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Comparison of the different formulas from which to derive PHI-based tunings

🔗djtrancendance <djtrancendance@...>

5/9/2009 7:51:29 AM

As I understand it, these are the main options (if I interpreted one wrong, specifically the logarithmic one, please let me know).

1) PHI^x/2^y
2) 2^(x/36) (since 2^(25/36) is very very close to the value of PHI)
3) 2^((1/PHI)^x) where x can = 0,1,2,3 but also -1,-2,-3... (logarithmic)
4) (1/PHI)^x + 1 and then taking the results and subtracting them from 2 to get a few extra notes as a "reverse golden section"(Temes formula, which I also use)

I have found 3) to be the most dissonant formula, it produces almost exclusively ratios like 1.53 and 1.30 whose harmonics are within 1.015-1.049 (IE roughly the zone of maximum harmonic entropy) relative to higher harmonics in the scale.

I have noticed 2), meanwhile, includes virtually every possible PHI-related note combination, both consonant, dissonant, and assonant. It ends up using inversions of PHI ratios (IE inverting the chord 1/PHI, 1, PHI, 2/PHI, PHI^2) to cover the entire 36TET scale with only a few cents error vs. PHI. So 36TET seems like an ideal basis for PHI scales. However, there are two major problem: one is that several of the 36 notes have harmonics within a sour 1.015-1.049 interval from either the nearest harmonics of another note in a chord or a tone in the scale on a subsequent octave...so you need to do a lot of searching by ear to find non-dissonant sounding tone combinations.

Meanwhile 1), phi^x/2^y can produce any note in any of the scales above with tremendous accuracy, plus more (it's basically an infinite spiral with an infinite number of tone)...but there's no way to choose which notes to use except by ear which makes it fairly unrealistic to use (at least if your goal is to use a combination of consonance and assonance in your scales to avoid dissonance).

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4) Seems like the best bet to me by far after trying to compose with all 4 of these methods. It takes a strategic subset of 2) (36TET) where in just about any combination of its 8 or so tones only one or two possible combinations of tone out of the 8 will be dissonant or fall with the 1.015-1.049 (so it mathematically solves most of the possible dissonance problems and produces either consonance or assonance for a huge majority of note combinations).
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Another problem (which Chris notes...and effects all PHI-based scales which use PHI exactly or near-exactly in intervals), it that PHI itself sounds a bit dissonant as an interval (so long as you use a harmonic timbre).
This means you can't play one phi-tave up from a note without nasty beating (although all the other intervals sound fine as does two PHi-taves up). I know this from experience because I've tried it several times.
Using 1.625 (13/8...about 7 cents off from PHI) gives virtually the same tone color yet alleviates most of the harsh beating that the root and PHI played at once cause. So I think it is fair to say it can help alleviate dissonance in most PHI scales without effecting the sound or symmetries in any notice-able way.

-Michael