Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum How to tune Tempered Octaves ?

Pat:

I understand that it is called a “Schubert”, not a Schumann, tuning. I have been moving in that direction lately, but it is more subtle than it may seems.

On a small piano 4:2 octaves will produce pure twelfths in the temperament. On a larger one they will produce pure twelfths an octave or so higher. Since there is little difference between a 4:2 and 6:3 octave on a large piano, there is even less to be gained by tuning a compromise between 4:2 and 6:3 octaves in the temperament. And once 4:2 octaves, with the help of iH, have produced pure twelfths when expanding the temperament upwards, isn’t there enough stretch by continuing pure twelfths?

And let me “beat you to the punch” and agree that a little difference can mean a lot. But I am not so sure the difference that extra wide octaves in the middle of the piano makes in the sound of the piano, provides what is needed to satisfy the ear pitch-wise as it provides what is desired to satisfy the ear contrast-wise. A piano that stands out can be heard easier.

In fact, I seriously wonder if by making the middle octaves extra wide if the ear is not indulged and expects even wider octaves on the extremes! I think there is more than what “meets the eye” in a Schubert tuning.

@Cashley: Good post, thanks - now I understand better what's bugging you I will reply ASAP.

@Jeff: You're right, and that's an interesting post you wrote. Maybe that's why so many temperament sequences just say something in the essence of "tune A3 to A4, and check with M3-M10. The upper interval should not be slower than the lower one." Thus, leaving room for octaves slightly wide of 4:2, but also for 4:2 itself. This would, just as you say, be different from instrument to instrument.

And thank you, "Schubert tuning" it is Now, what I've come to associate with that tuning is a very narrow temperament, then huge stretch towards the ends of the piano - some refer to this as "Carnegie Hall tuning" or use similar descriptive terms. In short, a tuning that might give sparkle to the upper end and depth to the lower end on 9 ft grands in a great hall, but that sounds VERY pitch-confused in, for example, using close-up mics on the piano (not to mention if it's a shorter grand.)

What you advocate though - using a narrower temperament octave and a more resonable stretching than I mentioned above - makes total sense. And I think you might be right about this:



I get that feeling from a 6:3-based ET temperament. I also don't particularly like the "looseness" of the harmony in the midrange in that kind of temperament stretch. To me, it doesn't hold together that well.

Yes, I seem to agree with all that above.

good reflexions.

In Chas the medium benefit from somehow close harmony, I suggest that the sparkle is due to the raised resonance.
Chas method is something like "let the piano find it itself, the tuner is only there to push in the good direction" that is something I like and the same idea I had while tuning, as many tuners say "let the piano tell you", but the 12/15 relation is taken in account, while only octaves and multiples are in a more classical approach.

That wasn't intended as a "conclusion". It was a statement of fact, and it actually concerned all octaves, not only the temperament octave. Right at the beginning of my post I said that any tuning must be stretched to accommodate inharmonicity. Then, later, I described what happens if inharmonicity is not accommodated: the bass will sound sharp and the treble flat.



Indeed, it is the answer.



I don't think we're moving in circles at all. You just have to find out what harmonicity is, and then you will see why/how it causes the bass to be sharp and the treble to be flat if the octaves are not stretched.



What do you mean by this? Inharmonicity is not a question, it is a phenomenon that can be measured, demonstrated and explained. But it seems as though you are not clear on what inharmonicity actually is?

Bill suggested earlier that one should read up on inharmonicity, e.g. on Wikipedia. Have you done so?



I would rather word it like this:
If F4 is tuned beatless to the second partial of F3 (2:1 octave) and if F5 is also tuned beatless to the second partial of F4 (2:1 octave), then F5 will be lower than the fourth partial of F3. Hence, the double octave will have a slight beat.



Yes, they can, and they were. So where is the problem, I'm wondering?

To me if the temperament is enlarged, the main result is that the railsback curve is straighter and stepper.


with a close medium, one will have to add stretch in the extremes in a more abrupt way - Ih tend to give a higher pitch to the note, than the pitch of its fundamental. Then, which pitch have to be tuned ? the one we hear (mix of all partials), or the fundamental, or a partial ? or some thing dispated on a screen ?


Ih is what gives relief (?) to piano tone, largeness
low iH scale have a straighter enveloppe and saturates less.

Mark & Cashley:

The following link is to an article that sums up the subject of octave stretch better than any I know of, but others may disagree with what it says - http://en.wikipedia.org/wiki/Piano_tuning#Stretched_octaves

Jeff, I read it and I agree that it is a good summary but that is all it is, a summary. It is like an introductory page to what would be a book on the subject. I noticed an external link at the bottom of the page that went to my website! Not that I mind but nobody ever asked me if they could put that there. The title, "Octave types and Distribution" is erroneous. The word "distribution" was only the file name I had given it in order to distinguish it from the parallel file I had made that had been edited for submission to the Journal. When I asked my webmaster to put the file on my website, he thought the word "distribution" belonged in the title. I saw that and had him take it off later but somehow it made it on to Wikipedia.

