole piano, minus the notes I had to fill in to get to a 12th in the first place.

Comments or suggestions are welcome. Also let me know if there is another better way to get to my first 12th after setting the temperament. I typically use m3-M6 below and then I have 10ths,5ths,4ths (the usual suspects) to check as I go up. I had been doing mindless octaves when I reached the first point that I could, with 10th and eventually 17th checks to make sure everything is smooth. But with this new method, there might not be a need to run 10th and 17ths up and down the extremes of the piano, as long as the temperament is pretty accurate and the 12ths are pure?

Thanks for any info you guys might have.
-Erich

Erich:

Stick with what works for you. I see no reason why starting with a 12, 13, 17 note temperament will not work for 12ths the same as mindless octaves as long as other checks are made. The choice for the width of the temperament octave should hopefully coincide with what is required to produce either pure 12ths or mindless octaves, depending on which you are choosing to expand the temperament with.

Some will agree and others will disagree with the following: On a decent sized piano, 4:2 octaves are appropriate for mindless octaves and between 4:2 and 6:3 octaves are appropriate for pure 12ths when setting the temperament.

And if you don't quite get the appropriate octave width (for either) small errors should show up if you listen for them and can be dealt with when expanding the temperament.

For me, at least, I am starting to think that what pure 12ths does is force me to have a more accurate tuning. When an interval is pure it shows errors in other intervals all the more. There are no compromises between two intervals that might mask an error somewhere else. I am not sure that the quality, that I get, from pure 12ths is so much due to the specific stretch, which is really very similar to other stretch schemes, as to a more accurate tuning.

What I have been doing lately is to tune a pure D3-A4 12th to start with, add in A3, D4 and the Gs, and then go around the circle of fifths E, B, etc. filling in octaves as I go. This ties things into the original 12th early in the sequence and seems to minimize corrections as the temperament is expanded.

I haven't been doing this for long. I had been tuning the temperament and expanding it with 4:2 octaves until the 12ths became pure and then continue from there. I think I like this better, most especially in the low bass, where there is a difference in how I was tuning.

Jeff,

Thanks for the info and advice. Could I ask you to please post a short version of your 12th temperament sequence? For instance you said you tune D3 first as a 12th from A4. Then do you tune A3 as compromise between 4:2 and 6:3 octave to A4? Same with D4 to D3 (between 4:2 and 6:3). How do you proceed using 12ths? Are you doing down a 12th, up an octave, down a 12th, up an octave..or some other scheme?

Thanks in advance. I will try my spinet tuning with as many 12ths as I can, just to see the results. I like trying new things on my little practice/teaching piano.

-Erich

I'd also like to encourage anybody curious enough to try tuning the high treble by making the note to be tuned 1) the root, 2) the 3rd, 3) the fifth of an arpeggiated chord. I started to use this method as I repeatedly had to go back and sharpen the treble when I played the piano for final checks. This method, to me, puts the high treble where it sound musically good. It is kind of a mirror method of what Bill does in the low bass (as witnessed several times by myself).

I think you might be very right here, Jeff.

I don't think so. The goal was to have specific M3's sound better than others, being most significant in determining the sound of a tonal key. The 5ths were then fitted to achieve that goal. Then theory kicked in and you had to describe the scheme as a division of the comma over the 12 5ths. Simplest is to have 4 5th off by 1/4' (Werckmeister 3 and Kirnberger 3), if that isn't smooth enough we get into 6 5ths off by 1/6' (Young) and more refined some 5ths off by 16' and some by 1/2' (Neidhardt, Sorge, and others).

How those actual 5ths sounded was immaterial, as long as they were not too offensive, 1/4' being the limit in practice.

Kees

I automatically read this into being the whole tuning, not just the temperament. Might be my bad.

... and this gets me curious - when did the idea of using a set, narrow temperament and to "carbon copy" that order to the outer octaves possibly arise?

In the old days the temperament was the whole tuning due to the lack of inharmonicity.

Kees

Kees: And you still doubt it? (that is, Jeff's thesis.)

Not sure what you mean.

Kees

I would assume from having essentially zero inharmonicity keyboard instruments (organ, harpsichord) where an octave is an octave. Not sure about the inharmonicity of clavichords with their typically short low tension strings. But then on a clavichord you can control pitch with key pressure, making tuning somewhat irrelevant (withing reasonable limits) as the tuning is up to the performer by controlling key pressure.

Kees

Yes, I start off as you mentioned: D3-A4 pure, A3-A4 and D3-D4 a little wide of 4:2. I like to include the fourth above the lower note when tuning an octave. The fourth need not be accurately tuned, just somewhere close, as in the first few steps. The 4th and 5th beat the same and produces the smoothest sound when the octave is 4:2, so I can tell when the octave is just a hair wide. This also tunes the fourth just a bit faster than the fifth. But then I stay within the D3-A4 twelfth. I do not tune any other twelfths until the temperament is completed. Then when the temperament is expanded with twelfths, other tests may show where things can be improved (What’s new? )

For pianos with D3 unwound (P4 checks are omitted, they’re everywhere!):

A4 to Fork
D3 to A4 P12
A3 to A4 (wide of 4:2)
D4 to D3 and A3 and A4
G3 to D3 and D4
E4 to A3 (first M6 check)
E3 to E4
B3 to E3 (first M3 check)
F#4 to B3
F#3 to F#4
C#4 to F#3 (first inside/outside check: A3-C#4, G3-E4)
G#4 to C#4
G#3 to G#4
D#4 to G#3 (first chromatic M3s: D4-F#4, D#4-G4, E4-G#4)
D#3 to D#4 (better, slower chromatic M3s: D3-F#3, D#3-G3, E3-G#3)
A#3 to D#3
F3 to A#3
F4 to F3
C4 to F3

I am not completely certain, But I think the inside/outside test results in the M3 beating just a hair slower than the M6. I mean like no more than ¼ bps, not as much as ½ bps which is what a RBIs chromatic progression would be.

If D3 is wound, well you have to make adjustments. The first dependable, slow chromatic M3s would be with A#3. If the piano is a console or a spinet or a petite grand, straight 4:2 octaves are good above the break, but (usually…) wider when spanning a break. When in doubt, make sure the 2:1 octave beats about ½ bps wide with the M10-M17 test. And any RBI test involving a wound string must be used with caution.

Well, this is what I am trying for now, anyway.

PROOF READING CORRECTIONS ARE WELCOME!!!

Thanks for the sequence Jeff. I was wondering if the Stopper method uses a similar approach, or if he uses his 12th spanner tool to tune nothing but 12ths even in the temperament phase. The temperament would thus have to be pretty wide (span wise, not stretch wise) to circle back around to A4 or A3 using large 12th intervals only. Or is this simply not done for reasons unbeknownst to me?

Just to clarify your 4th check, let's say you are tuning A3 to A4. Do you put the D4 between them to test for the octave being wide of 4:2? In other words, A3-D4 4th beats slightly faster than D4-A4 5th? I just used these notes as an example - of course it could be any two notes (that are an octave apart) with a middle note making a 4th to the lower and 5th to the upper, right?

Is there a reason you don't use a 5th to the lower and 4th to the upper? Maybe it doesn't provide the same beating check you prefer.

Thanks again for your insight.

-Erich

Erich:

You would have to ask Mr. Stopper about his sequence. I do not know it. I do know that he considers his method to be more than just pure 12ths.

