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I started working on tuning quite some time back from the American School. I've searched through threads here and seen that others have had the same experience about learning to tune only with 4ths and 5ths through the American School.

I'd like to know how to check thirds and several other things to be able to set a temperament by ear. I don't like feeling reliant on my tuner all the time. It would be nice to set a decent equal temperament without it.

Could anyone give me some tips?

Thanks in advance.

Check out the Randy Potter School, which by all accounts is better than the American school. Or get a copy of Arthur Reblitz's book about tuning, repair, and rebuilding. There is a large portion of it dedicated to ways to check the various intervals. If you join the PTG, you will receive the PTG Journal monthly. It is invaluable. There are often articles about tuning. There was a recent series that ran for several issues that gave detailed instructions on this very subject. You might look into back-ordering these issues.

Good luck, and enjoy the journey!

Dave Stahl

I don't want to be discouraging, but learning to tune a piano can demand an enormous amount of time and is very likely to deliver a lot of frustration if you are doing it on your own. I was very lucky to find a PTG technician who took me on as an apprentice.

Dave Stahl's advice on the tuning articles in the PTG Journal is apt. You will discover that tuning a piano well does not mean tuning just a good temperament octave, but that the middle three octaves of the piano, say from F3 to F6, will all be tuned so that all the included intervals fit as perfectly as in a temperament.

If you do not have a lot of spare time and frustration tolerance and want to be able to enjoy playing your piano while you are learning to tune, I would suggest the following: that you have a good tuner tune your piano and that, after the piano is tuned very well, you listen to it and apply all the interval checks which are mentioned in the various articles on tuning. Knowing what a good tuning sounds like and understanding why it is a good tuning will be a great deal of help to you.

Once this good tuning is well-understood, you can follow its devolution as your piano starts going out-of-tune. This would be a good opportunity to try your skills, for example in unison tuning, and will allow you to work at the critical skills of setting pins and settling strungs, without necessarily making your piano completely unplayable.

The unfortunate truth about learning to tune is that you can't do it by tuning only one piano. You have to learn to distinguish between tuning errors, scaling design problems and mechanical problems which affect the beats that you hear. You can only do this with expert advice and/or by tuning lots of different pianos and learning to make the compromises that tuning requires.

My American School instructions taught the relationships of 3rds and 6ths, and called them "proofs" to check the fourths and fifths. I'm sure your course work did too. An experienced temperment setter can start from any note, use any intervals as a base, and come up with a decent temperment (on a piano with good scaling). Tweaking a temperment for a concert level tuning requires this ability.

I pretty much always tune by ear, and have tuned over 10,000 pianos like this. I was taught to tune from C 523.3 and once I have that tuned to the fork, I tune the E above - and listen for a very fast beat. After a while you can tell when it's right - too slow and you can tell, too fast and you can tell. From there, I can go any way I want - normally to an A below, then the F below that etc etc. It's as good a place to start as any. But - it's all a compromise and each piano is different. The more you tune, the more you understand this!

Simon
www.pianotech.biz

I learned to tune with fourths and fifths, then check and clean up with thirds and sixths, making sure the thirds progressed evenly and retuning the fourths and fifths accordingly.

Many tuners tune temperatments using thirds and sixths to start with, but this is difficult to learn for a newbie, so I understand why they teach fourths and fifths.

It took me a couple of months just to learn to set an OK temperament and about 6 months to a decent complete tuning. I worked in a shop so had lots of instruments to practice on and probably tuned 50 before my first really acceptable complete tuning (other, more experienced tuners in the shop would critique me and clean up my attempts, showing me my errors). After 100 tunings on different instruments, I began to have some confidence.

It takes a lot of dedication and much tuning on many different instruments to develop this vastly underrated, very difficult skill and art.

I tune to a C-523.2... fork, then tune middle C and C below that. Then I tune E below middle C, which beats about 5 bps to the C below it, and G#, which beats about 6 bps with that E. G# beats about 8 bps with middle C, and since the beats double every octave, middle C to E beats about 10 bps. Once I've done that, I fill in the fifths and fourths between each major third. The test is that all intervals beat increasing geometrically as you go up the scale. There are lots of bad tuners whose thirds don't get faster as you go up the scale. Don't be one of them.

A major third, an octave and a third, and two octaves and a third beat at the same rate. This is a good way to make sure that your octaves are good.

Incidentally, I'm talking about beats on harmonics. They are clear enough to hear, but there is also a beat on the fundamentals. The beat on the fundamental is the difference tone. If you play a major third, the difference tone is the fourth below the fundamental. It is quite strong in some areas of the piano. You can't play a major third without getting a major triad (in the second position).

