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It depends on why you believe octaves should be stretched. There are a lot of people who make octaves wider as they go up the scale because they have tin ears. Even Mozart recognized and mocked that tendency.

The whole idea of tempering octaves makes no sense. The softest possible octave is pure, so what else could you be tempering?

Well, 2:1 octaves make octaves go wider going up the scale cent-wise. But I know what you mean and do so delibrately with 4:2 octaves until the point that 3:1 twelfths are reached, which then produces narrowing 4:2, but wideneing 2:1 octaves. Not a trace of inharmoniousness to these tin ears, though.

Whatever that means.

I see this fact as follows:

Problems begin when one starts measurements of frequencies. As long as we hear to tune, everybody knows what a clean octave is. But when we take measurements of the frequencies we find that fundamnetals of the two notes of an aurally clean octave are not in the ratio of 2:1 but in a greater ratio. So that clean octave is not really pure, in fact it is stretched.

Tooner will call this "natural stretch" because the octave sounds clean.

But problems do not finish here. There are in fact no pure octaves at all, that is why I talk about clean octaves.

In fact when playing the notes all pairs of nearly coincident partials are sounding simultaneously and they don't match with each other. So is iH, if you tune a 4:2 octave then 6:3 will be narrow and 2:1 will be wide, or whatever else depending on the iH of the strings involved. And you can have anything but a pure octave.

So what can we do? We do our best to make it sound good. That implies making decisions about what is the best compromise in terms of harmoniousness. That is the meaning of "tempered octave", an octave where the fundamentals of the notes are not in a 2:1 ratio but which sounds as we want.

So in fact we can only tune tempered octaves.

Now, if we accept that the perfect pure octave does not exist in piano tuning, then how much we must temper octaves? Natural stretch? Beyond natural stretch? It is a matter of priorities we give to all other intervals in our tuning.

Onlypure favors 12ths, mindless octaves tunes a balance between 15ths and 12ths, some prefer to tune double octaves and even triple octaves, some others strive to clean single octaves, etc...

But in that respect all octaves, no matter how we tune them, are stretched.

Gadzar:

You may like this tidbit. When an octave is tuned so that it is an exact compromise between the 4:2 and 6:3 octave, the 2:1 octave also beats at the same rate! Pure enough for 'ya???

Of course, the pitch of a piano string does not remain the same as the note decays, let alone its harmonics, so none of this pseudo-physics and pseudo-math makes any sense. The beats do not remain the same, because the wave-form varies with the decay. From a practical sense, you cannot distinguish fractional beats once they beat more slowly than the note decays, and probably a lot sooner than that. So it may be that you can get an octave to be "pure" within those limitations.

For me: Pure octave = Beatless octave.

Even in such a case where 2:1 would beat at the same rate than 6:3 and 4:2, they are beating all 3 of them, so this octave can by no means be a pure one!

And I am not so sure if 2:1 will beat at the same rate than 6:3 and 4:2. That depends on the real frequencies of partials 1, 2 and 3 of the upper string, and partials 2, 4 and 6 of the lower one. I would be surprised if they all match in such a perfect way.

Again, I am talking about real strings, made of steel (and cooper)! I am not talking about a simulation in a computer using the Young formula for estimating each partial of theoretical perfect strings, regardless of all imperfections of the real world!

BDB:

Seriously, you are right. Until the theory of inharmonicity was popular I do not think there was ever a mention multiple beating of octaves (although I did notice multiple beating of 5ths during my tuning lessons, which was nothing new to my instructor.)

As you say, if it sounds good, it is good.

OK, you have an ETD. Try it out and let us know. It does give frequencies of partials, doesn't it?

I need no ETD to know what is known by every piano tuner of our times! Every body is tuning using more or less stretch. Nobody tunes nowadays without stretch. Even if they don't notice they are doing so.

iH has existed long before and octaves were beating then as much as now at all their coincidental partials even if tuners did not speak about multiple beating octaves.

Tooner, I have a question for you:

How does your ideal simulated piano sound like if entirely tuned to balanced 6:3/4:2 = 2:1 octaves?

For what I know about the real world pianos I tune: the low bass will sound sharp, the tenor will sound good, the treble will sound sharp, too stretched, and the high treble can't be tuned that way because 6:3 octaves are not audible.

But in a computer all is possible, so how would it sound?

Hi all !

Yes older tuners did not worry about the iH theory. eventually many pianos had less iH (just a guess, but the strings where softer, out of the pre 1900 period where they harden them a lot)

so the high beats where certainly less noticeable, and in any case they did not learn to listen to them, I never see mention of 8:4 ,12:6 , etc in literature . I have to look at that again, but only basic checks where used.

Gadzar, (Zapata !) you asked about tuning with sustain pedal engaged, I had a colleague retired concert tuner who did that in the 90s he told me when he was tired, but it worked fine, and showed me that natural resonance can be used with all the piano, not only octaves doubles etc.
i like to use that when tuning Chas, in the treble because the resonant quality is easy to recognize. I use the first inversion of the minor chord and look for the resonant spot a 12th above, and it work for me. in the basses I have not find a similar trick, but Alfredo showed me the use of the M14th 2 octaves less one tone)which is a well sustained beat and have a particular "body", very useful for low basses

my colleague was mostly using 4ths in the basses, and a M third at the top of the octave for the treble, then check a sustained mix of major minor chord .
I did not find that particularly relaxing but checking major/minor against notes 2 octaves above/below is a method as another. his tunings where very lively and appreciated, the piano fall by itself in some kind of resonant spot, that is what I like with the Chas method, very similar by some aspects.

best wishes to all, dont fight !