But don’t be afraid to try what you propose. I gave that sort of thing a try a while ago and decided it was too many notes to go back and correct if the octaves or resulting fifths were not exactly the right size.

Here is how the P4-P5 test works for the 4:2 octave, and why the P5-P4 does not work:

Let’s take A3-D4-A4 as an example. All the intervals involved have a common nearly coincident partial at A5. The A3-D4 P4 is a 4:3 interval with the nearly coincident partial at A5. The D4-A4 P5 is a 3:2 interval with the nearly coincident partial at A5. And the A3-A4 P8 is a 4:2 interval (among others…) with the nearly coincident partial at A6.

D4 is the test note for the 4:2 octave. It can be a little higher or a little lower and the 4th and 5th will still beat at the same speed as long as the A3-A4 octave is beatless at the 4:2 partial match. And if the A3-A4 is not a beatless 4:2 octave, the difference in the beatspeeds of the P4 and P5 will equal the beatspeed of the 4:2 octave.

But with A3-E4-A4 there is no common nearly coincident partial. For the A3-E4 P5 3:2 interval the nearly coincident partial is E5, and for the E4-A4 4:3 P4 interval the nearly coincident partial is E6. Neither of these is common with the A3-A4 P8 4:2 nearly coincident partial which is A5. So E4 is not a test note.

I feel this could be explained simpler. If you don’t understand my explanation, try Mr. Bremmer’s website. He probably worded it better.

Considering this, it makes sense to tune pure 12ths with a fifths and octaves sequence. The 3:2 P5 and 2:1 P8 intervals are the test intervals for the 3:1 P12. And then when tuning across a break, the 2:1 octave must take precedence over the 4:2 octave.

In Jim Coleman, Sr.'s great class in Kansas City, he reminded us to use 19ths in the treble as a test (12th + M5). I've always tested 17ths, but I find that 19ths helps me dial in the pitches even better. If the 19th is noisy, a slight adjustment quiets it.

--Cy--

Jeff, thanks for the explanation of your test notes. It makes sense to me now.

I might just try that pure 12th only sequence for the temperament on my practice spinet piano, just for the heck of it. If it ends up sounding weird, I'm the only one who will hear it in the practice room.

Thanks again for all the info.

Cy, I will have to try the 19th idea for the treble. I too used only 17ths going up and typically when I finished the piano, I would end up in want of more brightness or stretch at the top. Maybe the 19ths will help me stretch it even more and get some more 'sparkle' out of the high treble. Thanks for the post.

-Erich

Thanks Bill & Cy for bring up the 19ths. I had not thought about them much, but now I have.

There is much to be said for having checks that span a large part of the keyboard. The 19ths certainly do that.

At first glance, it would seem that 19th would cause less stretch than 12ths, but actually the 19ths cause greater stretch. If you look at the intervals each being stretched by 2 cents, because of the fifth that is involved, and this 2 cents being applied evenly over the entire interval, then it would seem that the 12th would cause greater stretch. But this is ignoring the effects of iH.

If instead we look at the partials involved (1:3 for the 12th and 1:6 for the 19th remembering that the iH increases the frequency by the square of the partial number) we get a different answer. In the case of the 12th the square of 3 is 9, but in the case of the 19th the square of 6 is 36. iH stretches the 19th 4 times as much as it does a 12th! Forgive me for omitting some other less important factors in order to make the point clear.

Please do not think that this makes one interval superior to the other. I just want to point out that if 19ths are tuned pure, then the 12ths are tuned wide.

Erich:

You're welcome!

I sure hope to hear what you think of the result.

When you tune the 19th and then come down an octave, you stretch the 19th to be pure at partials 6:1, but if you make the octave 4:2 for example, it would be less stretch than the 6:1 19th, so it would make the 12th wide.

I also thought about maybe tying something on my elbow that I can use to play the lower note and the other note I could play with my fingers, so that I could play and listen to the large intervals and tune them at the same time. Haven't tried it, but I should.

I think the 19th may be only useful, aurally, as a check. With an ETD I suppose there is no limit to what size interval may be tuned.

How high up do you hear beats in the treble? Do you tune the 12ths to what a pure 12th should sound like by the sound of the interval or do you actually hear the beats in the high treble?

Somewhere in the 7th octave I can no longer hear the beating of the 12ths. It depends on the piano. Then, yes, I tune the 12ths to sound right. This makes it important to listen to the single octaves to check that they are equally brilliant. If I tune the single octaves in the 7th octave to be smooth, then they sound flat to me.

Hi Jeff and all,

Sorry to bump this thread up again, but Jeff, you wanted to know how I liked the pure 12th tuning if I ended up trying it.

Well, I'm happy to report that I not only like it, I love it! I recently tried the pure 12ths out on my practice teaching spinet at the music school where I work. It has never sounded so good and pure! I then tried it out on a customer's baby grand and it sounded much better than the time I had left it before.

My final test was a request by one of our piano/choir directors to tune two pianos side by side for a September 11th memorial concert last weekend at his church. The second piano would only be used on the last piece of the program, a rousing rendition of "Stars and Stripes Forever" for 2 pianos, 8 hands.

When I arrived at the church, I immediately spotted the 7 foot Japanese Kawai on stage. But to my surprise, next to it was a small Kawai upright! I was informed that the church only had 2 pianos available: the grand piano for services and the upright for choir-room rehearsals. Plus, at the back of the stage sat a big organ to be used on some numbers as well! I walked up to both pianos and they were both a good 20 cents flat. Needless to say, I had my work cut out or me.

This might be a good side thread topic, but I tuned both pianos only to A440, so that both of their A's matched my A440 pitch source. And that's it - for the remainder I tuned each piano to sound good by itself. In my earlier days, I would have thought to make the upright match the grand, but techs and the Reblitz book have advised against this. Basically you just tune them as best you can and hope for the best when they play together right?

Anyway, I started with the grand. To quickly review, I use a hybrid tuning method so to speak. I really have enjoyed the Stebbins 5 stacked contiguous Major-3rds method so graciously posted by Bill. This gets me a nice big stable scaffold that sounds good for that particular size piano ("let the piano tell you" method). I then follow Bill's ET via Marpurg sequence to get the F3-F4 temperament octave. I tune the remainder from F4 to A4 (just 3 notes left there, F#4,G4,G#4) then downward from F3 (just 2 notes left there, E3,D#3).

Jeff, this is where I deviate from my old practices. When I reach D3, I now have my first pure 12th I can tune from A4 (A-440). I continue down to the last octave or where it becomes single bass strings, all pure 12ths. Then just for tension stability's sake, I go back right above A4 and I now have pure 12ths to tune from all the way up until the last octave where I can't hear the 12th beating/pure very well (like you had mentioned earlier). I do the last bass octave with a combo of 12ths, double octaves, and octave plus minor 7ths. On certain pianos I can hear one or the other, or sometimes on poor pianos, none at all. In that case, I just go with the double octave and nice sounding 12th and what sounds good on a big 2-handed major chord.

Like you mentioned, I like the final treble octave to sparkle and not be flat. So I intentionally make the double octave 'roll' wide, or beat slowly, and make sure the 12th sounds nice and bright and clean. That usually does the trick.

When I was finished, I really liked the sound of the grand and the upright as well. They both sounded very crisp and pure. As to how they sounded together, well, I just have to hope for the best. I will talk to the choir director/pianists today at work and see what they thought.