Check these intervals up and down the piano for a good job. It does take lots of practice to hear what you need to hear, but it takes lots of practice to get the string to stay where you left it. Practice a lot, and you will get both techniques down. Just don't try to take shortcuts. Do the best that you can, and try to do it better all the time, and don't worry about how much time it takes when you start doing it. It will get faster and easier if you do it right. If you aren't careful as you learn, you will never get it right, and it will never be as easy as learning to do it right in the first place.

I also first learned to tune through the American School and while the information there is not wrong, it is not complete enough and the approach offered can easily lead you very far astray. Here's the reason: Historically, keyboard instrument tuners used the very same pattern offered in the American manual and many people are still being taught that same method today.

Unfortunately, the pattern being taught would be far more useful for a class of temperaments, generally called "Historical Temperaments" but which fall into several categories: Meantone, Modified Meantone, Well-Tempered and Quasi Equal ("quasi" means "almost"). These temperaments were used in past centuries but many piano technicians of today (including myself and some others who write in this forum) use them because we believe that, properly and judiciously chosen, they can make the piano have a more musically appealing sound.

What typically happens when the American school method is tried is that an error or miscalculation of exactly how much tempering a 4th or 5th should have is consistently made. The errors I used to make when I first started tuning in the early 1970's were that I took the instruction literally and used my watch to try to make my 5ths beat "3 beats in 5 seconds" and my 4ths beat at one beat per second. While this is theoretically correct, it is actually too much tempering because the octave, even in the mid range must be stretched slightly to end up with the overall best sound. The 5ths must beat a little slower and the 4ths can beat a little faster than the theoretical amounts.

So, I proceeded painstakingly through the cycle of 5ths sequence until I got to the end of it and inevitably discovered that the last 5th and the F3-F4 octave were irreconcilable. So, I backed up through the cycle until things got worked out, more or less, somewhere in the middle of the cycle. While I was never satisfied with this, I knew it was somehow wrong, people seemed to like my tunings and said that I made the piano sound "warm".

Well, after all, the word "temper" comes from the Latin word which means, "to warm". What I didn't realize was that I was really creating a crude form of Well-Temperament and that for the kind of pianos and customers I was getting at that time, it made them sound just fine, indeed. Today, in fact, I use a highly refined and very mild form of Well-Temperament as my usual way of tuning all pianos for all occasions. Combined with the unique way in which I construct the octaves, it creates a sound which seems to really sing, is bright and clear, harmonious and melodious and just somehow seems more "in tune" over all than if I did it the way which is considered conventional at this time.

The most common mistake made these days however is just the opposite of what I used to do. Technicians don't like the sound of a tempered 5th, so they try to "cheat" on them and proceed through the cycle which is presented in the American school, C4-F3, C4-G3, G3-D4, D4-A3, A3-E4, E4-B3, etc. , and get the 4ths and 5ths either pure or nearly so. Just as I did, they find the last 5th and the F3-F4 octave irreconcilable, so they back up through the cycle and even it out.

The results of either error will create Major 3rds which are quite uneven. But the M3rds in Historical Temperaments (HT) are supposed to be uneven, aligning themselves with the cycle of 5ths rather than progressing smoothly and chromatically as they would in a true Equal Temperament (ET). This is what gives the various keys their "color" in the HT's. The problem is that so many people make the mistake of trying to get their 4ths & 5ths among the white keys too pure and thus it forces them to temper the 4ths & 5ths among the black keys more than would be appropriate for ET.

Unknowingly, unwittingly and unintentionally, this creates a crude and backwards version of an HT and has come to be called "Reverse Well". Just as the key "colors" of true HT's have a clearly heard effect on the sound of the piano and all music played upon it, so does Reverse Well but instead of being an appropriate enhancement, it ends up making everything sound out of focus, "off" and simply put, "out of tune". This is quite unfortunate but also very common. I have heard it on professionally tuned pianos for concerts and even on some recordings.

Often the amount of backwards error is very slight, so the damage is minimal but sometimes the reverse error is quite glaring. There is even one older gentleman in my city who tunes a few pianos here and there who habitually tunes what could be best described as a "Reverse 1/4 Meantone". The chords among the black keys are beautifully harmonious but the simple keys of C, G, D, A and F sound so sour as to be unusable. Frankly, I don't understand how anyone would accept this and many don't that's how I know about it but still some people must accept it because he is still working and has been the entire time I have lived here, over 25 years.

So, the problem with the American school method is that you must tune 5 notes, C4-F3-G3-D4-A3 before you ever have a M3rd to even listen to. Then, if it is too fast or too slow, how would you know where the miscalculations are? Were they in each 4th & 5th? Were they in some but not others? How fast should that F3-A3 3rd beat? Then, You tune another 5th, A3-E4 and you have another 3rd to listen to but it has absolutely no relation to the F3-A3 3rd. Then you tune a 4th from E4 to B3 and now you have a G3-B3 3rd but again, very little if anything to establish a pattern or any way of sorting out your errors. There are some 6ths which become available, yes, but they also don't help much at this point with establishing an even progression or sorting out where errors may be.

