Thanks Rafael, the problem with the Steinway sequence is essentially the same as with the Braide-White or any similar 4ths & 5ths sequence, as you have seen. It is a problem of cumulative error and/or error canceling out error. When you arrive at an interval to test and find there is an error, you cannot know which previous intervals may have been in error or by how much. It is also possible to arrive at a test interval which seems correct but the previous intervals which lead to it are erroneous but the errors canceled themselves out.

In the latter case, any future intervals tuned from faulty placement will produce cumulative error. That can lead to a long and confusing process of finding and correcting small errors which requires a high degree of skill. Novices rarely have sufficient skill for such a challenge.

Having said that, a Steinway tuner such as one at the factory or who mostly or exclusively tunes Steinway pianos, especially the larger grands, works with only scale design that with practice, becomes familiar. The technician learns how to estimate correctly at each step and is therefore successful. Still, many Steinway tuners produce a slightly irregular temperament but refuse to admit it or recognize it. Their priorities are first with unisons, then octaves, then perceived evenness of 4ths & 5ths. A little irregularity among the rapidly beating intervals is unimportant to them. Franz Mohr actually says that, I have heard him say it on at least 3 occasions, so I know he believes that. I recall one tuning he did for a recital at a PTG convention where the key of Ab was quite pronounced, as if he had done some kind of well or meantone temperament deliberately.

When questioned bout this, he did not seem to have a firm or ready answer. He would only say that among Steinway tuners, they all recognized that each tuned slightly differently. While that could mean differences in the amount of octave stretch, he did seem to recognize that temperament was also one of those differences. He said that Steinway technicians all strive for the same kind of sound because one tuner may often follow another. If each tuning were radically different from another, that would present a problem in itself. That is what they believe and one reason why they strongly maintain ET as the one and only temperament they will ever use.

However, I have also heard Steinway technicians say that when an artist is not completely satisfied, they may call on a tuner known to tune with special "color". They called that tuner, "Rembrandt" (implying artistic "color"). That again can involve octave size but in my view, clearly implies temperament manipulation. The Steinway people could offer no details about what differences there may be.

I can only conclude that there is some degree of contradiction between what Steinway has as a policy and what they really may do in practice. That can be and is probably true for voicing and regulation techniques as well. In the end, they consider piano technology an art which must be practiced and years of experience go a long way towards credibility over written or verbal descriptions of what any technician may do at any given moment.

I found myself very busy this past week and have a full day today and a piano recital to attend tonight. I would like to write a plan for tuning the EBVT and EBVT III by the Direct Interval method. Although it has been said to me that the SAT III is the most useful ETD for tuning with that method, I believe any of the four most popular devices and programs are capable of it.

The advantage is that the ETD can find and capture the exact desired pitch of each note. The tuner will have made the decision in each case, not a calculated program. Once the exact pitches are known, the ETD can be used to really solidify the tuning to the degree which it is capable, 1/10 or 1/100 of a cent from one end of the piano to the other. When each and every full unison (not just a single string) can be verified to hold that exact pitch, the effect of the perfected tuning will be truly stunning. Prepare for a pipe organ effect as you have never experienced it before!

Thanks Bill, we will be wiating for that post!

With the Verituner Pocket PC, which I have, there is no direct interval tuning mode. The Verituner 100, the box, does has a Direct Interval tuning mode.

But the pocket PC version has a feature designed to measure tunings and store the results and that can be used to make direct interval tuning.

It's easy done once you know how to do it, and it becomes faster when you have stored the partial you will use for each note.

For example, to tune F5 as an equal beating 12th-doubleoctave, i.e. F3-F5 double octave beats wide the same as A#3-F5 12th beats narrow, you have to:

1. Set the ETD to measure the first partial of F5.

2. Play F3 and take note, mentally, of the reading in cents. That gives you the offset of the 4th partial of F3.

3. Play A#3 and take note of the reading in cents. That gives you the offset of the 3th partial of A#3.

4. Tune F5 at the mean of the two previous readings.

So for example lets say that in step 2 you measure F3 at 1.3 cents and in step 3 you measure A#3 at 5.5 cents, then you have to tune F5 at (1.3+5.5)/2= 3.4 cents.

And that way you tune an exactly equal beating double octave/octave + fifth interval, i.e. a "mindless octave".