I find that I can use the pure 12th method to quickly expand the temperament as soon as a 12th interval becomes available inside the temperament tuning procedure. I used to rely on major 6ths, minor 3rds for the bass, 10ths for the treble, 17ths for the low bass and 17ths for the high treble, but those take longer to evaluate in my experience. It's easy to just 'stick the 12th pure' and move on to the next note. When I'm done, rarely is there a non-progressive 10th or 17th, if at all.

Having said that, the initial ET via Marpurg sequence has to be good for a solid foundation. Then, using pure 12ths, it's hard to miss!

One final note: whenever I used 17ths in the treble in the past, it was never as sharp or bright as I would have hoped when I got done and sat down to play my tuned piano. But with the pure 12ths, it really makes the treble bright and clear, without the added effort of comparing fast beats!

Thanks Jeff and everyone for all the suggestions and input to this topic. I may want to look into one of yours or Mr. Stopper's 12th-spanners to save even more time expanding the temperament.

-Erich

Erich:

Thanks for the update. My experiences are very much like yours.

I am waiting for an odd opportunity to tune two very different Petite Grands together. If it should arise I plan on doing the tunings for free in exchange for a recording of at least part of the performance, which I would provide a link to.

One piano is a Howard with a reasonable break around F3. The other is a Whitney with a ridiculous break around C3. They are both about 4’ 6” long. It would be interesting to see how they sound together both tuned to pure twelfths.

Hey Jeff,
That would be awesome to hear not just one, but two pianos tuned by 12ths!

I have been thinking of doing something similar. I have access to one of those cool Zoom hand-held recorders with stereo microphones and the ability to convert to an internet friendly mp3 file.

Although on this recording I would first go through the temperament so everyone could hear the normal temperament checks (chromatic 3rds, 6ths), followed by some chromatic pure 12ths all the way down and then all the way up. Then maybe I would play some classical snippets and jazz/pop snippets so listeners could hear the overall sound and finished product.

Doing this though requires that I go over the piano again and record it that very day as it is left in an un-air conditioned building overnight here in record-hot Texas. Needless to say, some of the unisons have drifted on me in the first couple of nights. I want the recording to be a good example of the 12th tuning and not have people think "man those unisons are out".

Anyway, good luck with getting your recording together. I will try to piece something together on my end. I hope the techs around here would not be too harsh on my tuning 'skillz' (I've been at it for 10 years and still learning), but instead just be able to hear a 12th-tuned piano up close and personal as if they were sitting down after tuning it themselves and performing the interval tests. Also it would be nice to hear the overall piano sound with a couple of pieces played on a pure 12th tuning, and compare it to other methods of expanding the temperament.

-Erich

Erich,

Did you choose the octave size in the temperament in any specific way to match the pure 12ths tuning? I ask this because I'm wondering about the progression of (for example) M3s from within the temperament region into the pure 12ths regions.

E.g. D#3-G3 is in the extended temperament section, but D3-F#3 is in the pure 12ths region. Are they progressive?

Hi Mark,

Thanks for the good questions. I must confess I had not thought of the octave size vs. pure 12th's making progressive chromatic 3rds right outside the temperament area issue before. Having said this however, I did not notice any discrepancies arising from the first available 12th to be tuned (D3 from A4) vs. the progression of the same D3 to its upper Major 3rd (D3-F#3) in relation to the chromatic third right above it (D#3-G3) originating from the traditional temperament sequence. In other words, they still turned out right for me...so far.

Like I told Jeff, I have only tried this new sequence on a total of 4 pianos and now that you mention this question, maybe I got lucky with those particular models. Or, it could be that this will always be the case and turn out right if I put a good Major 3rd scaffold + honest ET temperament that sounds good for each particular model that I encounter. I use a hybrid method of Stebbins "let the piano tell you" and Bill's ET via Marpurg along with his suggestions on how to tune the first A's (A3 to A4).

In my experience tuning all sizes of pianos, I have noticed that some finer, larger pianos can handle the zone where Bill states the optimum octave size is right between a 4:2 and a 6:3. I'm sure you'll recall that this means that if the F3-A4 Major 10th beats slightly faster than the F3-A3 Major 3rd AND the C4-A4 Major 6th beats slightly slower than the A3-C4 minor 3rd, then we are in the optimum zone.

However when I try something like that on my little teaching spinet piano, the A's beat or 'roll' (as I like to say) noticeably - so much so that I do not want to accept it. So I default to the 4:2 and make the F-A 3rd and F-A 10th match. In fact, on the 2-piano concert I mentioned above, the Kawai grand was able to handle the larger optimum octave, but the Kawai upright could not. I have yet to encounter a piano that cannot handle the narrower 4:2, although I'm sure one of the experts around these forums have come across such a beast. And once I encountered a beautiful Steinway that handled the 6:3 with ease - it was toward the upper end of that optimum zone (where the A-C and the C-A could match without harm).

Once I see what kind of octave the piano can handle, I use the Stebbins method for getting the ladder of 5 contiguous Major 3rds. That could have something to do with the outcome (progressiveness)also. My best answer would be if my following Marpurg sequence is good, then the slight deviation from a normal expanding (D3-F#3 and D#3-G3) vs. the resulting D3 of pure 12th expansion (which might be slightly wider due to the pure nature with the A4 above it) would be fractional, or fractions of a cent. I don't believe it would be enough to derail the progressiveness of the chromatic Major 3rds.

Maybe Jeff can come to the rescue on this one, because I only have scratched the surface with the theoretical numbers and math behind the pure 12th method. I know that Jeff has used it more and probably pondered on it much longer than I have been able to.

Thanks again for the question!

-Erich

Yes I have studied this. The math and my experience agree with each other.

I had been tuning a 4:2 octave in the temperament and also expanding the temperament upward with 4:2 octaves until the 12ths became pure. Then I would continue expanding the temperament upward with pure 12ths. What I noticed was that the smaller the piano, the sooner 4:2 octaves created pure 12ths. Turning it around the other way, tuning pure twelfths in the temperament will produce a wider octave type in larger pianos (where there is less of a difference between octave types) and a narrower octave type in small pianos (where there is more of a difference between octave types). And this is exactly what you want when choosing an octave type for the size of the piano! So now I just strive for pure 12ths. It’s a slam-dunk.

For a picture perfect 12th tuning you would need picture perfect octaves, and 4ths and 5ths. But if they are not picture perfect to start with, any noticeable errors can be dealt with as the temperament is expanded with 12ths. Reading into Erich’s experience it seems that the octave type(s) that he chose were plenty close enough because the M3s were progressive. But I would not expect them to be progressive when transiting a break.

So 4:2 temperament octaves for spinets and consoles, slightly wider for studios and smaller grands, and between 4:2 and 6:3 for larger grands should work well. And the octave types will be larger lower in the piano and smaller higher in the piano with pure 12ths. Again this is exactly what you want! I think every well tuned piano has an area where the 12ths are pure anyway. And they will seem pretty pure in the rest of the piano. There may not be that much difference in how the tuning turns out. The biggest difference may just be in how the tuner goes about it.

Well said, Jeff. I knew the math was there, I just couldn't wrap my brain around it. Yes, with my choosing the octave type that suited the particular piano size at the moment, I still ended up with progressive thirds once I reached that first D3 to be tuned as a pure 12th from the A4 above it.