So, the answer is to abandon the use of a Cycle of 5ths sequence for ET tuning, either the American school sequence or any other which follows a 4ths & 5ths pattern along the cycle of 5ths. The best pattern to use is also the easiest to remember:

"UP A 3RD, UP A 3RD, DOWN A 5TH".

Starting with your A3 that you got from A4 or directly from the tuning fork, estimate the F3-A3 3rd. It is about 7 cycles per second but all you need to do at this point is guess. Then you estimate the A3-C#4 3rd, making it a little faster than your first F3-A3 3rd. Now tune the F3-F4 octave (make it just begin to beat on the wide side, don't tune it too pure). Now, listen to these 3 3rds (which are called "contiguous 3rds". You should hear that they get progressively faster while ascending and slower when descending.

For any two contiguous 3rds, the higher one should beat slightly faster than the lower one. If they are exactly the same, it is not quite right, if the upper one is slower than the lower, it is definitely not right. There should be only a slight increase in speed between any two contiguous 3rds. Actually the difference is a known mathematical constant, a ratio of 4:5. For every 4 beats (not beats per second, just beats) of the lower 3rd, there should be 5 in the upper. Therefore, if the upper 3rd beats very much faster, the relationship is also not correct. The rediscovery of this principal in the latter part of the 20th Century has been the cornerstone to vast improvements in both aural and electronic tuning. People today tune much better than they did just 30 years ago.

What you must first do is work this F3-A3-C#4-F4 series over until you've got it properly balanced. It will be difficult at first to master and it will be more difficult to determine that it is exactly right and far easier to know that it is wrong. When you've got this first series of contiguous 3rds tuned properly, the rest of the temperament octave will be relatively easy, you will now be tuning all 4ths & 5ths and checking with M3rds instead of tuning them. You'll be tuning from notes you are relatively sure of rather than possibly compounding your error by tuning from a note which hasn't been verified. Plus, you can be absolutely confident that you will not make the Reverse Well error. Furthermore, you should be able to pass the mandatory aural portion of the PTG Tuning Exam.

After getting F3-A3-C#4-F4, remember the "DOWN A 5TH" part of the pattern. Go down a 5th from C#4 and tune F#3 and temper the F#3 5th the way you think it should sound. Now, move "UP A 3RD" to A#3 but instead of trying to tune more 3rds, and especially from a note you just tuned but had no way of verifying, tune that note from one you are already very sure of, either F3 or F4. Make both that F3-Bb3 4th and the Bb3-F4 5th sound right to your ears.

Now you will have an F#3-A#3 major 3rd. If you play the F3-A3 M3rd, then the F#3-A#3 M3rd in sequence, the F#3-A#3 M3rd should sound very slightly faster than the F3-A3 M3rd (less difference than in the 4:5 ratio of contiguous 3rds). Continue, "UP A 3RD" to D4 but tune it as a 4th from the most reliable note you have, the note you started with, A3. Now you have another set of contiguous 3rds and another pair of chromatic 3rds.

Now go "DOWN A 5TH" to G3 and tune it as a 5th from D4. Go 'UP A 3RD" to B3 and tune it as a 4th from F#3. Continue 'UP A 3RD" to D#4 and tune it as a 4th from A#3. Now you have another set of contiguous 3rds and several chromatic 3rds plus many other intervals such as 6ths and minor 3rds which you can play to test for evenness. Always go back to the 4:5 ratio contiguous 3rds test to identify where your error is. It will inevitably point to a 4th or 5th which is either too pure or too tempered.

Now you have just 3 more notes to tune: Go "DOWN A 5TH" from D#4 to G#3 and tune it as a 4th from C#4. Go "UP A 3RD" to C4 and tune it as a 5th from F3 but also check it as a 4th from F4 and a 4th from G3. It should sound satisfactory to each test and the Ab3-C4 M3rd should progress chromatically from those above and below it. Now, there is just one more note to tune, go "UP A 3RD" to E4 and tune it as a 5th from A3 but remember to temper the 5th and check it as a 4th B3-E4 and to check the progression of both contiguous and chromatic 3rds.

Now if you play up and down pairs of 4ths, 5ths, 3rds and 6ths, you should be able to spot any unevenness. Don't just guess where the error is, use the principal tool of the ratio of 4:5 contiguous 3rds to point to the error.

While it is possible to tune a smooth ET from the American school pattern, I know of people who do and I know I can too, it takes a lot of skill and experience to be able to do it and many people just never do get to that level and don't know what they're doing wrong. These days, they turn to the Electronic Tuning Device for help and they often do tune well with one but still don't have a clear idea of what tuning is all about and can't pass the PTG Tuning Exam.