For tuning a given interval in the temperament you proceed exactly the same way. Note that you have to convert beats to cents. For example F3-A3 beating at 6 bps means that

4*A3-5*F3=6 bps so
F3 = (4*A3-6)/5 if A3 = 220 hz then
F3 = (4*220-6)/5 = 874/5 = 174.8 hz and so

M3(F3-A3) = 1200*log2(4*220/5/174.8)= 11.84 cents


Now we can tune F3:

1. Set the ETD to measure the first partial of A5 (which is the coincident partial of F3 and A3)

2. Play A3 and take note of the cents reading, say it is for example 6.3 cents

3. Tune F3 to 6.3-11.8 = -5.5 cents.

And we have the M3 F3-A3 exactly 11.8 cents wide = 6 bps wide.


If I have some free time tomorrow sunday I would make the cents calcul for the entire EBVT temperament and I'll post the results here.

Tooner,

I hope you will be reading this. As you can see with an ETD you can do things that are not possible without it.

I checked. White's figures are all based on A-440. Why did you think otherwise?

Tooner,

In the book of Owen Jorgensen "Tuning", page 689, section 218 called William Braid White's Bearing Plan of 1915, the first step of the sequence says:

Tune C to 517.3 Hz.(International Pitch based on A = 435 Hz)

I don't have the Braid White's book so I trust Mr. Owen Jorgensen.

I have a copy of White's book. It's a reprint of the fifth edition (1946), "revised and enlarged". In this edition, the figures are based on A4=440Hz.

According to my copy, there were 5 editions: 1917, 1927, 1938, 1943, 1946 (and 32 reprints of the last one, at least until 1987). White mentions "the old 435 pitch" in some parts of the book, and offers two plans to "set the bearings": one starting from C, and one from A ("some times convenient to tune to some wind instrument").

I think he used A4=435 in the past (from a C fork), that makes sense historically, and I also think the above mentioned details support this, at least indirectly.

In this case, Mr. Jorgensen's historical overview of tuning seems to deal with the foundation of White's work, in other editions figures were adapted to "the modern standard".

In this edition he mentions using a Conn Chromatic Stroboscope, "with a tolerance of 1 cent" (his assumption of "given the proper source of electric current supply" doesn't make me feel all that comfortable).

He mentions the benefits of using a device... back in freaking 1946! Why not use a modern device today? :p

Erus,

Thanks for enlighting us.

I've once had a copy of White's book in my hands, unfortunately I did not get a copy of it. I don't want to buy the book, I think it has only a historical value and what I have from Mr. Jorgensen covers the Braid White's works on tuning pianos and much more.

When I decided to learn to tune, I went into a piano shop and asked the owner if he could teach me to tune. I paid him the first lesson, the fee of a tuning, and he took out his copy of "Modern Piano Tuning and Allied Arts" and began teaching me temperament setting. It was the only lesson I took with him as he asked me to return when I was able to set the temperament. After that I found the Randy Potter School of Piano Technology and never returned to take the second lesson. I think if I were to teach tuning to someone I will begin with unisons not temperament setting, but I think he started like this because he sees temperament setting as the more important skill in piano tuning.

So Tooner, can you say me what edition of Braid White's book you have?

In a comparative scenario it would be interesting to hear two identical pieces played on a Yamaha Disklavier or Bosendorfer CEUS system with the pianos tuned in the different temperaments. This would give a direct comparison with most of the parameters equalized. I wonder if Bill has access to one of these instruments for this purpose?

White's book is certainly a very important book in the history of piano technology (it is a huge step forward compared to J. Cree fischer's, for example). However, A LOT of the research he quotes has been surpassed (both in physical and in historical matters).

I found it only useful as a historical reference. I got my copy from a set of books I found at ebay (the most interesting and useful book from the set turned out to be one about the harpsichord).

What I've found the most useful book until now is "Tuning" by Owen Jorgensen.

One can experiment tuning all these amazing tunings and temperaments form Pitagoras to CM3s.

I wish I was able to play music from several historic periods to listen to the actual music those systems were designed for.

However one can listen to different temperaments by playing intervals to get an idea of what was achieved in the past.

It is a great excercise on tuning different types of fiths used mainly in the meantone temperaments. Meantone is the most used temperament in human history, more than well temperaments and ET.

It is amazing to listen to Bach's music in a piano tuned as it was in Bach's time.

