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I have asked twice what those different beat rates are supposed to sound like, and I have gotten no response.

Beats occur according to when the maxima and minima of two interfering waves coincide. When the maxima and minima of a piano tone occur is affected by the shape of the wave, which determines the partials, but it does not mean that the maxima and minima of the partials can interfere with each other to the extent that they differ from those of the fundamentals. It just means that the beats will not be nice, clean, regular sine waves themselves, just as the fundamentals are not a nice, clean, regular sine waves.

There are other things that affect how beats sound. There is the difference in volume between any two notes, which is why beats on a piano never approach silence at their minima, and there is the decay which overwhelms the sound of the beat as the intervals come closer to coinciding. These make "beatless" a relative term. I would like someone to demonstrate how octaves can be beatless on a piano. If it is such an important phenomenon, someone should be able to record it so we could listen to it and hear exactly what it is.

In addition, the iH is not consistent across the break, and, the lower partials in the bass are anomalous, due to bridge rocking or some other not well researched effects. Any tuning paradigm must include those factors - in the case of aural tuners , they use their ears, in the of ETDs, compromises must be made.

OK, go on. Lets suppose that you determine beat rates for a chosen model. Then how do you propose to extend this to actually tuning a piano?
Have you actually tuned a piano like this?

Unless I am completely wrong, it is not possible to use any size octave other than precisely 2:1 octaves for instruments that do not exhibit iH, such as a pipe organ. As a result, on a piano, the iH is 'a priori', after which a stretch can be imposed.

Well, no.

If I understand, Mr. Capurso designed CHAS to be valid in instruments without iH.

Fine aural tuning predates a complete understanding of inharmonicity. There was a time when piano techs didn't even know inharmonicity existed and yet they tuned pianos as best they could using the beat rates of the model, because that was all they had.

You say, 2:1, 4:2, and 6:3 can not all be zero, which is obviously true. However, what does that have to do with artfully tuning a clean octave aurally? The whole sound of an octave can have a clean sound. And the octave can be tuned clean while tuning 4ths that beat much the same as 5ths in the 4th-5th test of the 4:2 octave.

Well, that's a problem then.

Take linearly stretched octaves of 2.005 (as I believe is the CHAS model) of say 100 Hz-200.5-402.0025-806.015 on a organ. Because the partials are, in fact, harmonic, you now have the following beating sequence - 100-200-200.5-300-400-401-402.0025-500-600-601.5-700-800-801-802-804.05.

Now, if the octave were precisely 2:1, the sequence would be 100-200-300-400-500-600-700-800.

Which do you think will sound better?

Of course I have tuned a piano like this. And I have said how to go about it, by artfully tuning as closely as possible to the beat rates of the model and by doing so, to give the illusion of zero inharmonicity.

For my PTG Atlanta institute class, I have prepared tunings of the Pianoteq modeled piano to demonstrate equal temperaments at various recognizable widths.

As as been mentioned elsewhere on Piano World, Pianoteq supports the Scala tuning protocol, and it even supports 88 note tunings, and so I have used Scala to tune the Pianoteq piano note by note to a high degree of accuracy, and to many different tunings.

Obviously, it would be better to demonstrate these tunings with real pianos and that can be done, but with Pianoteq in a classroom situation, all the tunings will be utterly stable, and will be immediately available and comparable at the click of a trackpad button.

This works well for a decaying sound envelope. Not so easy with a constant amplitude sound.

ET forces the existence of a semitone ratio. Usually this ratio is supposed to be 12th root of 2, as the ratio of the octave is 2.

But iH in pianos creates multiple kinds of octaves: 2:1, 4:2, 6:3, 8:4, etc.

Ths the magic octave ratio of 2 is blown off.

In real pianos there are no pure octaves. They beat and they have multiple beat rates, one for each pair of quasi coincident partials.

The same happens for all the intervals, they all have multiple simultaneous beat rates. In P5s for example we have two distinctive beat rates namely 3:2 and 6:4
Theoretically there is an infinite number of quasi coincident partials for each kind of interval. Thus for the P4 we have 4:3, 8:6, 16:12, etc... In the real world only 2 or 3 pair of quasi coincident partials are audible at some specific locations of the piano's scale.

IMO there is no fixed ratio for the semitone that can be taken to tune a puano from A0 to C8.