Wikipedia can be an excellent source for basic information but inevitably, it is a hodge-podge of information put up by whoever wants to put something up. I have no idea who wrote that article although it is basically fine. I noted that they used Jorgensen's book, "Tuning" and Helmholtz as sources.

At the end of the paragraph on stretching, it implies that the octaves cannot be stretched using inharmonicity enough to satisfy what the ear really wants to hear. I disagree with that because I have done it for a couple of decades now. It isn't right for every situation but the formula I used on GP's piano is one I often use: F#4-E5: octave equal beating with the 5th below the top note of the octave. F5-E6: double octave equal beating with octave-5th below the top note of the double octave. F6-B6: pure 6:1 octaves (the note being tuned is exactly in tune with the 6th partial of the note two octaves and a 5th below it). C7-E7: pure 8:1 octaves. F7-B7: pure 12:1 octaves. C8: pure 16:1 octave.

You may notice that as the scale is ascended, it keeps reaching back to the temperament octave for the reference. The partials involved at F5-E6 are 4th and 3rd. Then, they jump to 6, 8, 12 and 16. The higher the partial, the sharper the pitch. Using these partial selections for the high treble really will satisfy the ear's desire to hear those pitches much higher than they would be theoretically. When the stretched is always "pushed" as described above, the beating between single octaves in the high treble also remains in a tolerable range.

Hi Bill:

The reason I posted the link is because Mark and Cashley seem to be missing something very basic about stretch.

Yes, I agree that it is good as only a summation, and within the context of the stretch available with 2:1 octaves, I agree with the article: the stretch produced by inharmonic tones with 2:1 octaves is not sufficient to satisfy the ear pitch-wise in the extremes of the piano.

Of course, using higher partial matches will satisfy (or even offend such as with 8:4 octaves) the ear pitch-wise, but that would be beyond a summary article.

But when I consider the stretch scheme that you mention, there are only 6 notes (C7-E7 and C8) that use inharmonicity exclusively to obtain stretch. The remainder of the stretch scheme also uses fifths or extensions of fifths and would cause stretch even with harmonic tones because the fifths are being tempered less. This not a criticism of your scheme. I am just pointing out that it stretches by the use of more than a piano’s iH.

I can't speak for Cashley, but what might I be missing?

Mark:

Your statement: “I don't think we're moving in circles at all. You just have to find out what harmonicity is, and then you will see why/how it causes the bass to be sharp and the treble to be flat if the octaves are not stretched.” seems odd. Are you saying that iH makes the pitch of notes sound different and that the pitch must be changed differently than harmonic tones so that they sound in-tune melodically? Actually, I have wondered about this, but it does not seem to be the question at hand.

The Topics with Cashley can be confusing because so many different conversations are going on at once. Instead of jumping into the subject that you two are discussing I thought it best to just post a link to give you two more common ground.

I'd like to try to contribute with some simple charts of how inharmonicity affects octave tuning. My charts are rudimental, because they deal with inharmonicity as a standard factor, applied equally to all strings. We all know that this is not the case, some strings have more inharmonicity than others. Still, the graphs are clear enough to make the main points - much in line with what Mark described earlier.

1) Theoretically, the partials line up mathematically correct, and if this was true on the piano too, there would be no need for stretch (besides from subjective choices, of course).

Here is a graph comparing theoretical partials with partials affected by inharmonicity. Inharmonicity is due to the stiffness off the wire. Jack Stebbins thaught me to imagine the piece of wire closest to the termination points, and I agree - it really has a rough time trying to move.

The result of that stiffness is that it shrinks the actual wavelengths, and because high frequencies have shorter wavelenghts that are more sensitive to outer factors, they suffer more than the lower ones. Thus, the first partial is almost unaffected, the 2nd partial a bit more, the 3rd even more, and so on.

Here is my first graph, showing a theoretical line-up of partials vs. an actual (although exaggerated) lineup caused by IH:




This shows the piano tuner's octave dilemma right there. We can seldom, if ever, make more than one partial match.

Mark earlier described the results of theoretical tuning, ie matching the fundamental (1st partial) of the higher note in the octave with the 2nd partial of the lower octave. Let's see what happens when we do that to the octave C4-C5, when inharmonicity has come into play:



The second partial of C4 matches nicely with the 1st partial of C5, but above that we have problems. All crucial overtones (partials) above that 2:1 relationship are going to be hopelessly flat on our upper C5. Hence, C5 is going to sound flat compared to C4.

What about octaves downwards? Let's try to make a 2:1 match between C3 and C4:



... and the problem is not solved, only reversed. Again, the 2:1 match between C3 and C4 is going to sound perfectly beatless, but inharmonicity messes with the higher partials. above the 2nd partial, C3's partials are going to be higher than those of C4, causing the overall pitch perception of the listener to experience C3 as higher in pitch than it should be for a sound-wisely perfectly pure octave.

Cashley, I hope this graphs help you to see what Mark already said: a 2:1 theoretical tuning will leave the bass sharp, and the treble flat.

Many thanks, Pat, this serves very nicely to better explain my statement that Jeff questioned in the post before yours. I can't add anything more. (And I think I'll pipe down now, seeing that my posts are based on reading, not tuning (yet).)