Unfortunately this book is no more available. I got a used copy at a total of about US $500.00 including shipment and taxes from a bookstore I found in the Internet.

For me it is a very valuable treasure! It is 700 pages long, absolutly rigurous and respectful of the authentic systems of tuning and temprering systems. It is written in a scientific way. Very interesting topics about the evolution of tuning systems.

It has also a very complete appendice on iH and how it affects the piano temperament.

Obviously I have not had enough time to read it completely but I love to read it time to time (when I have nothing more urgent to do).

It was expensif but it is worth every cent I've spend on it.

Thank you for the question Emmery. I don't really have such access but I know the university here has Disklaviers. This has been proposed before. However, I think the logic itself is flawed. When a musician plays a piano in a different temperament, they play the piano differently too. If you had been at the piano recital I was at last night where the piano was tuned in 1/7 comma meantone, you would have seen and heard that first hand.

The 1/7 CMT is a much stronger temperament than either EBVT. Yet, the artist embraced it, playing a set of Brahms, a flashy piece by Liszt and then all of the Rachmaninoff preludes. Much more music was played in the remote keys, those which are supposedly unusable than was played in the mild keys.

The pianist spoke at intermission, saying that he had not planned to do it but felt compelled to say how for the first time in his life, the music that he had learned, practiced and performed so often "spoke" to him in a wondrous way. He said that the piano "told" him the expression to use as he went along.

I'm afraid that if such an experiment were done, it would serve to "prove" only what advocates of ET and ET only wish to prove: that music would sound clumsy and out of tune in anything but ET.

I only wish that you all could have heard and experienced what I did last night. The audience was spell bound. There was certainly no one who got up, muttering about an out of tune piano, saying it made their skin crawl or any other such nonsense. The artist received instantaneous standing ovations after each set. He offered a piece by Scriabin for an encore.

All of this only a very strong indeed, 18th Century historical temperament, complete with the "wolf" key of A-flat. (a 9 cent wide 5th, beating about 4 beats per second). If what the majority of piano technicians, musicians, music educators, etc., believe is to be believed, what did, in fact occur, could not have happened. It would have been thoroughly unacceptable. People would have walked out; there would have been cat calls from the audience. There would be a scathing review in the newspaper today about an out of tune piano and a disgruntled and embarrassed pianist.

What happened instead was a performance of the very highest caliber that the audience had ever experienced.

"Unfortunately this book is no more available. I got a used copy at a total of about US $500.00 including shipment and taxes from a bookstore I found in the Internet."

You were lucky to get it Rafael. I bought my first edition for $75 and had the author autograph it. I knew what the value of it would be at that time but most technicians ignored it. A second printing did go out but most of the copies ended up in half price, "remainders" type bookstores. Then it was junk but now it is one of the most highly sought books on tuning there is.

I have the 1946 edition of Piano Tuning and Allied Arts, the one where Dr. Bill recommends getting a strobe tuner if you can't manage to follow his directions. The only time I look at it is when I want to point out what's missing, obsolete and wrong with it as a tuning manual.

I've heard some "comparisons" rigged in favour of ET: who chooses to play music from the late 18th century, in 1/4 meantone, knowingly using the wolf in what are supposed to be the most consonant harmonies? They could just use a "slendro" scale and take things to the next level (of idiocy).

I've also heard some serious recordings of Scarlatti's sonatas in meantone. They just come to life, and the contrast between sections is amazing. A truly revealing experience. It's amazing how many effects are lost when played in ET. (Some sonatas do sound better in the harpsichord and clavichord, but a lot are just great when played on a piano).

I think it's harder to rig comparisons for milder temperaments, but still possible.

I guess some romantic music with clever modulations would be very nice in EBVT. I've heard Chopin played in romantic temperaments, and key colour can be a really nice thing.

Thanks Erus, I have also heard Chopin, Debussy and Jazz in 1/7 comma meantone at the same venue. The artists eat it up. None of them has complained or even questioned the technician about it. He NEVER tells them though. He just presents the piano he has prepared for them and they play it. He says that if he were to tell them the tuning is MEAN tone, then they would start listening to the way it is tuned rather than performing music.

He always provides a splendid piano. The truly fine pianists accept and expect differences from one piano to the next. Apparently, they are all able to recognize the particular power that the 1/7 CMT has and they take it for a ride.