With the grand, it was less noticeable, but the D-F# was a hair (we're talking fractions of a cent) wider on the smaller piano. In other words, if I was to tune by my old method which is fitting a D to the F# so that the third is to my ear progressive, vs. what I end up with when tuning a pure 12th, is very slight on the small piano. But like you said, it must be negligible because the thirds still came out ok right out of the temperament gate. Does this jive with your experiences or maths? I can't remember now where the particular break was for that smaller piano, but it might have been around that area (maybe D#3). Also it went from steel wire tri-chord on F3 to wound bi-chord on E3 as I went down. That probably also plays into it, but I can't remember all the theory behind it.

Regardless, it is a small enough difference (that I have experienced thus far) to not be noticed when playing chromatic intervals below and above the temperament. And like you said Jeff, you have to make a good honest pass at getting the ET temperament octave (F3-F4) right in the first place for the whole piano to work out later, expanding with 12ths.

Once I get into my hammer 'rhythm' with the 12ths, it really is easy. I do the usual checks (3rds, 6ths, then 10ths later, then 17ths later when I can't hear anything else), but I find that they all turn out even more precisely descending and ascending than if I was to try to tune those very intervals to be progressive by 'feel'.

When expanding with pure 12ths, it's hard to miss! And the result is a very clean, pure sounding tuning. I hope we answered your question satisfactorily.

-Erich

I am sure the D3 was wound on the small piano. The F#3 may or may not have been but was surely above the break. Anyway, as far as I am concerned, when wound strings are involved (especially on a small piano) all bets are off in regards to RBI progressions.

Here is something to consider. You know what pitch A4 will be. So you know where D3 will be. Why not tune D3 to A4 before the temperament and use D3 along with A4 while setting the temperament? This will get you closer to pure 12ths sooner.

Yes I was thinking along these same lines while typing the stuff about D3 being progressive. I should post my revised hybrid temperament tuning procedure involving the early setting of D3 to A4 as a pure 12th. Jeff, you may not use the 5 contiguous Major 3rd ladder or scaffold like I do, but one of the lowest notes that is tuned in getting that initial ladder is C#3, right below the D3.

A nice double check to make sure I'm not too far off is when I reach that C#3 note again when going down while tuning pure 12ths right below the temperament octave, is playing C#3 against G#4 and making sure it sounds pure. If it beats slowly due to being narrow, I will slightly sharpen G#4 so that this 12th sounds pure. The reason I do not want to flatten the C#3 right away is because in the Stebbins ladder sequence, the C#3 was supposed to be tuned as a 6:3 bass-type octave to C#4 in the first place. So, I'd rather not widen that octave anymore than necessary. Usually though, this is not even an issue because the pure 12th fits right into the mix, and I can continue on my merry way.

I'll try to post a hybrid sequence involving all this and see what you think, although you can probably already guess what I'm doing before I type it. If it benefits a newcomer to this method, then it is worth posting.

Erich:

Yes, please do post your sequence.

Just today I was thinking of the different ways that a 12th can be divided up by tuning intervals, sort of the way an octave can be divided up including with a ladder of CM3s.

With a 12th there are some interesting mixes like two 4ths and a M6. And three 4ths and a M3 may interest you if you are starting with a ladder of CM3s. So I am not sure that what I am doing now is really the best. I may try getting C4 in toward the beginning of the sequence instead of leaving it until the end. I just hate tuning the circle CCW, though!

Hi Jeff,

How detailed should I make the sequence? I could spell everything out step by step; or refer people to the articles that I use at each step. In other words, Stebbin's article on getting A4 from a pitch source, and also his method of getting the first 5 contiguous Major 3rd ladder. Then I could post a link to Bill's article describing how to use ET via Marpurg (which uses 4ths and 5ths only - sorta like the intervals you like). Then we have to tune D4 as a pure 12th to A4, then it's time to fill in the gaps.

There are only a couple of notes left to be done the old fashioned way and those are E3 and D#3 (could be tuned 6:3 octaves with 5th and 4th checks if we want). D3 and C#3 are done already, then we need a couple of more notes right below that (namely C3 and B3) that we could do with 6:3 octave (and 5th/4th checks as we did earlier). The reason we need these extra few bass notes becomes apparent, when we can then tune the remaining upper temperament gap right above F4, namely F#4,G4, and G#4. These we can get directly as 12ths from the C#3 (already tuned in the ladder), C3 and B3 that we just tuned.

We would at this point have the initial 3rd ladder, nice ET temperament via Marpurg, and the remaining notes just above and just below this area filled in. It would now be a matter of expanding through nothing but 12ths as we now have the reference pitch (all notes from B3 to A4) for every bass and treble 12th we want to tune.

Does this sound like a good plan? There are more options, like getting the F#4,G4,and G#4 first, then tuning down a 12th to get the bass notes right below D3. Maybe this would be more accurate?

Once we settle on a good way to fill in the missing gaps, I will give this hybrid method a test drive on my home piano.


-Erich

P.S.
I hope I'm using the right name, B3, to designate the B right below C3. Sorry if I got the octave number wrong.

I've followed this thread with interest and today I decided to tune my piano with pure 12ths all along the complete keyboard.

I used the following sequence

-A4 to fork
-D3 to A4 as a pure 12th, as beatless as possible
-A3 and D4 to both D3 and A4. That gives two octaves D3-D4 and A3-A4, two fifths D3-A3 and D4-A4 and one fourth A3-D4. By tuning octaves tempered between 6:3 and 4:2 one gets reasonable good sounding fifths and fourth. And that ensures I tune the right amount of stretch to have pure 12ths.

Now I divide both octaves D3-D4 and A3-A4 into contiguous major thirds.

I tuned:

F3 to A3 (M3) and D4 (M6)
F4 to F3 (P8)
C#4 to A3 (M3)and F4 (M3)

Now I refine the tuning of F3, F4 and C#4 to have a good progression of M3s.

Now I divide the octave D3-D4 into M3s. I tuned:

F#3 to D3 (M3) and C#4 (P5). (I had troubles here, see below).
A#3 to F#3 (M3), F3 (P4) and F4 (P5)
F#4 to F#3 (P8), A3 (M6), C#4 (P4), D4 (M3)

From here we have a lot of options to tune the notes left between D3 and A4. In the mood of the Sanderson/Baldassin sequence I tuned:

B3 to F#3 (P4) and F#4 (P5)
G#3 to C#4 (P4) and F4 (M6)
C4 to G#3 (M3), F3 (P5) and F4 (P4)
G3 to C4 (P4), B3 (M3) and D4 (P5)

Now I can fill the remaining notes up to A4: D#4, E4, G4, G#4 checking the M3s, P4s, P5s, M6s and P8s as available.

And the notes down to D3: E3 and D#3 in the same way, checking the progression of M3s (see below), P4s, P5s, M6s (see below) and P8s.

Now it is time to do our batery of tests in order to detect and correct errors before expanding the temperament by pure 12ths.

From here I tune pure 12ths up to C8 and down to A0. Making the usual tests and checks as usual but keeping the 12ths as beatless as possible, keeping an eye (ear) to have good sounding octaves, fifths, etc.

In the high treble it is difficult to hear the 12ths so I used pure 19ths instead.

The result was a very nice tenor, treble and high treble. A really in tune piano, very clean sound. But I had troubles with the break and bass.

I've got a jump in the beat rates of the M3s across the break. The major thirds with notes on both bridges were noticeably slower than their neighbors with their 2 notes on the same bridge.