The iH of a piano is not constant across the scale, it's curve has "hokey stick" shape.

Thus, I think the size of the semitone can not be constant but it must change from A0 to C8.

The fact here is that we can no more talk about an Equal Temperament.

Gadzar wrote:

"The fact here is that we can no more talk about an Equal Temperament."

Beat rates. Define equal temperament in terms of beat rates.

For me, the recently passed Bill Garlick provided the best definition of equal temperament for the modern world.

"In equal temperament on the modern piano there is not a single interval which is tuned just or perfect -- even including the octave. Due to the Comma of Pythagoras and inharmonicity all intervals must be contracted or expanded from perfect and will therefore beat... All the tempered intervals of equal temperament should gradually increase in beat speed evenly, ascending chromatically. It should be noted that such chromatically ascending progressions, particularly of M3rds and M6ths is a characteristic unique to equal temperament and distinguishes it from any other temperament."

-- Bill Garlick

Of course, but what does that have to do with exotic zero-ih models with weird roots?

Kees

"Of course, but what does that have to do with exotic zero-ih models with weird roots?"

Kees

I'm not so sure they are so weird.

They provide a good way of quantifying stretch, and executing their characteristic beat rate patterns evenly across the scale can promote coherent-sounding tunings.

It's impossible.

You can tune M3s in an even progression.
Or you can tune P5s in an even progression.

But in a real piano there are iH jumps in the scale and you can not tune an even progression of M3s and P5s.

You have a jump at the break, you have a jump when passing from wound strings to plain strings and you have a jump each time the diameter of plain strins changes.

So it is impossible to tune all intervals in even progressions of beat rates.

It sounds good in a vague sentence, but how do you propose to "execute their characteristic beat rate patterns evenly across the scale" exactly? Beat rates of which intervals?

Kees

Greetings,
I am going to guess that you never heard Bill Garlick tune a piano. He varied the sizes of his octaves as he got to them, after making an initial decision when he tempered the temperament octave, itself. It may be impossible for ideals to be met on paper, since the numbers are obviously indicating unevenness. However, in the real world,( where evenness is being measured sensually i.e., by listening), tuning can be perfected to the point where any inequity in the division of the octave is not discernible. This is assuming a reasonably scaled larger piano. It is possible to hide the errors in such a way that a clinical environment is needed to find them, but it is a rare ear that does this.

We spend a lot of time judging the evenness and accuracy of the tunings, but that last 1% is of little import to the musical world. ET has such a busy background that things have to be pretty uneven to be noticed in a musical setting. Even given that the thirds are in control of the tonal value of the triad, unless one is playing chromatic thirds and comparing them, equally tempered ones pass as identical if they are within a cent of each other. I use a lot of UT's, and the 12 or 15 cent thirds are never commented on. The 5 cent and 18 cent thirds, yes, but not those that vary by a cent.
Regards,

This is so true. A simple spreadsheet using 12-ET, which includes the measured iH of a given piano, will immediately reveal the variations in beat progressions. You can choose one interval to be monotonically increasing, but the others will then not be monotonic. Any theoretical ET model which does not involve iH will have all intervals increasing monotonically.

We had a long thread (or threads) on this subject a while ago which you may want to look up if you're interested. At some point it was proposed to relax these conditions to progressive M3 and M6 only. As it turned out nobody was able to actually tune a temperament octave with strictly progressive M3's, with the exception of Bernhard Stopper with a modified OnlyPure ETD.

Another remark I have is that that definition is incomplete, at least when taken to define what we commonly call ET. You have to put some restrictions on the octaves. Otherwise a semitone ratio of 2 would be ET according to this definition.

Finally the part "and inharmonicity" does not belong in the definition. Organs can also be tuned in ET>

Kees

Hi Ed,

Greetings.

I agree in all yo say. 100%.

And you guessed right, unfortunately I never heard Bill Garlick tune a piano.

Tuning a piano in ET is all about compromises. You compromise the progression of one interval to favor another one, and it is the balancing of these compromises that characterises a good tuning.

As you said the octaves are not the same size all along the scale of the piano and it is in that sense that I say ET does not exist. There are different sizes of the same interval in different places of the scale.

Here you go.

2:1
4:2
6:3
Equal beating 6:3 4:2

Inharmonicity according to measurements of Hellas Helsinki upright.

Kees