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I was just curious as Mark's filtered analysis seemed to indicate that all of 2:1 4:2 6:3 and possibly 8:4 were pure. I just wondered if the iH numbers were "close" to the magic value that maybe a few more partial pairs could lock, rather than the one and only one your analysis proves.

Paul.

F3 has false beat at partial 8, 6bps. Partial 8 is also very weak.
F4 has false beat at partial 4, 0.7 bps.

Looking at spectrogram for F3F4:
2:1 no beats
4:2 0.5 bps
6:3 0.5 bps
8:4 0.7 (could be false beat)

Tests bps:
F3G#3 8.9
G#3F4 9.5
F3C#4 10.5 (hard to see)
C#4F4 10.5

There were no 2:1 and 4:2 tests.

Kees

Thanks for the analysis Kees. I had a feeling today's recording was a tad worse than yesterday's. I wonder if any of the false beats came from the mutes? Nevertheless, it was interesting to see the same bps for the 4:2 and 6:3 partials. I don't suppose you could extract the iH values? I only have the entropy tuner and the iH values seem to be in quite different units to those used by Tunelab (which I don't have access to anymore).

Paul

Jeff,

You too were willing to give it a shot. I like that: "a change in timbre within the area that would be considered beatless or pure." We're getting a little closer but I'd be interested in even more description of your final decision to accept THAT octave sound (initially of course, and subject to change as circumstances dictate).


Rick,

You too made some interesting comments about "real world tuning", etc. Very true.

Does anyone ever have a circumstance when they would accept an actual "beating" octave in midsection of the piano? Just curious.

Very few seem to want to go out on a limb here on this subject. Interesting.

Pwg

Doesn't the entropy tuner measure each individual partial rather than fitting them to a one parameter model (the one parameter being the IH "constant")? Anyways the ratio of F4/F3 inharmonicity is independent of units and the is the determining factor.

Having pure 2:1 and 8:4 (according to test) and equal beating 4:2 and 6:3 is impossible with the usual model. You can get 2:1 and 8:4 to be both pure if IH ratio of F4 and F3 is 16.6 which is not realistic.

Kees

Hi Peter,

I wasn't sure how or who wrote the post quoted. I'll assume it was you.

I'll be more specific. I think we agree re:RBI tests. I think you thought I was implying just listen to the 6:3. No. We have to use the RBI tests. Those are what are more precise than just tweaking the octave by ear, like you said.

1) Tune a pure 4:2 (M3 = M10)
2) Test the 6:3 (m3 and M6)
- If the 6:3 is pure, leave it. (m3 = M6)
- If the 6:3 is barely narrow, tune it as a wide 4:2, narrow 6:3. (M3 < M10 and m3 > M6)
- If the 6:3 is very narrow, leave it as a pure 4:2, very narrow 6:3. (M3 = M10 and m3 >> M6)

= means "beats the same speed as"
< means "beats slower than", barely noticeable.
> means "beats faster than", barely noticeable.
>> means "beats much faster than", obvious.

I hope that is clearer.

P.S. I would leave a slow beat in an octave in the low midrange if the beatless octave would sacrifice the octave + fifth, octave + fourth, two octaves + fifth, and triple octave.

Here are some theoretical plots of the beat rates of the F3F4 octave at various partials, when 4:2 is tuned pure as a function of the inharmonicity ratio of F4 and F3.


Here's a zoom into the interesting region:



There seems to be a critical region near the ratio of 3.15 where 2:1, 4:2 and 6:3 are aurally pure. So on a piano which such a steep curve aurally pure octaves at 2:1 4:2 and 6:3 are possible. Other close match is Steinway D with ratio 2.7.

Kees
Note added: Looking at the IH data I have from various sources I only found a Mason&Hamlin which has an IH ratio close to that, 3.14, usually it is 2 or less. Other close one is Steinway D at 2.7.

This is my simple answer: I tune octaves as beatless as possible while also ensuring that other perfect expanded intervals are simultaneously as beatless as possible. Tuning is all compromises. Such is life. Nothing is perfect.

Um, I think I gave the answer:




I don't tune so an octave sounds like THAT. I tune for pure 12ths by listening to the ratio of the 4ths and 5ths. It is the pure 12ths that decide how an octave sounds, and they always sound great!