Really understanding what you're doing and being able to pass the PTG exam will make you a superior tuner and you will be able to take your art and craft to a yet higher level where you will know what you're doing when you attempt other kinds of temperaments and make your own decisions on octave stretching.

One year ago, I wrote a long article about all of this which I basically rephrased for you here today. If you or anyone would like a copy of that article (10 pages) it is still available by request and will be sent to you in the form of an attached WordPad document.

Good luck using this information to better your tuning skills.

Well said, Bill.

I've learned to tune almost exactly the way mr. Bremmer explaned.
Succesion was:
A3-F3
F3-D4
D4-A#3-F#3
F#3-D#4
D#4-B3-G3
G3-E4
E4-C4-G#3
G#3-F4

In fact,my master forbided me to even touch fifths and octaves.He said (and did it too)man don't nead them,if it get beatment progression right.
Maybe dificult at the begining,but the best way to do it(for me,at least).

Wow Bill,

I'm printing this one out so I can read it while not on the clock at work here. Great post.

Thanks!

Ingenious, Bill. Thank you! This is the first temperament that uses thirds only to establish the initial relationship of F-A and A-C#, then doing fourths and fifths from there, checking thirds. All other temperaments I've seen that start using thirds use thirds and sixths throughout, which is difficult. But this method is truly ingenious and I can easily see how it prevents "Reverse Well," which is always a problem with the fourths/fifths temperament.

Since you can tune historical temperaments, can you explain how you set a temperament for the 1799 Young and for the 1885 Broadwood's Best (or whatever you set that approximates these "romantic" temperaments where the beating of thirds follows the cycle of fifths -- pretty much perfectly in Young, less severely in Broadwood). I'm particularly interested in these temperaments as they were the ones most likely in use for much of the piano music that's in my repertoire.

Thanks for the kind words! I do indeed already have the answers for you on file which are the basis for upcoming PTG articles and ultimately a book. At this point, however, none of my drafts are copyrighted, so anyone is free to copy the material and use the ideas. PTG Journal articles are copyrighted but permission to reprint them is possible and easily obtained.

I have already posted both the 18th Cent WT and the EBVT on this forum but I will redo them as a part of this thread tomorrow. There wasn't much interest in them the first time, so I'm happy to throw them out there again.

That's also the reason I'm not at all concerned with copyright stuff. This material is intended for too limited of an audience to worry about someone stealing the ideas and making millions off of them. What the heck, it's hard enough to GIVE them away!

If you are familiar with John Travis' book Let's Tune Up you will recognize where Bill Bremmer's method came from. Mine came from the same source. I changed it, because it isn't efficient to have to rely on tuning all 12 notes in an octave until you can test whether your intervals are close or not. Using my method, you are filling in three major thirds within an octave, and then four fifths within a major third. Once you have filled in your first fifths, the others can be tested by comparing with a nearby third that you have already tuned. A signpost is never far off.

Whatever method you use, remember the principles: Fifths are slightly narrow, fourths are slightly wide, and major thirds are considerably wide. Every interval beats slightly faster the higher you go. You can use Bill's rule that adjacent major thirds beat 5;4 faster, and the cognate rules that adjacent fifths beat 3:2 faster and fourths 4:3, but I try to remember as few numbers as I can. It's better to remember the principle rather than the method. The priciples will serve you when the method leaves you hanging, as when you are tuning a piano with big changes in inharmonicity, like a spinet where the wound strings are in the middle of your temperament octave.

I think "temper" comes from the process of softening metal by heating, not from the Latin. Tempering was originally splitting the difference between harsh interval and and its adjacent pure interval. It made the harsh interval much softer.

Thanks, Bill. I, for one, am very interested to see how some of the historical temperaments are set, especially those used in the 19th century. I understand the theory of the Young temperament, for instance, but I've never seen written out the mechanics of setting this and other historical temperaments -- the intervals used and the tests done within the temperament.

I've had Owen Jorgensen's two books on search order at Amazon for 6 months, but none has turned up yet, so just seeing an exposition of some of these temperaments from the tuning perspective would be most interesting to me.

The "up a 3rd, up a 3rd, down a 5th" idea originally came from Oliver C. Faust in 1913 and to tune 3 contiguous 3rds to form an octave by Charles Ewing in the same year. This was approximately the same time that William Braide White wrote a book on tuning (1915) that uses the "bearing plan" (temperament sequence) that is offered in the American school. Charles Ewing also came up with the idea of using untuned notes to prove that a 4th or 5th was tempered the proper way.