And more disturbing than that, the other kinds of intervals, mainly octaves, were disrupted also, except for the 12ths of course.

I found it difficult to conciliate good sounding octaves with beatless 12ths, though I tuned pure 12ths in all cases, because it was the goal of this tuning. But the bass was not so clean.

In brief: the bass didn't match the tenor/treble.

I guess this is due to the fact that in the treble we only hear one unique kind of 12th: 3:1. While in the bass we can hear more than one kind of 12th (3:1, 6:2, 9:3, 12:4), so I think the pure 12th is not convenient to get a good sounding bass. I think we must compromise the 12ths in order to have better sounding 8ves, double octaves and in general all other kind of intervals.

And of course, by tuning pure 12ths across the break we upset the eveness in the progression of RBIs which use the 5th partial. And this 5th partial is completly left out of the game when tuning pure 12ths.

I think of piano tuning as the art of fudging conflictive intervals into a good sounding ensemble. In my opinion we can not favor one kind of interval at the expense of the other intervals. They are all important.


P. S. This is the first time I've got such a problem tuning the break. I understand now what Jeff is always talking about challenging pianos with jumps in the break. IMO the problem is not the pianos but the way of tuning them!

When I tune pianos as I usually do (tuning even progressions of RBIs and checking "good sounding" SBIs) I automatically and unconsciously avoid that kind of troubles. This problem is surely related with tuning even SBIs across the break, ignoring RBIs.

The result is: conflictive bass/treble passage.

Erich:

One semitone below C3 is B2.

Heck I dunno how much detail to put in. To me, a set of CM3s is a set of CM3s regardless of how they are tuned. You could always just give the basics and ask for questions.

Rafael:

Thanks for posting your experience. Hope you enjoyed giving it a try.

You mentioned that some of the octaves were "disrupted". This happens sometimes when I tune, also. What it means is there is also a 5th that is "disrupted". The SBI check for a P12 is the stacked 3:2 fifth and the 2:1 octave. If the octave is bad, the 5th is also. There was an error in tuning or something slipped. They both happen. Pure 12ths require a very accurate ET. I think that is one of it's strong points.

Jeff,
Thanks for the B2 note designation correction. I always forget the A0, A#0, and B0 at the bottom that start the counting system off funny. I'll learn one of these days!

Rafael,
Thank you for giving this 12th tuning a test drive. I agree that the final tuning sounds really good in the tenor and treble area without the usual time and effort of other expanding methods. I also have problems reconciling the 12ths in the low bass vs. other beating partials that are trying to grab my attention at the same time. You almost have to block everything else out which is harder (or perhaps impossible) on smaller pianos. Like you, I also use the 19ths when it gets too high in the upper treble to hear the 12ths. I had no problems so far with my experiences, but as I stated before, maybe I got lucky on the first 4 pianos I did this with.

Also, maybe the fact that I use SBIs for Bill's ET via Marpurg helps me get more in the zone so that the bass does not suffer when expanding later with 12ths. Have you given this ET temperament method a try before? I find that it gets my temperament super-accurate without any RBI tunings(like Bill said, a perfect 100 score on the exam if you do it faithfully).

Don't get me wrong, I'm not saying you should switch or that your sequence was less accurate, but for the purposes of finding out if this 12th method really can work across the board, it would seem that the initial temperament would have to be tuned with 5ths and 4ths (inverted 5ths) also. Maybe Jeff can "expand" on this temperament discussion.

-Erich

Erich:

I think the most appropriate way to set the “proper” ET for pure twelfths is with 5ths and octaves. This is because they are the tests for pure twelfth. Likewise if the goal is 4:2 octaves, the most appropriate way to set the temperament is with 4ths and 5ths. Again, this is because there is a P4/P5 test for 4:2 octaves. There was a Topic on this sort of thing not long ago. Doel posted some interesting graphs.

Now this is not to say that other temperament sequences cannot produce satisfactory results. But if you are using a sequence that includes setting a ladder of CM3s across a break, and then simply tune pure 12ths to expand the temperament, there can be problems. Sure, all the 12ths will be pure, but if some 5ths are busy, so will the contiguous octave above it!

The thing to remember is that the 3:2 5th and the 2:1 octave contiguously above it are both tempered the same amount. This is true regardless of the sequence used.

Jeff,

Yes I agree that if one relies on the CM3 ladder too heavily, then one can run into problems later while expanding, like Rafael apparently did. However, the only reason I use the Stebbins ladder at all is to get F3-A3 (theoretical 7 bps) and A3-C#4 (9 bps) as close as I can aurally for that particular piano. Then I abandon the 3rds and use the Marpurg which uses only 4th and 5ths. When I reach the C#3 right below D3, I can adjust if necessary. The 12ths take precedence and I will bump up G#4 or lower C#3 if it comes to that.

I need to try this hybrid method (CM3 ladder just to get F's and A's, Marpurg for ET, then 12ths) on my own piano this coming week. I will report my findings. If I run into problems because of the ladder, then I will abandon that possibly and just stick with the Marpurg. The Marpurg, for me anyway, allows the least 'rewinding' or backing up if there is an error discovered. I still think this could be a good diving board into the 12th tuning.

I'll let you know how it goes on my piano.

-Erich

Come on guys, I know how to tune a good ET using CM3s!

To correct the tuning all I had to do was lowering a little D3, in order to have a faster D3-F#3 M3 that fited in the progression and voila, the rest fell into place. But that was no longer a pure D3-A4 12th, nor pure 12ths across the break, nor pure 12ths in the bass!

The goal of this exercise was not a good ET but a "pure 12ths tuning".

The problem here is that in fact, as Jeff says, a pure 12th is a 2:1 8ve on top of a 3:2 5th, or a 4:2 8ve on top of a 6:4 5th, in both cases the octave is adecuate for the tenor and treble but is too narrow for the bass which needs wider than 4:2 octaves.

If we want 6:3 octaves in the bass that would need 9:3 12ths, which in turn need 9:6 5ths which as far as I know aren't audible. And I am afraid that 9:3 12ths do not sound pure. In other words, when we tune "pure 12ths" we are tuning 12ths narrower than 9:3, resulting in octaves narrower than 6:3.

But I did not waste my time! In fact I loved the sound of the piano in the tenor and treble. A very clean sound! I prefer it over what I usually tune (mindless octaves).

So, I guess I can tune a scheme with pure 12ths (and 19ths) in the treble with wider than pure 12ths in the bass.

Good thread .

Rafael:

I find that pure 12ths make wider and wider octaves the less iH there is, such as in the bass. Consider no iH at all. Pure 12ths would cause the octaves to be wider than a 2048:1024 octave, right?

But you may have had a problem with the particular piano you tried it on. You mentioned the M3s that straddled the break beat slower than those above and below. This would be indicative of a higher iH just below the break rather than the typical lower iH. So yes, it could cause a narrower octave type because the iH curve is inverted and if the iH is also high. I have been waiting to try pure 12ths on a Kimball console to see just what may happen. They have the same anomaly of higher iH below the wound/unwound break. I have one scheduled early next month. I may prefer narrower octaves if that is what ends up happening. I don’t know.