But perhaps you are asking about the FIRST octave that is tuned. THAT octave is the result of tuning up a 5th from the initial pure 12th. And if I don't care for the sound, I may adjust the 5th. I may use RBI tests to see what is going on, like is there a large difference between the 2:1 and 4:2. But it isn't about a precise sound of the octave. It is about the best sound of the 5th. the 5ths are much more "touchy".

And the same thing goes as the original 5th temperament is expanded until a full 12th is available: I will listen to the resulting octaves and use them to point out errors in the temperament and judge how the ratio between the beating of the 4ths and 5ths should progress. Generally the 5th need to progress faster. Experience is a great teacher, and I know that when I complete the first full 12th I'll be very close to the proper 4th and 5th ratio.

Peter, I am sure it is foreign to think of the temperament octave being made of 4ths and 5ths rather than the temperament octave being divided into 4ths and 5ths (and other intervals). But when you really consider the "old books", such as Dr. White's, that is what they proscribe: using SBIs to create an octave. If the resulting octave is good, the SBIs are correct.

So I gues MY answer to you question of how I decide to accept THAT octave is: it is the one that is acceptable with a pure 12th and a good sounding 5th.

Re: Kees graph showing pure 2:1/4:2/6:3 at iH ratio 3.15.

Quick calculations from Tremaine Parsons scale collection.
http://www.goptools.com/gallery.htm

Mason and Hamelin 6'4": 3.33
Steinway Upright: 2.65
Kimball Upright: 2.86
Bluthner 5'7": 2.55
Baldwin 9": 3.16

Mark, Chris, Jeff:

BINGO! Those are real world, practical descriptions of what we are required to do in tuning.

In other words would you all agree that what we are really doing is evaluating the MUSICAL qualities of ALL the intervals involved in order to make our decisions on octave width?

This is what I mean by the ART of what we do. Those who use ETDs (if they have not already learned to do so) need to incorporate this aspect into their work. They need to go beyond the numbers and LISTEN.

I am saying this for the benefit of those new or less experienced who are reading all this stuff. I know already that the seasoned pros know how to do it and do so regularly. But its important for newer ones to more fully understand the depth that we are trying to achieve here in tuning. This could be difficult for a complete non-musician, but it must become part of our MO if we are to truly succeed.

Also, I would suggest that we all go back to the very beginning of this thread and read it again before it morphed into a lot of technical stuff that is really hard to apply in reality. Some very good comments were made right at the outset.

I hope others will have the courage to also put in to words exactly what the are listening for so as DECIDE what to do with both octaves and other intervals. IOW What makes you smile and say "yes, I like that," and move on? And, what sacrifices are you willing to make if things don't seem to be working the way you envision?

Pwg

In those plots I posted I randomly fixed the lower IH (in tunelab units) to 1. 0.3 is more realistic, so all those beat rates on the vertical axis should be divided by 3 for a more realistic ballpark.

This will of course extend the "aurally pure" window around the 3.15 ratio.

So it seems there are plenty of pianos where those octaves can indeed be pure!

Kees

In my experience this should be a great move forward in understanding piano tuning and Inharmonicity. This is not the only forum where professional technicians have argued...well no, they didn't argue. This is argument. There was no place for arguement in some of the other discussions I have had with other professional piano technicians. In their opinion, it was just not possible to have pure 4:2 and pure 6:3 in a piano. End of story. I am happy that there are people like you Kees who are interested in advancing knowledge and not just keeping the status quo.

Yes, my numbers are for F3F4. The ratio increases as we move up, so more pianos would be tuneable as pure 4:2/6:3 if we looked at A3A4. It is not uncommon for me to find small octave spread (pure 4:2/6:3) for A3A4 and medium octave spread (wide 4:2/narrow 6:3) for F3F4.

I completely disagree. I think you are doing a big error. Not in your calculations. I'm sure your maths are OK. But you make an error when thinking that you can take B values to calculate beat rates of specific intervals.

I guess this is why I do not like the tunings I get from TuneLab while I like the tunings of Verituner. TuneLab computes a B value for a string by measuring its actual partials, and uses this B value to estimate the B values for other strings and finally to compute a tuning curve. Verituner measures each partial of each string and computes the tuning targets for all notes. It uses the actual measured partials to calculate the tuning.