While Braide White offered some "trials" in his plan, they are really not very useful. He stressed that the knowledge of science behind tuning was essential, that was, according to him, knowing about Helmholtz's calculations for the theoretical frequencies of each note of an equally tempered scale, but he offered no real or practical way to connect his tuning scheme to them. You just had to perfect ET through practice and experience.

Faust, on the other hand liked to use the rapidly beating intervals but didn't have a way of defining 4ths & 5ths. In short, Braide White was a "4ths & 5ths tuner" wile Faust was a "3rds & 6ths tuner". There are still such people today. It is not known why Braide White, Ewing and Faust could not get together and combine each other's ideas but perhaps it was professional jealousy. There were two tuners organizations at the time. Braide White belonged to one and Faust & Ewing to the other. Braide White's book caught on in popularity, however and in my opinion, unfortunately so because it spawned several generations of Reverse Well tuners and caused the first dependency on Electronic Tuning Devices (ETD) when Strobe Tuners were invented.

There were many approaches to temperament tuning but they were all imperfect. They created the class of temperaments called the Quasi Equal Temperaments (QET) ("quasi" means "almost"). Indeed, the "Marpurg" style QET which I posted on here a while back demonstrates that you can have "perfect" 3rds & 6ths but have half of the 4ths & 5ths pure and the other half tempered twice as much as they are supposed to be for ET.

It is true that John Travis used the "up a 3rd, "up a 3rd, down a 5th" idea in his book, Lets Tune Up but both he and Faust ignored the opportunity that was staring them right in the face, the ability to tune from a note which had some reasonable certainty to it rather than making the same mistake virtually all others in the past had made, that of compounding the error. All you have to do is ask yourself at each new note you tune, "From which note of which I am reasonably certain can I tune?" The more notes you tune, the more progressions of intervals become available.

John Travis (whom I had the pleasure of knowing late in his life) also seemed to be possessed with the idea that piano tuning must reflect Helmholtz's theoretical equally tempered scale. I have always found his section in the book, Let's Tune Up called "The Grabau-Travis Tuning Theory" to be interesting if not amusing. Actually, Travis includes a number of misconceptions in his book although they are mixed in with information which is perfectly sound, true and valid. It has taken decades to sort things out.

He makes the classic error of confusing and equating Equal Temperament and Well Tempered and claimed that "J.S. Bach ...tuned his own piano (sic) in ET and wrote the Well Tempered Clavier...". In Let's Tune Up he doesn't say it but in the previous publication, A Guide to Restringing, he cites the Grabau-Travis Tuning Theory where he claims that if you start on C# instead of C, "...[it] improves, if not corrects the temperament by making it more equal (sic), thus a superior tuning job".

The Grabau-Travis Tuning Theory chapter although sincerely written points to my own theory which I've been talking about for some years, the one about the Reverse Well error. The Grabau-Travis theory says that a typical tuner has the tendency to "err towards the just 5th" which I completely agree with. Between Grabau and Travis, they found that if they started the temperament sequence on a black key, "...you will still be making the errors, [but] the position of these errors will have changed and certain keys heretofore considered dull and lifeless will now sound brighter and the whole keyboard will sound better."

How could the temperament be equal but sound "dull and lifeless" by starting on one note and "brighter" and "better" by starting on another note if it is still equal? How could "brighter" and "better" be more equal? George Orwell had a lot of fun with such contradictory thinking in his novel, Animal Farm where after taking over, the pigs declared all animals to be equal but the pigs were more equal. I have seen and heard many examples in this and the last decade where people still want to believe that there is Key Color in ET (Travis also says so in his book). There seems to be the desire to have it both ways.

Travis also insists that octaves should be pure in piano tuning. He recognizes the existence of inharmonicity whereas Braide White does not but he doesn't seem to realize how much inharmonicity there is and that a tuner can use it to an advantage rather than fight it. What he claims in his book would be an all out unacceptable exaggeration (24 cents at the extreme high end of the keyboard) would be about what the Yamaha people in Japan today insist is as much as you can do. Most Steinway concert technicians these days will tune the C8 at least three times that sharp.

Thinking about octave stretching has done a complete 180 degree turn since Travis wrote, "In the extreme treble octaves, special care must be taken to avoid stretching, which is the natural tendency of the ear. The technician should not feel discouraged if it occurs, but he must be aware of what is going on and force his ear to hear the octave perfectly. In fact, when the octave is tuned correctly, it might even seem to sound "flat" by comparison to the prime octave or the 2-line octave. If stretching occurs, there will be audible beats in the high octaves which must be eliminated."

I bring all this up to point out how the state of the art of piano tuning is still evolving and as our understanding of the complex nature of piano tone increases, our acceptance of ideas which we may once have thought to be unacceptable or incorrect may have to change as we all learn how to make the piano sound better. I mentioned that I knew John Travis. I had the pleasure of attending tuning classes that he taught for PTG. Late in his life, he was in the audience where in 1995, I had tuned a 9 foot Baldwin grand at a PTG convention in 1/7 comma meantone and used the piano's own inharmonicity to plot out the upper octaves with my ETD. My C8 was +78 cents.