Rafael,

I of course did not imply that your temperament was sub-standard by any means. I was just referring to how you go about setting the initial octaves and 3rds (namely the F-A's and A-C# that I referred to above). As I understand it, a temperament can sound perfect using 3rds or it can sound perfect using circle of 5ths like Jeff does. But there will inevitably be differences between those temperaments (no matter how small) would there not? Maybe these very small differences play themselves out on a larger scale when blindly or mindlessly tuning a 12th expansion later? I don't know, that's my take on things anyway.

It seems to me that the 5ths need to be more accurate in the beginning for this all to work out. And Jeff really pays attention to the accuracy of his 5ths. Me, well I don't really hear them that well, and I was a RBI guy for 10 years. As long as the 5th didn't beat noticeably, I moved on to the next 3rd or 6th. Jeff would probably cringe at such a glossing over of 5ths! So, lately I rely on the Marpurg which requires you to make them perfect at the onset and later temper them. I find that my 4ths and 5ths are much better, and the 3rds, although progressive, take a back seat. There HAS to be deviations between my temperament and yours - just by the nature of how we arrive at the intervals. Whether or not they are big enough differences, maybe Jeff could say.

Also, I wanted to clarify something that you mentioned Rafael. You said that the 12ths are too narrow for 6:3 octaves. Does this mean that when I tune 6:3 bass octaves (as I always have done until now), my 12ths will be WIDE of pure? It's hard for me to imagine this being the case, but you and Jeff know much more about the partials and theory behind tuning than I do, so I defer to you. Please tell me if I am thinking correctly here.

Thanks. I will let you know how it comes out on my home piano this week, using CM3rd ladder, Marpurg, then 12ths only. One of these days I'll be able to record with a Zoom handheld recorder and post an mp3 on the forum for all to hear. I can do temperament tests, 12 tests, 10th and 17th tests up and down, and then play a couple of songs for everyone to judge. I believe Jeff wants to do something similar, but with an actual performance of 2 pianos and not the aural tests part.

-Erich

Jeff,

I have not made an analysis of what happens with pure 12ths and M3s at the break, you may be right. It's surprising that iH increases when going to the wound strings, as the goal of the windings is to increase the flexibility of the strings. So the top most wound string should have less iH than the lower plain string.

But in my tuning exercise what I've got was the M3s a cheval with the break slower than the rest.

The break in this piano is at D#3/E3. Wound strings in the bass, plain strings in the tenor.

The piano is an August Förster studio, 116 cm tall (45.7 inches).

M3s C3-E3, C#3-F3, D3-F#3 and D#3-G3 were slower than they should.

Rafael:

I checked some idealized tuning simulations of 4:2 octaves on a console and on a concert grand. Both happen to also have virtually pure 12ths in the bass. On the concert grand, the 6:3 octaves were also virtually pure. But on the console the 6:3 octaves were about ½ bps narrow. Not much, but it seems that on smaller pianos pure 12ths can produce narrow 6:3 octaves in the bass. I still haven’t decided if this is good or bad.

Your ETD should tell you what the iH is doing across the break on your piano.

I have been trying to understand just what you are proposing but am unable to.

”Then I abandon the 3rds and use the Marpurg which uses only 4th and 5ths.”

”If I run into problems because of the ladder, then I will abandon that possibly and just stick with the Marpurg.”

I take it that when you say “Marpurg” you mean Bill’s excellent ET via Marpurg. But this is based on a ladder of CM3s. The 3rds cannot be “abandoned.” If you are truly going to use only 4ths and 5ths I would expect this to be Dr. White’s sequence or a variation of it. I guess I don’t understand how you “reach the C#3 right below D3” in a sequence that tunes C# before F# or G#.

Hi Jeff,

I suppose that the ET via Marpurg has its roots in the CM3 ladder and like you said, if I'm not using Dr. White's sequence, I am using a CM3-based sequence. I guess I was sidetracked by the fact that Bill's sequence uses only 4ths and 5ths as opposed to the usual setting of F3-A3 (7bps), F3-D4 (8bps), A#3-D4 (9bps) that I used to always do. Of course these are all RBI's, but I forgot that the very 4ths and 5ths I am getting from the Marpurg ET are a direct result of their placement inside the initial ladder.

As I asked about above, could differences arise between say a 5th temperament like you do and a CM3rd temperament like we are doing, even though they all pass the usual tests? And could these differences affect the outcome, or are they possibly fractions of a cent and don't make a difference anyway?

Also, I had asked Rafael, but maybe he didn't catch that part of my post: On a normal 6:3 type bass tuning on a smaller piano, it appears to me as if blindly tuning octaves chromatically going down this way, would result in WIDE of pure 12ths. Am I thinking correctly here? This is because Rafael stated that the pure 12ths were not stretched enough to make a perfect 6:3 octaves down there. So, if I stretch them to the 'classical' 6:3 technique, will the 12ths become wide? Just curious.

Thanks,
Erich

Erich:

Thanks for the clarification, I think… Bill and others have suggested tuning a ladder of CM3s and then using it as a guideline for Dr. White’s sequence. Perhaps this is what you meant. In any case it is another option.

When setting a ladder of CM3s, the tolerance that is needed to have them be progressive is a number of cents wide. My #1 difficulty with them is how to tell if they are accurate enough for producing both M3s and M6s that are chromatically progressive when the rest of the notes are tuned. And when later in the sequence if I find what seems to be an error, I have no way to decide if it is the note I am tuning, the Fs or the C#s.

My #2 difficulty is what to do when crossing an ugly break and the only way to have the CM3s evenly progressive is to have some very wide and some very narrow octaves.

Interestingly Bill’s ET via Marpurg solves the first difficulty when a well scaled piano is involved, and may (I haven’t really tried it to make sure) solve the second difficulty on challenging pianos.

After the initial ladder of CM3s are tuned and then the six pure SBIs are tuned, there are 3 sets of chromatic M6s or m3s available to check all the intervals that have been tuned. If the SBIs are indeed pure, then any non-progressiveness in the M6s or m3s will show that improvements, or perhaps compromises, can be made to the original M3s or octaves.

So to answer your question, I think both the accuracy of CM3s and their appropriateness for challenging scales can be a problem.

As far as the 6:3 octaves causing wide 12ths in the bass on small pianos, it seems to be true theoretically. In the real world if I tune octaves in the bass, of small pianos, the 6:3 partial match is so strong I cannot help favoring it. But also, due to the partials often being pretty wacky on small pianos, the octaves don’t sound all that great anyway, no matter how they are tuned.

I am thinking it might be best to tune 12ths in the bass of small pianos, but temper them a little bit wide. I know that others hear the 6:2 12th stronger in the bass, which will produce wider octaves (probably wider than 6:3) so this is an option. I tend to just hear the 3:1 12ths, though. So as I get more experience I may widen them. We’ll have to see.

Tuned a Kawaii UST-5. The highest wound bichord was D3. The D3-F#3 M3 ended up beating the same as the F#3-A#3 M3. It was a bear to get right, but that's where it had to be for pure 12ths.

I am considering going back to a C-fork and forming the temperament between F3 and C5 in order to avoid most breaks.

Any thoughts?

Hi Jeff,

Sorry to hear about your tough break (pun intended). No seriously, your idea about F3 to C5 might be the answer. Dave B had a similar idea in his reply to the thread I started. Maybe this is something that we need to look into in order to address the breaks on smaller pianos.