I mentioned the 1/4 meantone example because those recordings were intentionally rigged, the music was transposed and specifically chosen to make the experience unpleasant.

I've never heard that kind of music with 1/7 meantone. There is certainly a difference between 1/7 and 1/4 comma.

Tuned a Kimball console piano this week using Tunelab and Bill's EBVT 3. This was my first time to tune for this individual. The owner is not one who is known for handing out the compliments. However, I saw her at a restaurant after church today, (She was not at home when I tuned). I asked her if she was pleased with how her piano turned out. She said, "Ryan, that piano has not sounded that good in YEARS!"

Ryan Hassell
Hassell's Piano Tuning
Farmington, MO

Gadzar:

OK, your information on White’s method is hearsay.

You posted: ”So for example lets say that in step 2 you measure F3 at 1.3 cents and in step 3 you measure A#3 at 5.5 cents, then you have to tune F5 at (1.3+5.5)/2= 3.4 cents.”

This would be close, but not quite. If this method was really correct, then two contiguous M3s that were both 400.00 cents wide would beat at the same speed, but of course they don’t. And Alfredo’s algorithm would be much simpler, but it is not.

And let me comment on beats and cents conversions without a long quote. You consistently point out the problem of not including the effects of iH when calculating frequencies and beat rates, and then fail to include iH in your own calculations!

I was thinking on what you said about how I discuss iH and cents without the use of an ETD. Young’s paper came to mind. iH was predicted long before it was measured. Young’s paper is an empirical proof of mathematical theory. I see nothing odd about calculating effects of iH rather than measuring the effects.

Something that I think you (and perhaps Bill…) get confused about is the difference between iH affecting frequencies and iH affecting beat rates. They are two very different things. Something that you may want to try with your ETD is to measure the frequencies of the partials of a well tuned temperament octave and calculate the actual beat rates. Not all the partials would need to be measured, just those necessary for the intervals. Then you could empirically determine if actual beat rates are significantly different from theoretical beat rates due to iH. You could also determine just how equal the temperament is.

I wish I could read Mr. Jorgensen’s section on how iH affects piano tuning. It may be misunderstood and perhaps misquoted….

Bill:

This Topic is a good example of why I no longer choose to have serious discussions with you.

You ask for opinions and when you get one you do not care for, start on a witch hunt that tries to implicate Dr. White into an equal temperament conspiracy theory. Anecdotes (which are really just stories to make a point and cannot be used to prove anything) are sprinkled with the word “fact” to try to lend credence to the idea that the use of Dr. White’s methods will make a tuner into a loser. When deriding Dr. White’s book you do not cite his work, but instead allude to things that he never wrote. Your use of sarcasm shows the desperation in your arguments. Scoffing at mathematics as a tool for understanding tuning theory is directed only at points that you don’t agree with. This shows that you are not interesting in the truth.

I choose to not be involved with your sophomoric debate tactics.

Tooner,



Tooner,

You don't get the point. I am not calculating partials. When I use an ETD, I am measuring partials, I am dealing with actual partials, i.e. inharmonic partials, so iH is already included and taken into account. That's why I said you to use an ETD. By using an ETD you can measure partials instead of having to calculate them.

Furtheremore I am talking about tuning EBVT III via direct interval tuning with Verituner Pocket PC. Mr. Capurso's algorithm has nothing to do here.

I guess you have no idea of what "direct interval tuning" is, because you have no ETD.

You say that what I've said of White's system is hearsay. I have read the statement "Tune C to 517.3 Hz." in Owen Jorgensen's book: Tuning, page 689. That is not "hearsay".

I give up!

I can not make you understand. Your mind is closed. You can not accept facts. If you think I am wrong, you can think so, that doesn't bother me.

I won't talk with you anymore, I will ignore your posts. If you talk to me I won't answer.

Here are some quotes from the 1946 edition of Piano Tuning and Allied Arts: On the use of the Conn Chromatic Stroboscope for measuring the accuracy of Piano Tuning. To tune a piano in ET...is a matter of arranging the scale...on the fundamental ratio for the ET semitone, namely 1:1.0594681. The practical methods have been worked out...but in any case, the margin of error in ordinary practice has often been too wide. It has often been found that, even with what is called a good tuning, a systematic error of more than 2 cents is commonly detected.