Now, this particular event produced some people who were very upset with what I had done and a small but very angry group approached the Baldwin rep and demanded that such a thing never be repeated and they got their way. But many people were impressed with the sound and wanted to know more about it. One of them who personally approached me with flowing compliments was John Travis. He didn't know about the temperament or what I had done with the octaves but he sure did like it. In his book, he insists that ET is the one and only and the best, including "color" and all, and that upper octaves should not and cannot have beats between them but when he heard a piano tuned in a very unequal temperament with seriously stretched octaves, he really liked it and wanted to congratulate the person who had done it.

Now, I've written too much to continue tonight but will get those two temperament ideas posted tomorrow night. They will show how to tune both an 18th Century style (like Thomas Young's) Well-Temperament and a late 19th Century style "Victorian" temperament from an A fork within the F3-F4 octave with the most precision possible by ear. That means, unequivocally, no guessing, same results each time. These, as I mentioned before, are part of a series I am writing for the Journal but getting early feedback on the ideas is always helpful.

Thank you, Bill. This is all very fascinating to me and I look forward to tomorrow's explanation of setting temperaments for those two historical temperaments.

I retired from piano work 20 years ago but always continued an avid interest in both tuning and rebuilding. After I had tuned maybe 100 pianos, I began to realize that tuning perfect octaves in the treble produced a dull, unmusical sound. I started tuning octaves to the 10th and 17th below, and that would often introduce a very slight beat in the upper octave, so I began introducing this slight beat (maybe one every 2-3 seconds) and found that the instrument started sounding more musical. I always learned to tune the bass listening to several partials, playing the octave above, the the third inversion of the chord which contains the next two partials and listening how the string responded to each partial and how harmonic it was. Slight adjustments would favor one partial or another, but there was often a sweet spot that seemed to blend them all quite nicely. It usually resulted in the original octave being a little flat, but no throbbing beating.

The point always was, in both treble and bass, to end up with a musical instrument rather than a lifeless but technically perfectly tuned instrument. Had I had the advantage of your handheld computers and tuning programs, it would have saved me a lot of headaches!

As it was I devised an amplifier for the extreme upper treble which consisted of a Fender guitar pick-up magnet with a clothes pin duct taped to it plugged into a RadioShack mini-amplifier. I'd clip the pick-up over the octave below, raise the key just to lift the damper, then play the key an octave higher and tune to the second fundamental of the lower note. It was crude, but it helped, especially in noisy rooms. I worked in a down-and-dirty shop where we were pushing several old uprights through there every week, so we got good at improvising, making tools, etc. I buddied up with a guy, a jazz player/technician, and we talked the store management into letting us rebuild a few grands -- action restorations, hammers, new pin blocks, restringing, refinishing cases (no new sounding boards, that was beyond us at the time), so I learned a helluva lot.

I worked very hard to tune good temperaments using fourths and fifths, checking the thirds and sixths, always having to work backward at the end, but having pretty good success at getting a relatively smooth progression of thirds, fourths, and fifths, though it sometimes took me 1/2 hour to get there. The ET temperament you outlined here makes so much obvious sense to me -- it's not at all hard to learn and avoids problems other temperament schemes pose.

All this is just by way of saying thanks, to let you know my curiosity isn't merely idle, and I looking forward to the next lesson!

A short note - in London over Thanksgiving, I was able to be present at a piano tuning by a tuner/technician taught by his Dad, who was taught at the old Bluthner factory pre-war - he learned by doing thirds, then fourths and fifths (widening, quickening) and checking against the thirds.

So sorry I couldn't get back to this sooner! 30 pianos this week, rehearsals for a concert next week and a concert tuning and prep tomorrow at 10 AM.

The problem I have see with most Historical Temperament (HT) tuning sequences is that just like the ones for Equal Temperament (ET) is that they give you beats per second numbers which you could have no way of verifying and even if you could, they aren't right anyway because they don't compensate for inharmonicity. That's why so many techs give up on doing them by ear and head straight for the Electronic Tuning Device (ETD) and dial in the numbers.

The problem with that is that you could easily make a transcription error and if the results were something really weird, it might turn you off to the idea really fast. You aqlso don't get used to listening to intervals and making compromises. You get totally dependent upon the ETD to do everything for you and you don't develop a good sense of judgment about what sounds good and what doesn't, what goes too far and what could go further without any harm.