I agree. In the instructions of all the ETDs that I have read, they advise to tune to the first reading when it varies in time.

Kamin also adivises to tune the attack of the unisons. I don't know if he means to avoid decay, but anyway it does avoid it.

And in general, it is considered as a good practice to repeat regularly the notes when tuning, not only because of decay but also for stability, string rendering and other issues.

But even, or mostly, if it sounds pure: it is stretched. You only have to measure the fundamentals ratio to confirm it.

Jeff,

Earlier, you wrote (colors added by me, for clarity):



But then you wrote:



Now what will it be? According to the second quote, one can either have three contiguous 2:1 octaves, or a beatless 8:1 triple octave, but one can't achieve both simultaneously. Hence, I conclude that:
1) 2:1 octaves are not sufficiently wide to accommodate inharmonicity (which seems in direct contradiction with the red part of first quote), and
2) octave stretching is therefore done not only to satisfy the human ear's pitch perception (see blue part), but also to accommodate iH.

Pardon my reading if it's wrong, but this is what I conclude from the second quote, as also illustrated by Patrick's drawings.

The attack have to be tuned because it deals with the enveloppe, but the thick part of the decay is also tuned to "body" the tone. There, partial coupling comes in play.(in unisson)

I guess that what is addmited is that 2:1 relation cover for iH correction. and all added sech is considered "artificial" , or result of the interval ratio used.

A large temPerament octave also helps to cover for mistakes when tuning above and under. with a purer initial octave is harder to realize progressive intervals, the added strech helps the intervals to jump in the ih spectra and sound purer

Gadzar:

My simulation only produces numbers, not sounds.

Thats the limit of the thing, but it is interesting anyway.

The fact is that in real word there are more parameters that comes in play than the theoretical or even real ih .
The notion of "acoustical justness" seem to describe that correctly, to me. It is the fact that the final jUstness may relate to a model. For instance "pure fifth tuning is non existent in regard to the iH theory, But acoustical pure fifth (or twelve) is a concept that can be grasped, and it have some existence .

The same apply to a pure octave, it is simply difficult to learn to recognize the good spot by ear, so the tuner is avidely asking for methods tricks, and checks to have a little control on that.

It certainly work to some point , but it also give bad habits, and lot of frustration, knowing that our tuning is only a bunch of compromises is not ve y good !

So, to me the quality of the octaves and other intervals is putting the resonance of the tuning at a more or less high level in the spectra. It is totally acoustic dependent, (room, voicing, kind of scale) unless you use a self refered ratio that auto replicates within the scale.

Hey, Mark:

I can see why what I said could be confusing because I was saying something very, very basic.

A piano tuned with 2:1 octaves have harmonically satisfying octaves. They are harmonically satisfying because they are 2:1 octaves. This is to say nothing about other intervals. But the same thing could be said about other intervals, like, “A piano tuned with pure fifths have harmonically satisfying fifths.”

So let’s look at your conclusions:

“ 1) 2:1 octaves are not sufficiently wide to accommodate inharmonicity”

Of course they are, for 2:1 octaves. But if accommodating any other interval produces a different stretch, then it is insufficient to accommodate 2:1 octaves.

“ 2) octave stretching is therefore done not only to satisfy the human ear's pitch perception (see blue part), but also to accommodate iH.”

Well it depends on what octave you are talking about, doesn’t it? Suppose I am talking about 12:6 octaves and lets say that is too much stretch to satisfy the ear and makes other intervals unharmonious. Then I could say, “Octave compressing is therefore done not only to satisfy the human ear's pitch perception, but also to accommodate iH.” Or if by octave we mean a 2:1 frequency ratio, then when tuning 2:1 octaves we can say “Octave stretching is therefore not sufficient to satisfy the human ear's pitch perception, but is sufficient to accommodate iH.”

Consider this. Even before iH was known by tuners, octaves where stretched beyond what was harmonious to satisfy the pitch sense of the human ear. That is what is still being done. Sure there are many stretch schemes that use the sharpened partials that are an effect of iH. But iH is a tool used to stretch octaves beyond what is harmonious. Stretch is not something used to "accommodate" iH (can iH be unhappy?).

I agree with the last sentence.
What is less useful than I believed is all that partial match progression to be used for makIng a skeleton for tuning.

Facts are real, but the difference is so small that a wrong basic concept can push a tuning in a no end way easily.

I heard lately a concert on a grand piano with a not so singing tone, and a very even, but harmonically inintelegible tuning. The piano was 'just' , by evidence, but at different moments I felt the pianist where lost in their speech, to the point they played a note for another.
Then, some unisons went out of tune.!
That kind of situation does not arise with a congruent tuning, that evolve in an harmonious way even if temperature change (I have no proof for that but I strongly believe it, a piano have some form of acoustical stability availeable, hopefully)

That is precisely my point. We (me included) are talking about sounds and you (me excluded) are talking about numbers!

But, even if you can try to imagine how your numbers will sound when transposed to a tuning, the fact is that you don't hear them!

So we do not talk the same language!

Fine, let’s talk about sounds.

I have never, ever heard a beating in a double or triple octave when tuning two or three stacked octaves to each sound as pure as possible. However, when tuning a triple octave this way, the upper note sounds a bit flat when compared to a chord three octaves lower.

What do you hear?