This somewhat distressing practical fact, together with others of parallel importance, has now been brought out clearly for the first time...since the monumental edition of Helmholtz. An ingeniously devised apparatus, making use of the optical stroboscopic effect, has been developed by the research engineers of C. G. Conn, Ltd.

The Conn Chromatic Stroboscope (now known as the Stroboconn)is a frequency measurer (sic) of great exactness, giving its readings visually and requiring no use of the ear whatsoever. In this way it is at once possible to tell whether any given tone on a piano is flat or sharp of its proper frequency at the 440 pitch. The control is absolute and takes care of all possible fluctuations of the outside power supply.

The Chromatic Stroboscope may be applied practically to the practical work of piano tuning in the field, given the proper source of electric current supply such is found in all great cities. The apparatus has many other uses too. For example it can be used to calculate and verify the amount of the stretching of Octaves, upwards or downwards, as commonly carried out in high treble and low bass. It can be used to construct a Mean-Tone Temperament or a Pythagorean scale. The Conn Chromatic Stroboscope is a very admirable piece of apparatus, and invaluable addition to the scientific equipment of the piano technician.

*********************************************************

In the next appendix, The Mean-Tone Temperament, the author says this: Now, on the one hand the Mean-Tone Temperament has the virtue of (1) giving smooth Major Thirds and Major Sixths, and (2) within the limits above stated producing an intonation delightfully "sweet". On the other hand (1) the Fifths and Fourths, even within the prescribed limits of tonality, are, to modern ears, unpleasantly distorted, while (2) outside the prescribed limits the howling of certain intervals, as may be seen from the data tabulated above, fully justifies the name of the "wolf" that used to be given to this awkward place in Mean-Tone tuning.

It is easy to see how, with the rapid expansion of musical art during the 17th and 18th centuries, the demand for something more widely useful than a Mean-Tone Temperament soon became irresistible. Sebastian Bach began to tune his clavichords in Equal Temperament so that his pupils might be able to play in all tonalities without frequent retunings (sic). The celebrated Well Tempered Clavier...was the first fruit of this famous experiment in Intonation.

***************************************************************

Do I really have to point out just why this book is obsolete? Even the spelling, punctuation and grammar do not meet today's standards. Obviously, many piano technicians read this material, bought and used strobe tuners in the belief that it would make their tunings scientifically accurate. It took the few people who really knew how to tune to point out to them that it did not.

The paragraph on "Sebastian Bach" reveals what others have written about extensively, The Conspiracy of ET; the deliberate confusion of ET with WT and the suppression of the knowledge of any possibility between ancient forms of tuning, 1/4 Comma Meantone, Pythagorean tuning and the glorious and all powerful ET. In his recent work of fiction, Temperament, Isacoff perpetuates this deliberate misinformation and claims that music as we know it today could not have developed had it not been for ET. Thank you, Owen Jorgensen for revealing the truth and exposing the lies.

Let's go back to the method of tuning an ET. Here are some of the suggestions:

"Of course it is evident that one cannot make use of the beats until one can hear them readily. It is at first a little difficult to hear them in piano strings. The difficulty may be overcome by listening to the beats evoked from the pipes or reeds of an organ. I therefore recommend the student to make use of a pipe organ, a reed organ, or even an accordion, for preliminary practice in detecting the presence of beats. All these instruments are tuned, of course, in ET, and the beatings set forth in Table 3 will be found in them."

It is little wonder why anyone reading this today and actually attempting to follow these suggestions would come to the conclusion that they cannot "hear" beats and have no clue about what ET really is or means.

Octaves: The author provides three different octave tests, one for each of 3 types of octaves of different sizes but suggests they are all equivalent: the 4th/5th test (as described reveals a 2:1 octave), the m3-M6 test (which reveals a 6:3 octave) and M3-M10 test (which reveals a 4:2 octave).

"Estimating Beat Frequencies. This has to be done of course, by ear and from what has already been said, it is easy to see that considerable practice is called for, if one's estimates of beat-frequencies are to be at all nearly accurate". He then provides the well known theoretical beat rates for 5ths, "3 beats in 5 seconds" and 4ths, "1 per second".

He goes on about the rapidly beating intervals: Faster beat rates such as those of Thirds and Sixths are not so readily imitable; but one can usually learn to tap out on a table (using drum sticks) five, six or seven taps per second, against the ticks of an accurate pendulum clock beating seconds".