There were however a number of temperament ideas in several classes, Meantone, Modified Meantone, Well Tempered and even Quasi Equal which Owen Jorgensen had assembled into a little known and very hard to find publication which was called, The Equal Beating Temperaments. These were special and different because you really could tune them by ear with the same accuracy and consistency that you could get from an ETD. Each interval is tuned either pure (beatless or zero beats) or one interval is made to beat exactly the same as another. The ear really can tell whether two M3rds beat exactly the same or not, for example, down to an accuracy of less than one cent.

The Thomas Young #1 Well Temperament is included in this publication and so is the Marpurg style Quasi Equal temperament. But in both cases the temperament sequence is from a C fork and does not constrain itself to the F3-F4 octave. Since most techs these days have learned to tune the temperament octave from an A fork and within the F3-F4 octave and because through experience and study, I have found the F3-F4 octave to be the ideal range to construct a temperament, I set out to find a way to tune the Thomas Young and the Marpurg this way. Also, there is no historically documented Victorian style temperament which uses an Equal Beating (EB) method for construction. I came up with a way for all three types.

The first, the 18th Century Style Well Temperament misses being an exact transposition of Thomas Young's temperament on paper but in practice, the musical effects would be virtually indistinguishable. I had Owen Jorgensen verify my work and his comment was that what it lacked in absolute perfect symmetry, it makes up for in his words, "extreme practicality".

It is ideal for classical music but can also serve well for many more modern types of music. It all depends on the taste and sensitivity of the performer. Be sure to try Bach's Well Tempered Clavier Music, any Mozart, Hayden, Beethoven and early 19th Century literature. It is also great for church hymns and any other 20th Century music in the easy keys with three accidentals or less. It begins to sound strained in music with 4, 5 or 6 accidentals. It can work for some people in modern music with complex harmonies such as Jazz and other contemporary forms.

You may see a graph of Young's temperament by clicking on the link below. With my version, the top of the curve is only slightly "clipped". (Jason Kanter hasn't put that graph on his website but I can send it by e-mail to any who request it.)

http://www.rollingball.com/images/Young.gif

Here is the entire temperament sequence:

Tuning a Representative 18th Century Well Tempered Tuning

(From an A4 @440 Pitch Source)

1. Tune A4 to A-440 pitch source.



2. Tune A3 from A4, a 6:3 octave.

Test for 6:3 octave: play A3 and C4, then C4 and A4. When both intervals beat

exactly the same, the octave is correct.



3. Temper F3 from A3, a widened 3rd to beat at 4 beats per second.



4. Tune C4 from F3, a pure 5th. The A3-C4 minor 3rd will beat faster than

the F3-A3 Major 3rd (at 6 beats per second).



5. Tune F4 from C4, a pure 4th. This will form a perfect 4:2 octave, F3-F4.



6. Tune Bb3(A#3), a pure 5th from F4. This will also make a pure 4th,

F3-Bb3(A#3).



7. Tune Eb4(D#4), a pure 4th from Bb3(A#3).



8. Tune Ab3(G#3), a pure 5th from Eb4(D#4).



9. Tune Db4(C#4), a pure 4th from Ab3(G#3).



10. Tune Gb3(F#3), a pure 5th from Db4(C#4).



11. Temper E4 from C4, a widened 3rd to beat at 4 beats per second.



12. Temporarily tune B3 to E4, a pure 4th.



13. Listen to the F#3-B3 4th and notice the beating.



14. Flatten B3 until both 4ths, F#3-B3 and B3-E4 beat exactly the same.



15. Temper G3 from B3, a widened 3rd to beat exactly the same as the two

previous 3rds, F3-A3 and C4-E4, (exactly 4 beats per second).



16. Temporarily tune D4 from A3, a pure 4th.

17. Notice the beating between G3 and D4.

18. Sharpen D4 until both the G3-D4 5th and the A3-D4 4th beat exactly the same.

2003 Bill Bremmer RPT


With either the above or the next temperament, you may used your own discretion to alter the temperament to suit either your own taste or sensitivity or that of your customer. You may make the initial 3rds slightly faster and may temper very slightly the intervals which are designated as "pure". There is nothing wrong with doing this and in fact has always been done throughout tuning history.

Here is the Victorian style temperament:

Equal Beating Victorian Temperament



1. Tune A4 to an A-440 pitch source.



2. Tune A3 from A4, an octave.



3. Temper F3 from A3, a widened M3, to beat at exactly 6 beats per second. This will be one beat per second slower than the same interval would be in Equal Temperament, (ET).



4. Tune F4 from F3, an octave.



5. Tune C4 from F3, a pure fifth. (Check that this will also create a pure 4th from C4 to F4.)



6. Temper E4 from C4, a widened M3 to beat also at 6 beats per second, (exactly the same as the F3-A3 M3.)



7. Temper G3 from E4, a widened M6 to beat also at 6 beats per second, (exactly the same as the F3-A3 M3 and the C4-E4 M3.)



