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Thanks. I added the 2:1/4:2 mix to the demo list. Can you still hear the 6:3 beats?

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

Kees

Jeff, I can see where you're going (I think): People are getting caught up in the numbers. Of course, purely mathematically, 2^(1/12) = 4^(1/24). But you're making a different point (I think).

"Pick an ET model, see what pure or equal beating intervals define it, apply those intervals to a tuning of an instrument with iH and the result will be a certain amount of stretch."

For example, if you decide to tune an ET inside a pure 3:1 P12, you tune a real, inharmonic, pure 3:1 P12 on this piano, and divide it into 19 equal (progressive) parts. This temperament would be called a "19th root of 3" ET for this specific piano, because the 3:1 P12 is beatless on this specific piano.

However, I see one problem. This notation only works for n:1 invertals, i.e. 2:1 octave, 3:1 P12, 4:1 double octave, etc.

How would you express an ET inside a 4:2 temperament octave or 6:3 temperament octave, or a 4:2/6:3 compromise? 12th root of {what}?

You got it, I think !!!

With the huge advantage they are easy to tune and they give you direct access to the 12th -

Can be confused with 6:3

funny that double conversation

Thanks for the recordings

Thanks for posting your figures. Is there a way you can vary your calculation to make all intervals progressive, or do the variations in iH preclude that?

I think you have already indicated you can't but can you give an example of iH values causing problems away from the break?

The iH seems to preclude the possibility of fully progressive intervals. I am running a test using progressive narrow fifths at the moment and it is proving difficult. I will post my results and the iH values as well.

I must add that I am using the classic iH formula with corrections for the lower six partials from A0 to about E4. I am working on a test using Robert Scott's modified iH formula as well. It shows great promise and may end up being the best fit for a generic iH formula that does not take into considerations bridge anomalies.

The classic formula iH curve is about as perfect as one can get above the bass bridge break - essentially straight, though on very close inspection you can see the string diameter change points across the compass.

Cheers

It will probably pass unnoticed because it is a little OT of the thread

How about the 36th root of 6 divided by the 19th root of 3 or something. (Or something... I'm not thinking clear today - Dentist. I just might win the Jack-o-Lantern look alike contest this fall!)

Here is an example using progressive P5s:



Note that I was able to create precise matches to the theoretical P5s. However, it is important to understand that the iH shown has been corrected for anomalies in the first six partials for most of the notes. If you were to use the data indicated without those corrections, the beat rates would be different.

Note also that the M3s are not progressive, but actually still a pretty good fit. The P4s are not even close.

This is only one of an infinite number of progressive P5 tunings, but it seems to show promise.

I have more trouble getting a coherent sound using progressive M3s than using progressive P5s. This may come from my tuning mostly UTs in the past, where I am listening to the P5s and taking whatever M3s result.

Beats are slowing audibly. I am not sure I understand how you compute. It is only one level of partial match .?

i was said the best way to use credible partials was yo examiné each string and use the most accurate one for the display (etd)

Doh!

(2/1)^(1/12), 2:1 octaves
(4/2)^(1/12), 4:2 octaves
(6/3)^(1/12), 6:3 octaves
(((4/2)+(6/3))/2)^(1/12), equal beating 4:2, 6:3 octaves

Prout:

Something seems wrong. Maybe it has to do with: "the iH shown has been corrected for anomalies in the first six partials for most of the notes."

First, since the iH is progressive, I would expect that it is possible to have all RBIs progressive if not the SBIs as well.

When I use your iH and offsets for A3 and A4 I get an octave wider than 6:3. But according to you table, both the A3-D4 P4 and the D4-A4 P5 have identical beatrates of 0.99; the octave must be a pure 4:2.

Any explanation? Is it that the individual partials are that whacky?

Jeff:
Yes, the individual partials are wacky, which gives a pure, beatless 2:1 and 4:2 A3-A4 octave with a narrow 6:3 A3-A4 by 1.4 bps.

For that test I chose to use the theoretical values of the P5 and P4 on A3-D4-A4 as a starting point which resulted in the above data.

I could have compromised and stretched the octave a bit to create a 0.5bps wide 4:2 and a 0.5bps narrow 6:3 which yields a slower P5 by 0.3bps and a faster P4 by 0.3bps, but when I start doing that the tuning tends toward Reverse Well.

Keep those fifths narrow.

Prout:

Then you are demonstrating what non-typical partials do to beatrate progressions. I don't see the point. You are not showing how iH affects beatrate progressions, just what "background noise" does. Is that your point? It may be a good one.

In a way, I guess that is my point. Unless my methodology for measuring the individual partials is wrong, it seems that there are statistically significant departures of the various partials from the theoretical values when using either the standard iH formula or R. Scott's formula. Those anomalies force the tuner to make a decision about how to tune a particular piano.

So the question still is, as has been discussed on this forum so many times, what is the best way to tune an ET that is consistent across the compass and also coherent?

Coherence, to me, as a pianist, seems to come mostly from the interaction of octaves and fifths, especially when using the damper pedal. A single note played with the damper lifted, needs to have the rest of the piano helping it. Having tuned various temperaments on my piano, each one still had to have the octaves and fifths sounding well together.

Cheers.

I would call that resonance. I think a piano resonates more when the strongest partials are near to equal beating. The lowest partials are of course the strongest, until you get low enough in the scale that the soundboard does not vibrate that slow. Then the strongest partials are higher ones.

My answer to your question is pure 12ths with prioritizing the P5 across the breaks. And including the octave above the lower note when tuning wound strings. Others have other answers.

Good word - 'Resonance'

I would like to think that everything comes into 'tune', but, in reality, it is the calm 'airiness' of the piano that is so satisfying when well tuned.

Thanks for your thoughts.

For me the problem when dealing with a jump in the iH is that the relative distance between partials changes abruptly. If we pass from wound to plain to strings the higher iH in the plain string makes its partials be more distant to each other. If we chose to tune based on the 4th partial we split in two the deviation of partial 5 (M3s) and partial 3 (P5s).

Prout, haven't you cosidered a tuning based on a smoth progresdion of 4:2 octaves or 4:3 fourths?

IMO, such a tuning will give more coherent M3s and good P5s.

prout, how did you get those figures for those tables? Did you measure them on your piano with your software? Are they repeatable?

Also, I suppose you derived the iH values from the partial data. If so then they will be subject to an error term.

I would be interested in seeing similar figures comparing chromatic M6th given evenly progressive M3rds.

How far up do you usually take the 4:2 octaves?

I usually start with 6:3 on A0 transitioning smoothly to 4:2 by A2 and then transitioning to 2:1 above A4. Even with that conservative treble stretch I am 26 cents sharp at C8.