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Bill:

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

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

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

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

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

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

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

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

For calculating the difference of two notes in cents:

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

For calculating the theoretical frequency:

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

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

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

For example:

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

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

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

Doel:

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

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

All:

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

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

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

And from page 85:

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

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

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

You must mean 0.3 cents.

Here is the link, again. Look for yourself:

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

Hello.

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

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

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

Regards, a.c.

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

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

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

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

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

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

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

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

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

Very interesting!

Have you tried this tuning? How does it sound?

It is easy to measure if you have an ETD.

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

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

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

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

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

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

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

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

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


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


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

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

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

Regards, a.c.

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



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




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

Alfredo,

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

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

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

Unright,

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

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

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

Or this other one:

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

Bye.

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

Kees

Kees,

Excellent demonstration!

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

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

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

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

Kees