8. Temper B3 from G3, a widened M3 to beat also at 6 beats per second, (exactly the same as the F3-A3 M3, the C4-E4 M3and the G3-E4 M6.) (Check a again for four Rapidly Beating Intervals (RBI), F3-A3, G3-B3, G3-E4 and C4-E4, all beating exactly the same at 6 beats per second). This will also naturally create a pure fourth from B3-E4.



9. Temporarily tune D4 from G3, a pure fifth. Notice the strong beat in the resultant A3-D4 fourth, then flatten D4 until both the G3-D4 fifth and the A3-D4 fourth beat exactly the same (a little more tempering than in ET).



10. Temporarily tune A sharp 3 (B flat 3) from F3, a pure fourth. Now sharpen A sharp 3 until the F3-A sharp 3 (B flat 3) fourth beats at one beat per second.



11. Notice the resultant rapid beat in the A sharp 3 (B flat 3)-D4 M3. Now temper C sharp 4 from A3, a widened M3 until the A3-C sharp M3 beats exactly the same as the A sharp 3 (B flat 3)-D4 M3.



12. Tune F sharp 3 from C sharp 4, a pure fifth.



13. Tune G sharp 3 from C sharp 4, a pure fourth.



14. Temporarily tune D sharp 4 from G sharp 3, a pure fifth then notice whatever beating there is between the A sharp 3 and D sharp 3 fourth. Now flatten D sharp 4 until both the G sharp 3-D sharp 4 fifth and the A sharp 3-D sharp 4 fourth beat exactly the same.



Bill Bremmer RPT

Madison, Wisconsin

May 2004

(Originally designed 1992)


I use the above tuning as my default tuning for virtually all kinds of music although I sometimes do mitigate it slightly. If further mitigation is desired or the customer really wants the sound of ET 3rds and 6ths, I use the following "Marpurg" Style Quasi Equal Temperament. The tests for pure 4ths & 5ths attributed to Charles Ewing in 1913 are included.


Here is the step by step bearing plan for the "Marpurg" style Quasi Equal Temperament:



1. Tune A4 to A-440 pitch source.



2. Tune A3 from A4, an octave.



3. Tune the series of contiguous major thirds, F3-A3, A3-C sharp 4, C sharp (D flat) 4-F4, exactly the same as when tuning standard equal temperament. (See my article on tuning in the March, 2004 issue of the Journal for detailed instructions on steps 1-3).



4. From each of the notes within the F3-F4 octave already tuned, tune whichever available 4th or 5th there is within the F3-F4 octave perfectly pure.



a. From F3, tune A sharp (B flat)3, a pure fourth and C4, a pure fifth.



b. From A3, tune D4, a pure fourth and E4, a pure fifth.



c. From C sharp 4, tune F sharp 3, a pure fifth and G sharp 3, a pure fourth.



5. The remaining 3 notes, G3, B3 and D sharp (E flat) 4 are tuned as Equal beating from the available 4ths or 5ths within the F3-F4 octave.



a. Temporarily tune G3 from C4, a pure fourth. Then, flatten G3 until both the G3-C4 fourth and the G3-D4 fifth beat exactly the same.



b. Temporarily tune B3 from E4, a pure fourth. Then, flatten B3 until both the B3-E4 fourth and the Fsharp 3-B3 fourth beat exactly the same.



5c. Temporarily tune D sharp 4 from G sharp 3, a pure 5th. Then, flatten D sharp 4 until the G sharp 3-D sharp 4 fifth and the Asharp 3-D sharp 4 fourth beat eaxactly the same.



The tests for pure fouths and fifths



To test for a pure fifth, find the note which makes the interval into a minor chord (a minor third above the lower note of the interval). When the lower minor third and the upper major third beat exactly the same, the interval is a pure fifth. Alternatively, you may choose to drop that note down one octave and perform a major third-major tenth test. This alternative tests the lower set of audible coincident partials and is often easier to hear.



To test for a pure 4th, find the note which is a M3 below the bottom note of the interval. When the M3 and M6 beat exactly the same, interval is a pure fourth.



These two pure interval tests may be used in ET to prove a fourth of fifth is tempered, not pure or widened by mistake. In both cases, if the upper interval beats slightly faster than the lower, the interval is properly tempered. These tests were published by Charles Ewing in 1913 but also were not included in Braide White's teaching. It again took until sometime in the 1970s before PTG began to teach them and before they were widely known.



Sources: "The Equal Beating Temperaments - A Handbook..." 1981 by Owen Jorgensen. "Tuning..." 1991 by Owen Jorgensen.



Bill Bremmer RPT

Madison, Wisconsin

March, 2004

Mr. Bremmer:




one for the enlightenment,
one for the clarity,
one for the time spent.