Yes but we hear all the partials together, plus fundamental, and there is some attraction due to whatever effect. (the pitch is not really stable, it varies between attack and sustain, and iH of the note (not iH of the wire) varies with the voicing)

So a note can tend sharp or flat, depending of the harmonic context.

The fact that one level of partials is lined does not preclude of justness it only influence the beats produced by partials.

That approach is just to give explanations on the basis of the problem, and why pure multiples of fundamental will not give the right frequency (it does that pretty well, nice sketches, pat)

Due to that, we always try to avoid beats in octaves, while another approach (an easier one)is to accept them.

the next trouble is due to the human ear that is asking for extra high treble (due to the lack of perception of the high frequencies probably)So a piano tuning may have a way to fall in a high treble stretched in the most natural way.

Correct.


Now wait a minute! Who says that it is the higher partials that tell the ear what pitch a note is at? Someone else could say that it is the fundamental that tells the ear what pitch a note is at, and tuning to higher partials makes the higher notes too high.

So let’s take a look at what is held as a standard way to tune large pianos differently than small pianos. Large pianos are generally tuned with higher partial matches and smaller pianos with lower partial matches. This indicates that the ear does not decide what the pitch of a note is by the higher partials. Otherwise the same partial matches would be used for every size of piano.

Or we can listen to theoretical harmonic tones, such as on a common electronic organ. Play a very low note, and then a note as many octaves higher as possible. Now play the lowest note again, but this time play the note a semitone higher as many octaves as possible. If your ear determines pitch like most other ears, the note a semitone higher will sound more on pitch when compared to the lower note. If the higher partials is how the ear determines pitch, the lower note would sound more on pitch.

There are two kinds of stretch. One is due to inharmonicity and is required to make intervals harmonic (partials line up). The other is due to how the human ear perceives pitch. Tuning 2:1 octaves on a piano is perfectly sufficient for stretching the theoretical pitches for harmonic octaves, but is not enough to satisfy the pitch sense of the ear in the extremes of the piano. The article that I posted a link to explains this very simply.

Jeff,

Are you saying that if you tune, for example, C5-C4 2:1, then C6-C5 2:1, ditto C7 and C8, that C7 and C8 will have no audible beat(s) with respectively the 4th and 8th partials of C4?

No, I am not saying that, but they could very well be too fast and weak to be audible. (Especially considering it would be the 8th and 16th partials, respectively )

But let me ask you a question. Are saying that hearing beats and perceiving pitch are the same thing?

OK, then let's choose another example. My understanding is that the 8th partial of a low bass string can be quite audible. From what I've heard on my piano, the 7th one is quite audible in the extreme bass, so I would expect the 8th to be audible as well.

So, what would happen if you tune contiguous 2:1 octaves all the way down from C4 to C1 - would there be no beat between C1 (partial 8) and C4 (fundamental)? If, like you say, 2:1 octaves have sufficient stretch to cater for inharmonicity, then there shouldn't be a beat. No psycho-accoustic matters, just matching of partials. So, would there be a beat or not?



No. That's exactly the point. I've read about the psycho-accoustics of high notes, but all my points here are directed at inharmonicity only. To wit: from what I've read hitherto, 2:1 octaves are actually not sufficiently wide to cater for inharmonicity, i.e. they are not able to prevent beats between the low bass and mid-section (or mid-section and high treble) - in other words, to put it as succinctly as I can:

My understanding hitherto was that even before the psycho-accoustics of high treble notes come into play, 2:1 octaves cannot get a piano's extremes in tune with the rest of the instrument. They will actually, physically, beat, because of mismatched parials - irrespective of my perception of pitch.

Was this understanding wrong? (Question aimed not only at Jeff.)

Mark:

It is a two edged sword. If 2:1 octaves are not sufficiently wide to prevent beats between the low bass and mid-section ... THEN ... intervals between the low bass and midsection are not sufficiently narrow to prevent beats in the 2:1 octaves.

In your example, C1-C4 would beat about 2 bps on a typical piano when 2:1 octaves are tuned. And if C1-C4 were beatless, the 2:1 octaves could be tuned to each beat about 1/3 bps.

It is often mentioned about the need to use higher partial matches to prevent beating in large intervals such as double and triple octaves. Even if this was not the case (and I don't think it is a big deal), or if these intervals are not played, the ear still wants the stretch (or even more) to satisfy its sense of pitch not only regardless of iH or beating, but (I believe) in spite of it!

For instance, mindless octaves are often tuned, not for the sake of pure double octaves (these are deliberately stretched wide) but for the sake of satisfying the ear pitch-wise. And pure twelfths and even pure fifths are tuned for the same reason in spite of any beating that may occur in octaves or double octaves.

The reason for stretch is to satisfy the ear. Inharmonicity helps stretched intervals be more harmonious.

I think that should be "Stretch helps inharmonic intervals be more harmonious."

Not when the stretch is past the point of harmoniousness. When the purpose is to have the pitch pleasing to the ear, (regardless of the amount of inharmoniousness) the more iH, the more harmonious the interval.

"Inharmoniousness" what a word! I wonder what it could score in scrabble, if it is possible …