I have a lot of troubles trying to express my ideas in english, so I'll illustrate this concept with a concrete example.


Here are the B's I've just measured of three pianos which are now at my shop:

August Förster Studio






Hamilton (Baldwin) Studio






Steinway & Sons Concert Grand






Ratios for these 3 pianos are:

August Förster = 2.18
Hamilton = 2.24
S&S = 2.56

Maybe these ratios are in the desired range to make it possible, theoretically speaking, to tune beatless 4:2 and beatless 6:3 octaves. But, and this is a big but, the actual partials as measured by TuneLab and showed in the pictures above give the following widths for the 6:3 octave when 4:2 is tuned pure:

August Förster

4:2 = 0 cents
6:3 = 2.90-4.24+5.96-8.69 = -4.07 cents

4:2 pure, 6:3 narrow by 4.1 cents



Hamilton

4:2 = 0 cents
6:3 = 3.02-3.08+4.91-7.49 = -2.64

4:2 pure, 6:3 narrow by 2.6 cents



Steinway & Sons

4:2 = 0 cents
6:3 = 2.38-2.71+4.19-5.27 = -1.41 cents

4:2 pure, 6:3 narrow by 1.4 cents


None of these pianos can be tuned with pure 4:2 and pure 6:3 octaves, despite their B values may be in an appropriate ratio range.

I've measured with Verituner the octave spread of these pianos for A3A4 with similar conclusions.

And more important: I've tuned these pianos, not once, but several times, the August Förster many times, and there is no way to tune them with pure 4:2 and pure 6:3 octaves in the temperament area.


We are faced here to the fact that theory is using a mathematical model that may work fine in general, to calculate a tuning curve that can make the piano to sound in tune, but which can not be applied to specifical isolated pairs of partials, i.e. 4:2 and 6:3 of the F3F4 octave. This mathematical model works fine for a tuning aid such as TuneLab, but is not adequate to conclude that:

"there are plenty of pianos where those octaves can indeed be pure!"

Irregularities in the distribution of partials can not be ignored by using a B value which represents the inharmonicity of a string as a whole and is no other than a "mean amount of iH" assumed for all partials of this string.


When tuning an octave at a piano you won't hear the calculated partials from estimated or computed B values.


To make such a study and take such a conclusion, you should use actual measured partials because that's what you hear when tuning an octave, this is obvious, for me at least.

I could be wrong but I think you making a sign error. For the S&S example we have from your post:

S&S
IH(F3) = 0.232
IH(F3) = 0.596
ratio = 2.56

Partial offsets:
F3_4 2.71
F3_6 5.27
F4_2 2.38
F4_3 4.19

To make 4:2 match raise F4 by (2.71-2.38).
So now F4_3 is at 4.19 + (2.71-2.38).
F3_6 - F4_3 = 5.27 -4.19 -2.71+2.38 = 0.75 cent.

Taking F3 at 174.6Hz, 6:3 bps is 6*174.6Hz*log(2)/1200*0.75 = 0.45bps.

Calculating just from the tunelab IH model I get bps=0.4bps which is very close.

On another note I did an extensive analysis of Prout's data set quite a while ago and concluded the partials fit the usual IH model very accurately with almost no exceptions.

Kees

Wouldn't the 1:3.15 ratio hold true only if the Tremaine Parsons iH curves were determined using Tunelab's equations? If he used Young's equations, which I think likely, then wouldn't the ratio need to be 1:4?

Good point. Tunelab IH values and Young IH values approximately differ by a constant, though the constant varies somewhat over the scale. If you accept that the tunelab model is more accurate then you could simply take the Young ratio as an approximation to the tunelab IH ratio and 3.15 will still be the best estimate.

Kees

You are right, I made a sign error. The correct widths for the 6:3 octaves are:

August Förster 6:3 = -1.66 cents
Hamilton 6:3 = -2.52 cents
S&S = - 0.75 cents

None of these octaves can be considered as beatless.

As is correctly predicted by the tunelab inharmonicity model, as the IH ratios are not close enough to 3.15. Though the S&S 6:3 beat rate of 0.45 is pretty good.

So I consider your point that these IH models can't predict beat rates correctly and that the individual partials need to be considered refuted.

Kees