Just as scientific theories need refinement (gravity and light come to mind), this Pure 12th expansion theory just needs a little tweaking. I have faith in it because once you get out of the problematic temperament gate, it is really easy to tune 12ths (as you said, a slam dunk). It requires the least checks that I know of, although mindless octaves are a close choice. So, there are alternatives, but it would be nice if this method worked out for most folks, due to the fact that it would be easy to pick up for experienced and novice tuners alike.

-Erich

Need some guidance. I'm a little lost\uncertain about two aspects of this perfect M12 or equal beating M12 tuning.

1. Assuming a temperament set from A440, if I first set the tenor and bass to perfect 12ths derived from the bearing range, when I then move to tune the notes immediately above A440, should I try to tune perfect 12th's derived from the tenor?

Doing so creates a stretch for the first octave above middle C. Seems natural and it sounds good, but it worries me. I like David P's perception that the human ear is sensitive to even a slight stretch in that area. And the fundamental is strong in this octave above the middle C octave. To account for iH, yes, a stretch may seem desirable, but shouldn't the amplitude of the partials be considered? (This may be a theoretical question. If it sounds good, do it, is probably the answer, and this stretch does sound good, but ignoring the loudness of the loudest partial bothers me.Is the situation simple: although the 1st partial has the highest amplitude, the 2nd partial is also so audible in that range that it has to be taken into account?)

2. To try to create an equal beating octave and 12th, which partials should be used? Complex, to me, since there are different sets of possible partials on each pair of notes: Equal beating on 3:1 and 4:2 or 6:2 and 2:1 or...

What I've been trying to do is create an equal beating between the partials of both notes in the M12 first. But that still leaves open the question of which partials to listen to for all three notes. Seems as though there are several possibilities, each of which could create a valid and good but different tuning?

(Somehow, my user name got lost or changed. My fault. I've posted here in the past as Jake Jackson.)

JJ (Seems like a good choice to refer to both your old and new names ):

Once upon a time I used to tune octaves to sound pure. I imagine this did what you propose: set the octave according to the level of the partials. What would happen is the treble would sound flat.

It may seem that pure 12ths stretch the tuning by 2 cents over 19 semi-tones or about 1.2 cents per octave. This would only be true if there was no iH. Because there is iH, which naturally stretches a tuning, expanding the temperament with pure 12ths will stretch the tuning according to the iH but very differently than how specific octaves would.

On a small piano, the temperament will have 4:2 or even smaller octaves, but on a large piano they will be closer to 6:3 octaves. This is what I desire. I am not much worried about the strength of partials.

Erich:

After “monday morning quarterbacking” the UST-5 tuning, I think a mistake I made was trying to make the octaves in the low part of the unwound strings too wide. I know better, but did it anyway. I use an up a fifth up a fifth down an octave sequence. The more iH, the narrower the octaves. I did get it correct the second time around. It is just not second nature to me yet, like 4ths and 5ths with 4:2 octaves was. But then I didn’t tune as large a jump in the M3 beatrates then, which I now believe to be more appropriate.

I have a Whitney petite grand scheduled for tomorrow that has the break way, way too low. I won’t be dealing with a break in the D3-A4 temperament, but the iH will be dripping off the strings. I think nearly pure fifths and pure sounding octaves in the lower part of the temperament will do the trick.

Update:

I am not sure what to think about Whitney petite grands. I run across them now and then. The octaves needed to be just a hair wide in the temperament after all - only noticeable when actually tuning. And the treble worked out well with 12ths, except I am thinking that the top octave needs a little more than pure twelfths. On some pianos, like this one for some reason, I can hear 12ths up to about G7.

But extending the tuning across the break was a shocker! This is a 4’6” piano with a C3/C#3 break and all unwound trichords above the break. The D3-A4 temperament worked ok. But when C3 was tuned as a perfect 12th to G4, the C3-C4 octave was horribly wide, much wider than 6:3. So for the next 5 notes or so going down I made a compromise between the octave and the 12th – wide octave, narrow 12th. Surprisingly the M3s were pretty darn progressive across the break. So I guess those octaves ended up close to 8:4. Not much could be done with the lower bass monochords…

So, the adage of there being a time and place for everything holds true for twelfths. Yeah, I now think they are not quite enough in the top octave and should be tempered wide if used. And when there are outrageous breaks involved you just have to smooth things out a bit.

Ya know what? Maybe I am looking at this the wrong way. Maybe I should have taken the bull by the horns and have set a F2-C4 temperament. Then the 5ths octaves and 12ths would have all been at thier best. Hmmmm

Hi Jeff,

Yes, the 12th expansion method might require some tweaks like you did (namely the compromise between the 12th and octave). I think due to the plethora of scaling designs, breaks, etc., that any expansion sequence needs some 'breathing room' to be effective. If one just blindly says "I'm going to use 12ths exclusively" with an iron-fist attitude, that person will eventually run into problems at the breaks on certain pianos.

If however the tuner allows some compromises initially out of the temperament area, eventually the 12ths will be able to be tuned as they were intended in this method.

I'm glad that you also mentioned that the last octave of the high-treble might need additional sharpening (and thus wider-than-pure 12ths), because I was thinking the same thing. It ultimately depends on the customer though, because some pianists prefer really clean octaves up there, while others tell you to stretch/brighten the heck out of it!

I find that using just pure 12ths all the way up to the top gives a bright enough sound for my taste - sort of like a vintage sound that I hear on old jazz/classical recordings (Bill Evans, Rubenstein, etc.). If I know ahead of time that someone prefers a super-bright high treble, I will leave the 12ths behind at that point and deliberately tune wide, beating octaves way up there (maybe 1 bps). I choose the octave because I can reach it with the left hand while tuning with the right. I find that when I finish and get ready to pack up, the piano has settled; those octaves seem to settle and blend in better (less noticeable beating), and the treble is still at its brightest. Do you use any similar techniques/helpful short-cuts?

Erich:

But I thought you do not use 12ths for the extreme ends of the piano. Pure 19ths naturally produce more stretch in the treble because of iH.

As far as expanding across the break, if you tuned a temperament above a break and expanded downwards and then tuned the same sort of temperament below a break and expanded upwards, I think you would have two different tunings. Different intervals would be out of whack.

For instance, if you tuned 6:3 octaves (which have the m3-M6 test) when tuning the temperament above the break the m3s will be progressive across and above the break. But when tuning the temperament below the break, the M6s will be progressive across and below the break. There was a Topic on this sort of thing a little while back.

So I am thinking that it may be best, though perhaps difficult, to deliberately set the temperament spanning the break. It would be necessary to make some compromises, but the compromises should be made only to smooth out the test intervals for the interval that the temperament spans. For a 6:3 octave, this would be the m3s and M6s. For a pure 12th, this would be the 5ths and 8ves. Then the 12ths would be able to remain pure when expanding with the compromises that already have been made within the temperament.

Anyway, this is the line I am thinking along right now. I'll have to see how it works out.

Oh yes, I meant to say 19ths. But still, is that enough stretch for the brightest possible high treble? Or would wide 19ths be possibly called for? I don't know - I was just wondering if you have any tricks to doing that last octave.

Let me know how your different temperament areas work out when you get a chance to try them. I am curious to find out whether that will do the trick.

