Why Tenney Height could also use prime limit cues (esp. regarding equivalence with the 3/1 period)

>"But to my ears, 8:10:12:15 is more consonant than 8:11:13:14, so I don't use odd-limit much."

A few things to consider
1) 8:11:13:14 has more odd harmonics, which lessens the chance of (typically stronger) even harmonics in a timbre matching

2) 8:10:12:15 has many lower odd-limit dyads than 8:11:13:14 (13/8 vs. 3/2, 5/4 vs. 11/8, 14/11 vs. 3/2).  In this case I think prime limit really does have a good point so far as consonance...not only should octave equivalence be able to be reduced before Tenney Height, but equivalence at the 3/1 period as well.

  In many ways, it seems to says TRIADIC odd limit can be trumped, in some cases, by DYADIC odd-limit...rather than that odd-limit is inferior to judging consonance than prime limit.

   On the other hand, I don't think prime limit-type reductions should be ignored in Tenney Height.
I will admit 9-odd-limit often sounds more consonant than 7-odd-limit for dyadic consonance (so long as the 9-odd-limit consonance does not include a 7 in the ratio).  IE 9/8 sounds more consonant to me than 8/7.  I assume this has something, again, to do with odd timbres...and the fact that a fairly loud third harmonic (3/1) will also intersect at 9/1.

  Personally, I think something like 9/7 (even) should be allowed have any number that's a multiple of 3 reduced before taking a Tenney Height calculation IE instead of taking 9*7 = 63, you'd take 9/3 = 3*7 = 21.  Also, I think there should be a penalty for the prime factors of 7 (odd factors with weaker harmonics, a 2/1 penalty), which would make it, say 3*(7*2) = 42.  This would make it a bit more dissonant than a 7/5 (7*5 = 35) and a fairly more dissonant than 5/3 (5 * 3 = 15)...but still not so much higher you'd think "how could ratios with such different dissonance values all be considered fourths?!"

I think 11/9 is another good example...using this method would make it 11 * 3 = 33, a tad less dissonant than 9/7, a tad more dissonant than a 6/5 (6*5 = 30) minor third and a significant bit more dissonant than a 5/4 major third.  IE it would actually give it a very "third-ish" dissonance level, rather than the bizarre level of  11*9 = 99, which doesn't seem third-like at all.

Even with 22/15...take the octave inverse of 15/11 and reduce by 3. This would give 11*5 = 55. And 14/9 would become 9/7 (octave inverse)...which, again, would give 42. Listening to these two...I think it's clear why they are both almost equally useful to use as alternatives to fifths. Meanwhile 16/11 could be reduced to 11/8, which gives 88 as a dissonance level...and faithfully summarizes how it sounds edgier than 22/15 despite having a lower numerator and denominator (Igs warned me a long time ago how harsh 16/11 sounds vs. 22/15...despite how much Tenney Height may say otherwise).

On Fri, Aug 19, 2011 at 3:04 PM, Michael <djtrancendance@...> wrote:
>
> >"But to my ears, 8:10:12:15 is more consonant than 8:11:13:14, so I don't use odd-limit much."
>
> A few things to consider
> 1) 8:11:13:14 has more odd harmonics, which lessens the chance of (typically stronger) even harmonics in a timbre matching

Why should even harmonics be stronger? I'd think that odd harmonics
should typically be stronger.

>   In many ways, it seems to says TRIADIC odd limit can be trumped, in some cases, by DYADIC odd-limit...rather than that odd-limit is inferior to judging consonance than prime limit.

Except that 8:10:12:15 has that 15/8 dyad, which has an odd limit of
15. Maybe average dyadic concordance would do the trick.

>    On the other hand, I don't think prime limit-type reductions should be ignored in Tenney Height.
> I will admit 9-odd-limit often sounds more consonant than 7-odd-limit for dyadic consonance (so long as the 9-odd-limit consonance does not include a 7 in the ratio).  IE 9/8 sounds more consonant to me than 8/7.  I assume this has something, again, to do with odd timbres...and the fact that a fairly loud third harmonic (3/1) will also intersect at 9/1.

Right.

>   Personally, I think something like 9/7 (even) should be allowed have any number that's a multiple of 3 reduced before taking a Tenney Height calculation IE instead of taking 9*7 = 63, you'd take 9/3 = 3*7 = 21.  Also, I think there should be a penalty for the prime factors of 7 (odd factors with weaker harmonics, a 2/1 penalty), which would make it, say 3*(7*2) = 42.  This would make it a bit more dissonant than a 7/5 (7*5 = 35) and a fairly more dissonant than 5/3 (5 * 3 = 15)...but still not so much higher you'd think "how could ratios with such different dissonance values all be considered fourths?!"

How about 45/32?

-Mike