Doesn't it make sense therefore to use a plan that calls for only one estimate and no "counting" of beats? What about beats which are faster than 7 seconds? How do you count those against a grandfather clock with a drumstick on your kitchen table? What about the beat rates that fall in between whole numbers as are listed on the chart of theoretical beat rates? How do you count them? How do you adjust them for inharmonicity by the very small amount they must be adjusted?

Then, there is the infamous "Bearing Plan": Scheme of "Laying the Bearings in ET between F3 and F4 by 4ths and 5ths with tests".

It goes like this: TUNE (to the fork, no problem). TUNE, TUNE TUNE (which means, estimate, then build another estimate upon that estimate, then build yet another estimate upon the first two estimates). Test. A beat rate of 8.0 beats per second is provided. How do you count that accurately against a grandfather clock with a drumstick on your kitchen table? What do you do if the test is not exactly 8 bps? Do you TUNE TUNE TUNE again backwards? TUNE (estimate). Test, 7 beats per second. TUNE (estimate). Test, 9.0 bps. Test, 10.5 bps. TUNE (estimate). Test, 8.0 bps. Test, 13.0 bps. Test, Major triad, if the chord sounds good, it must be right. But what if that chord sounds a little funny, then what do you do? TUNE (estimate). Test, 10.0 bps. Test, inverted major triad, if it sounds good upside down, it must be right but if it doesn't what do you do? TUNE (estimate), test, test, test, 9.0 bps and more triads. TUNE (estimate) 5.0 bps. Test, test, test, 8.5, 9.5 bps then another chord. TUNE (estimate). Test, test, test, test. 8.5, 10.0, 14.0, then another chord. I sure hope these chords are sounding as good as those on the accordion. TUNE (estimate). Test, test, test, test, test. 7.5, 9.5, 10.5, 12.5 then another chord. This time, compare it to the same chord on your reed organ. Make sure you pump the bellows evenly with your feet (no specification for that, just do your best). TUNE (estimate). Test, test, test, test, test, 11.0, 16.0, 9.5 ,0.0. The last, is the pure octave that you can test three different ways for three different sizes but it should still come out "pure".

Wouldn't a method that involves only one estimate, no counting of beats and no tests but which can and does yield perfect or nearly perfect results each time be just a little more practical? Wouldn't it also be more practical to use tests upon the completion of the sequence and test by comparing intervals rather than counting beats?

***************************************
So, you would like to become a piano technician? You don't need to join any organization or go to any school, I have an old book, the classic method that people learned from during the entire 20th Century and it is still being highly recommended on the internet in places like the Piano World Forums. The instructions for just what to do are all written out in clear and easy to understand terms. It only costs $9.95. The gooseneck tuning lever is only $19.95 and includes a felt strip and two rubber mutes. Such a deal! For an investment of only about $30.00, you can soon be earning $100.00 per hour or more!

(Two weeks later) Oh, you couldn't understand or follow those instructions? You don't have a grandfather clock, a pipe or reed organ or an accordion? I see you do have a drumstick, though. Oh, you would like to by a Stroboconn! Sure thing! You won't have to use your ear at all, just like it says in that book! I have an old used one here in perfect condition, only $50.00. Good luck!

Gadzar:

Read or not and reply or not. Saying so-and-so wrote that so-and-so wrote something is not the same thing as reading it yourself.



The problem with this math is that the fundamental frequency is multiplied by the partial number to determine the frequency of the partial without including the effect of iH. Young’s paper explains how to do this. The odd thing is that although iH definitely affects the frequencies, it makes very little difference to beat rates, especially in the temperament. And that is my point when criticism is made of using theoretical beat rates to tune with. It is impossible to aurally tune by frequencies, so the inaccuracy of theoretical frequencies is not important in itself. What is important is whether the theoretical beat rates are accurate, and oddly enough they are.

No need to believe me, the math will prove it. And there is no reason for me to show the math. If someone can comprehend it, they can perform it themselves, and should do so rather than take my word for it. If they cannot comprehend it, showing the math will do no good. These people can believe what they want. I am not the only one to say that the effect of iH on theoretical beat rates is insignificant, especially in the temperament.