Thanks,
-Erich

Erich:

“Brightest possible stretch” would be just before you hear KERBANG! Reminds me of a riddle an Amishman once gave me. “How do you know when it is time to plant corn?” My best answer was to plant corn in Columbia County a week before they plant it in Sullivan County (about 90 minutes north.) The real answer is, now you have to understand that Amish humor is a little different, “You know it is time to plant corn when you get up in the morning and your wife’s bare butt is sticking out from under the covers and you put your hand on it and it isn’t cold. Ahgagagaga!” Believe it or not, it took me three days to get the answer out of him. See, I wasn’t married at the time so he figured I didn’t need to know...

Anyhoo, I was curious about the difference in stretch so I crunched some simple scenarios. If the F5/F6 octave is tuned as a 50/50 compromise between a 2:1 and 4:2 octave the difference between C8 being tuned as a pure 12th and a pure 19th is about 38 cents on a Baldwin Spinet and 30 cents on a Baldwin L with the 19th being the greater stretch.

What I do isn’t what I would call a trick, just comparing (as much as I can…) what is heard objectively with what is heard subjectively. As I ascend into the upper octaves, I will play the major tonic in the temperament and then the note I have tuned. Subjectively, I will listen and decide how it sounds. If I think it is definitely flat, I will objectively temper the 12ths a bit wide.

But the whole decision process does get “fuzzy”. I think somewhere in the top octave I am hearing the 12ths more melodically than harmonically (even though the notes are played simultaneously), which is where, I think, the extra stretch is needed. Listening to the chords and tuned notes a bit lower down is important, I think, to get a feel of how the treble is really sounding so as to keep the same quality going up. By melodically: I mean subjectively, separately pitch-wise. By harmonically: I mean objectively, beating interval-wise.

Maybe in the top octave I hear what I believe it should sound like if I actually could hear the beats when playing the 12th. When I am very honest with myself, it needs more stretch melodically than what I come up with harmonically.

Thanks Jeff for the info.. and the joke! Yes, you kinda have to be married to fully appreciate that one.

I had no idea that the 12ths and especially 19ths created that much stretch. See, that's why I'm glad to have forum fellows like you to keep me educated on such matters! I'm thinking now that the 19th would make a sufficiently bright treble.

One more question - if you deliberately widen the 12th a little, will the 19th be wide of pure automatically, or not quite yet, due to its increased size? Also, I would assume it would not be necessary to make the 19th wide of pure (probably at that point, the single octaves might beat too much). I guess I am wanting to know the limit, so that I can approach it, but not pass it (just for peace of mind). In other words, I want the top C to be a sharp C, but I don't want it to be a C#.

-Erich

Erich:

Thanks for the words of appreciation. You are very welcome!

This Forum has been great for me, too. There are many things about tuning theory that have been posted here that were contrary to what I was hearing. When I learned how to crunch the numbers I found that my ears were correct. But the value in all this was in learning what stretch is really about and different ways of controlling it. The latest factor is the 19ths, which I have only very recently looked at. I have to wonder just how valuable they are for aural tuning, though. They are ok for a check I suppose.

Anyway, to answer your question, it depends on where in the scale as to the difference between 12ths and 19ths. It should be apparent that with no iH 12ths would produce a tiny bit more stretch than 19th. But with iH 19ths produce more stretch. And the more iH the more stretch, such as in the high treble. So for much of the piano the 12ths would be tempered a little wide to produce pure 19th. But higher in the treble the 12th would have to be tuned very wide. More than what would be considered tempering.

Where the limit is, is of course subjective. The 12th is a totally unique interval because of the combination of its span and the partials involved. No they do not have to be pure, and I am learning that in some cases they should not be. If I was to give a rule of thumb, and that is what is seems that you are looking for, if the 12ths are obviously beating the stretch is wrong. Of course in the very high treble where you can’t hear beating I suppose almost anything goes.

A few years back, when I was expanding with pure sounding octaves, I found that if I played an ascending G arpeggio everything sounded fine. But if I played a descending G arpeggio and then repeated the top G7, it sounded very flat. In fact G#7 would sound much better. That does not happen with 12ths, but I think it is best to widen the 12ths a bit in the top octave. 19th would do the trick up there, too, I think. It just isn’t easy aurally.

There is an odd thing about the relationship of 12ths and 4:1 double octaves. If pure 12ths are tuned to beat wide at the same speed as the 4:1 is narrow (mindless octaves), they switch places in the last half octave in the treble. It is like the 12ths “run out of steam” or something. This is just another indication that in the last octave 12th may not produce quite enough stretch.

I probably won’t post again until Monday. I have some concrete work to do. They never found Jimmy Hoffa, and they won’t find the latest Amishman that told me a lousy joke either. Bwhahahaha.

Thank you for the advice - I will try widening he 12ths a bit (but not obviously beating) in the last octave and see how it sounds. That's interesting about the 12ths and double octaves trading places at the top. Ah, the wonders of mathematics and music (and the wonders of how we can't make music fit perfectly with the mathematics).

Something is wrong. The 3:1 is narrow and the 4:1 wide (and equal beating).

Kees

Yes, I was thinking the same thing - I think that's what Jeff meant to say.

Yep, I got it bass ackwards, sorry.

But what do you mean by they (3:1 and 4:1) "switch places"? Don't they just get further apart?

Kees

No. In the 7th octave they both become beatless and then the 12ths are wide while the 15th are narrow. Not that I can hear it. The interesting thing is that because this relationship is based on the inferred P4, it means in the 5th octave the P4s become narrow. But again, not that I can hear it. There are some graphs and discussions about it in the CHAS Topic.

Something is wrong somewhere. For example if I compute a tuning curve (using tunelab) based on pure 4:1 everywhere, the 3:1 are narrow, the higher up you go the narrower they are.

Why do you think if you tune pure 4:1 at some point the 3:1 becomes less narrow and eventually pure?

Kees

Two reasons, and I could be wrong. But what do you get when tuning 12ths or mindless octaves. There may be a difference.

First, that is what I calculated with my simulation. Now there is a difference between the calculation in my simulation and Tunelab. I use the original square of the partial numbers as in Young's paper while Tunelab uses an empirical table.

Second, when you really listen to the 4ths, the beatrate less than doubles each octave. It seems much more likely to me that they would reach a greatest value and then start decreasing and eventually becoming narrow than they would find a greatest value and stay there. It is even less likely to me that the beatrate progression would be a straight line. NOTHING in tuning is naturally a straight line! And if the 4ths become narrow the 12ths and 15ths must "switch places."

All the beatrates I have investigated show a curve that crosses the just intonation line when they less than double in each octave. M3s and M10s, have this character. A am sure the M17 would too if the piano had more notes. Ever wonder how the M17 is useful up to the 7th octave? If the M3 family doubled each octave, or even were on a 4:5 CM3 curve (which is less than double each octave…), they would be unusable.

Perhaps the iH curve has more to do with it than we think. Here are all the A's for the ideal curve that I used. It was based on based a Walter Console Verituner file that was kindly given to me and is similar to iH curves from Pscale for that size piano:

A0 0.27
A1 0.19
A2 0.13
A3 0.20
A4 0.52
A5 1.37
A6 3.60
A7 9.43

Enjoy!

With those particular IH constants 4:1 is wide when 3:1 is pure and 3:1 is narrow when 4:1 is pure. In an imaginary tuning with pure 3:1 everywhere and 4:1 everywhere for the second. And of course with mindless octaves 3:1 is narrow and 4:1 wide.

One of us is doing something wrong, could be me. The tunelab constants are not far from Young's ideal values. I guess it would be hard to show me your simulation.

Kees