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I finally see what you've been trying to relay here. It's interesting and I've spent some time with it, although I don't think about it the same. It seems like there's a good deal of incorrect info on this above.

If you take 6:3 octaves down to the break, the spread will be greater over all intervals. The M3rd will have a faster descending beatrate than if you approached the break with 4:2 octaves.

Contracting to 4:2 octaves right as you get to the break, will then smooth the M3rds in most cases. The drop in iH will be counteracted on both sides instead of being compensated for on only one.

This is the principal trick with breaks.

I don't see why the octave test is important to you. I see this as just dealing with the knowledge of expansion and contraction over variable inharmonicity.

With an expanded temperament, M3rds will be faster and 5ths nearer pure. As you approach the break, the octave will be contracted to maintain M3rd progression, but since 6:3 or 8:4 is used prior, you won't have to go to a 2:1 to maintain progression. The 5ths crossing over will be more narrow, so stretch compromises between the 3rds and 5ths will be necessary. The 5ths are always more important in my opinion.

Fourths will come out great, as nearly pure. Minor thirds will speed up, but that's a small price to pay, especially since the transition region for m3rds is the least of all intervals (4 steps).

This brings up another interesting subject. I personally think the integrity of the 5ths, 8ves and 12ths take priority for serious musicians. This means there WILL be a bump in the M3rd beatrate if these more important intervals are to take priority. The bump can be smoothed, but problems happen if the M3rds are given more importance than they are due.

The whole thing with progressive RBIs across a break is more of a mind exersize. But we can use the same mind exersize with 5ths and octaves. If we want the most consistent 5ths and 2:1 octaves across a break, then tuning with pure 12th is appropiate. The internal test for a pure 12th is the P5-P8 test. This is one of the rationales that I use for justifying tuning with P12s. Another rationale is that pure 12th result in narrower octave types when the iH is high and wider octave types when the iH is low.

The iH for F2 is 0.1008 and for F#2 is 0.1436, but, due to string/bridge/soundboard anomalies, the first partial of F#2 is 12.4 cents flat of theoretical, yet still produces the precise expected partials as if it were not flat.

You don't.



Please explain - is your machine telling you that F#2 P1 should be 12.4 cents flat, or is P1 flatter than predicted by the rest of the spectrum (P2 is at ~1212.6c, and the rest is regular)? If so, that would make sense.

My measurements show P1 at 91.36Hz (which should be 92.07), P2 at 184.18Hz, P3 at 276.37Hz and so on, using P0 at 92.064Hz. If I tune P1 to 92.07, the upper partials beat like crazy.

Yes, from that spectrum you will have a heck of a time getting decent sounding octaves. I'd recommend you tuning slowly rolling octaves above the break in an extremely expanded pure 5ths temperament. This way, when you traverse the break you may have a chance at getting decent 4ths/5ths. The octaves should be expected to roll, with a gradual reduction, ending in a good, clean bass.

I responded too quickly initially. How are you measuring this?

If this is actually a sharp second partial, which would make more sense, and the notes near F#2 mirror this behaviour, then depending on the strength of this partial you will have an especially problematic tuning. The above advice still applies as the solution of best fit.

Actually, the second partial and all the succeeding partials up through P32 match with the theoretical P0. It is P1 that is really flat.

Really interesting, Prout.



Are you calling P0 the fundamental frequency? If so, then you are describing a flat 2nd partial (P1) in F#2? I am unsure what you are saying here.

Sounds like you need to replace the strings for F#2.

prout,

Are you using this formula for IH calculations?


"where n is the partial number (n = 1 for the fundamental) and B is the inharmonicity coefficient"

from this site

Tunewerk, I thought it the other way around. P0 being the nominal frequency f0 used for the calculations, and P1 being the first partial using n=1 in the formula. Like that, P1 will be slightly sharp, as with the rest of the partials.

For example, the note A4=nominal 440Hz. According the the equation P1 = 440.01.

Did I get this wrong prout?

Good to straighten that out; I haven't worked with that terminology in awhile. I did indeed get the same numbers predicting a 92.06Hz P0 from the inharmonicity equation.

This means that from the equation perspective, P1 is flat, but from a tuning perspective, I would consider P2 - P[n] to be sharp.

I know the equation perspective is more correct, since all the other partials share a common inharmonicity coefficient and P1 exhibits negative inharmonicity. The critical, simple thing to keep in mind when tuning however, is that there is a greater space between P1 and P2 than the rest of the series for F#2 because of this negative iH.

I'd expand the octaves above F#2 as much as they could handle, as F#2 has even more iH in a sense, given the flat fundamental. The F#2 octave should roll slightly if things are lining up correctly, but the high frequencies will be correct.

Across the break, I'd immediately start tightening the octaves to control the roll of the 4ths and smooth the M3rds. The 5ths should be good.

Approaching intervals involving the F#2 on the other side will be tricky, as the note will be flat relatively. Expand the octave here a little more to segue into wider expansion over the break into the bass.

Well that complicates the theory when 'B' varies across the partials!

I wonder if prout can derive individual 'B's for each partial using other 'P's spectral frequencies, and then find out just how much iH can vary for given strings. I think that a high level of precision would be required of the frequency spectrum to calculate reliable 'B's.

Or perhaps I should watch my P's and Q's

Prout

A passing thought. Have you checked the offending F#2 string and its termination points to see if you can spot anything that might cause the fundamental to be flat?