I know that there are members on this Forum that are capable of this and have studied it. I have wondered why they do not join in. It could be to avoid the frustration that I sometimes experience. I guess I am not over being mislead by this myth and having false self-doubts. My ears said one thing, this myth said another. By doing the math, I proved to myself the falsehood of this myth and regained confidence in what I hear. I would like to prevent the same thing from happening to others.

Good post, Bill!

It is worst than I remembered it!

Tooner,

For the sake of exacteness, I will answer this last time, it is worthless I know but even so you can have it:

With actual numbers, as measured on a Knabe Upright I have in my shop for repair:

The actual frequency of the fundamental of A3 does not matter, we measure the actual frequency of the fourth partial of A3.

Offset of fourth partial of A3 = + 1.13 cents

Frequency of fourth partial of A3 = 440*2^((37-49+24+1.13/100)/12) = 880.574575 Hz.

So we must tune the fifth partial of F3 to 874.574575 Hz. (6 hz lower) in order to have 6 bps.

Offset of fifth partial of F3 = 1200*log2(874.574575/(440*2^(33-49+28)/12))) = -10.83653794 cents.

So:

Width of F3-A3 M3 = 1.13 - (-10.83653794) = 11.83653794 cents

After rounding decimals we have:

Width of F3-A3 = 11.84 cents, isn't it?

Sorry, I was interrupted, but i'm back.

For the equal beating 15th-12th we have, in the same Knabe upright, we measure actual partials:

offset of 4th partial of F3 = 0.67 cents

frequency of 4th partial of F3 = 440*2^((33-49+24+0.67/100)/12) = 698.726823 Hz.

offset of 3rd partial of A#3 = 1.69 cents

frequency of 3rd partial of A#3 = 440*2^((38-49+19+1.69/100)/12) = 699.138617 Hz.

So, in order to have equal beating between F3-F5 and A#3-F5 we must tune F5 at the mean frequency:

1st partial of F5 = (698.726823+699.138617)/2 = 698.932720 Hz

and we have:

double octave F3-F5 beat rate = 698.932720 - 698.726823 = 0.20589687 bps

12th A#3-F5 beat rate = 699.138619 - 698.932720 = 0.20589687 bps


Tooner,

So with this system of direct interval tuning you can tune exactly what you want.

Without an ETD you can't do this because you don't have the measured partials involved in the intervals you want to tune.

I hope this time you will give me some credit for my calculations!

This time I will finish my post!

Instead of doing all that frequencies calculations you can go directly with the offsets in cents, the error introduced is negligible, really negligible, as we'll see:

offset of 4th partial of F3 = 0.67 cents
offset of 3rd partial of A#3 = 1.69 cents
mean = (0.67+1.69)/2 = 1.18 cents

Now you tune F5 to +1.18 cents offset, without any further calculations, just as I have said in a previous post.

The error introduced is:

1st partial of F5 = 440*2^((57-49+1.18/100)/12) = 698.932689 hz.

instead of the theoretically correct value of 698.932720 Hz.

error = 698.932689 - 698.932720 = -0.000031 Hz.
error = 1200*log2(698.932689/698.932720) = -0.000077 cents.


The best human tuner can only adjust tenths of a cent so there is no error at all.

You folks have inspired me...

Some time ago, a number of my books and other items were destroyed when I had raw sewage back up into my basement. Of course, my homeowner's insurance refused to cover anything.

My copy of WBW's PT&TAA was destroyed.

Today, I finally went to Amazon and ordered another copy... along with a couple more books:

William Braid White's "Piano Tuning & the Allied Arts"
William Braid White's "Theory & Practice of Piano Construction"
and
Alfred Howe's " Scientific Piano Tuning & Servicing"

I was shocked to see how much my copy of Owen Jorgensen's "Tuning the Historical Temperaments by Ear" is now worth.

For some time I've been thinking about starting a thread (I don't want to hijack this one )regarding modern books that are still in print that deal with modern tuning theory from a mathematical perspective.

Joe,

I don't know if we are talking about the same book of Owen Jorgensen.

What I bought is: (I reproduce all it has in the front cover)

-TUNING-
Containing
The perfection of Eigheenth-Century Temperament
The Lost Art of Nineteenth-Century Temperament
and
The Science of Equal Temperament
(grand piano illustration)
Complete with
Instructions for Aural and Electronic Tuning
OWEN H. JORGENSEN

Edited in 1991, by Michigan State University.

This is a huge red book with letters in gold which has 798 pages.

I think what you have is another book of the same author: "Tuning the Historical Temperaments by Ear".

Hi Rafael,

I'm sure they're different books.

I "think" Tuning the Historical Temperaments by Ear was the first book he published on the subject.

Thanks,
-Joe

You're right, Joe. I looked them up on Amazon and was amazed it was possible at all to buy a used one of either. It is interesting to note that Jorgensen's books command a hefty price of $425.00 or more, Braide White's lastest edition, 1953, $14.00 and Isacoff's work of fiction, $1.66.

The ET via Marpurg method I developed for tuning ET was based upon research from two of Jorgensen's publications: The Equal Beating Temperaments and Tuning...The Science of ET. The latter contains two pages of theoretical beat rate information for ET in a far more readable format than the Braide-White book. So, if I ever need that information, I prefer to use Jorgensen's book. It also contains many other insights into tuning ET which Braide-White's book does not have. It also does not suggest using a pipe or reed organ or an accordion as a true reference for ET the way Braide-White's book does.

Finally, Jorgensen's book tells the truth that has to be told which in my opinion, makes Braide-White's book completely invalid and utterly useless for learning how to tune a piano. "Inharmonicity: the quality whereby the harmonic series deviate from theory by different degrees on various tones of a musical instrument. This in turn causes the intervals to deviate from theory and become distorted...Smaller or cheaper pianos contain more inharmonicity than larger or more expensive ones."

Two factors affect beat rates when tuning in ET: the actual amount of inharmonicity present and the choice of the size of octave within which the temperament is constructed. Braide-White's book offers three different octave tests, each one would produce a different sized octave. So, not only does BW not tell us that these exact beat rates we must use are wrong on one count but also wrong when you consider that he doesn't seem to know anything about octave types.

If you own Braide-White's book, read it for a good laugh, then sell it to some sucker good and cheap. It is thoroughly useless, incomplete, incorrect and obsolete.

I've seen Mr. Jorgensen's book cited in A LOT of serious resources on musical acoustics. I am sure it will be cited and used for a long time, it is a massive piece of research, in a rather obscure field.

An interesting thing: Mr. Jorgensen got into the piano tuning world with Braid White's correspondence course.

Bremmer is being deceitful. I read the first paragraph of his edited quote of Dr. White and see no reason to be mislead any further.

Bremmer quoted:

“The practical methods have been worked out...but in any case, the margin of error in ordinary practice has often been too wide. It has often been found that, even with what is called a good tuning, a systematic error of more than 2 cents is commonly detected.”

Since the emphasis in this Topic has been on methods for setting the temperament it would seem that this edited quote indicates that Dr. White acknowledges that his method is not accurate. But if we read what is substituted between the word “out” and “but” by dots AND WHAT WAS COMPLETELY OMITTED IN OTHER PLACES, Dr. White’s meaning (rather than what Bremmer wishes others to believe his meaning is) can be understood.

From Piano Tuning and Allied Arts by William Braid White, Mus. D. Fifth edition 1946, Appendix III, page 228 (omitted test in bold):

“The practical methods that have been worked out for the purpose of enabling tuners to realize frequencies in accordance with these ratios, can be brought to a high level of accuracy; but in any case, the margin of error in ordinary practice, has often been too wide. By methods now to be described, it has often been found that, even with what is called a good tuning, a systematic error of more than 2 cents throughout five of the seven octaves of the keyboard, is commonly detected.”

Now a reader can make up their own mind what it is that Dr. White means. What it means to me is that his method can be used to create an accurate tuning, and is accurately created in two octaves, but sometimes not in the other five. Which two could be correct and the other five wrong? Obviously, the first two octaves that are tuned, including the temperament.

I will be notifying the Moderator about Bremmer's attempt to deceive.

Let me quote the second paragraph of the preface to the fifth edition so that it will be clear that Dr. White’s method does not include estimates (bold added for emphasis):

”I have taken the opportunity thus presented, to go over the entire test with great care, clarifying obscure passages, simplifying and condensing some of the explanations, correcting errors that had crept in, and above all, expounding in detail various practical checks and tests whereby the accuracy of the tuner’s work may, as he proceeds, be effectively determined, step by step. Most of these tests have been worked out in the course of my own theoretical and practical studies.”

Having preferences for a particular method is understandable. And honest disagreement should be encouraged. But deliberate deceit is outrageous.