Finally:an existing historical theory/paper relevant to "my" PHI scales(Temes)..

...Not to say the scales are "mine" any more, it's up to public opinion how much they borrow from Lorne Temes' Golden Section scales (which actually are generated in a very similar fashion).
But, more importantly, this guy (Lorne) actually explains the phenomena I call(ed) "proportionate beating", although he does not mention a formal term for it (we really need one...)

Here are my thoughts about his very useful PHI scales paper on
http://www.anaphoria.com/temes.PDF

Temes>"There are, in fact, though, intervals which sound relatively consonant, but for which none of the harmonics coincide"
This is quite true for the scale I "ran into" as well...every harmonic either coincides or is just far enough away to avoid beating.
Although I don't make a point of going for >only< avoiding overlap...I go for the goal of either overlapping almost exactly (which produces a chorus effect) or being at least a 1.05 ratio away from the nearest overtone (which makes controlled beating).

Temes>"This number (PHI) we shall denote by 1 + 1/PHI..."
Indeed, he (unlike anyone in the other paper's I've read) uses the same convention I do: of deriving the golden section from the
inverse of PHI rather than PHI itself.

Temes> "Now sums and differences ought to have an implicit role in any phenomena involving the way frequencies combine...feed any two frequencies into a non-linear device (like the ear) and out comes, along with the original tone, both their sum and difference!"

If I understood this right...this also holds for where x = 1/x + 2 IE the silver ratio. So indeed, the extra sum and difference tones seem to be a universal coincidence that can be applied to any noble/fractal number that solves an equation in the form x = 1/x + b. This lends a lot of understanding to why the Silver Ratio scale seems to work so well also...and could serve as a "professional" explanation for the phenomena of "proportionate beating" (Carl, do you hear this...) :-)

Temes> "(trying to) get a scale which was nicely distributed, it's degrees neither too plentiful nor crowded"
In other words Temes, like me, was trying to fit in as many tones as possible while obeying the critical band. This is further proof that this type of scale >does< comply with Helmholtz and Plomp and Llevelt's psycho-acoustic theories for root tones along with harmonics (the relationship with harmonics/overtones was discussed above).

Temes> "Observe that...nor does anyone know for certain what either the fundamental interval or the proposed scale sound like"
This might explain how...why I came up with several different notes in the scale I found then Temes did, although they are both built on the same formula.

Here is the scale of Temes' that turned out closest to my own:
What I came up with
833 = PHI (I often round this to 1.625 to preserve consonance)
733.86 = 1.527934 octave inverse (just as with his scale!)
560 = 1.38
366.89 = 1.23606
235.76 = 1.14589
149.35 = 1.0901
93.839903905 = 1.0557

Temes-mix (closest match)
833.09030
733.82000
639.93400
560.06700
466.18100
366.90970
273.02400

Notes in common between the two:
833
733
560
366
(that's 4 of 7 notes in common with what I came up with using the (1 + 1/PHI)^2...more than any scale I have run into so far and close enough to say Temes could have easily run into the scale I did using his methods).

Considering Temes never had the technology to actually try these scales with (and regardless of how much people think Temes scale matches with mine)...I think it's fair to say the rest of us can work with our ears and advanced tuning software, VSTs, etc. ...and finish what he started.

Finally, though, a paper that seriously makes sense with the noble/fractal scale generation system I have made (thank you Cameron for showing me Temes' quite relevant work with PHI scales)! :-)

-Michael

>Hi Mike,

I was just getting into that Temes paper you sent and looking forward to a possible solution to the limiting problem of PHI, but it seemed to give the values and just peter out. Would have liked some more reasoning behind the choice of those numbers (have you tried them? How do they sound?). Still, I found the intro an interesting possibility about least shared harmonics being the "most dissonant". However, isn't that true of ALL irrational numbers, that they have no harmonics in common? For example, if we take the harmonics of 1,2,3,...then, permitting equal tone, the flat-fifth or sqrt 2 will have the series 2^(1/2),2 x 2^(1/2), 3 x 2^(1/2) etc...While this is as yet very superficial and I haven't yet taken into account entropy or proximity of harmonics up the series, it is at least mathematically true that they have no harmonics in common. It is why irrationals can never have a GCD frequency and hence a resultant tonic (and as a professional mathematician Temes really should be trained to know this. But like all, they tend to impose maths onto the musical material rather than first deducing what goes on in practice. But you've heard all this before from me). Of course 12TET should then also 'theoretically' be atonal, which I wouldn't strictly accept, and which is why I'm still interested in Carl-Erlich's entropy and the idea of 'almost periodic functions', approximate whole-numbers etc.

One other thing did catch my interest though about his discovery of something close to the semitone. I was thinking about PHI in 36TET, that PHI squared gives 50 = 3 (mod36) which is 8ve equivalent to the semitone. Hence, PHI^3/4 = 1.059017, very close to 1.05946 (Is this common knowledge cause I didn't know?). This suggests that (albeit remote) PHI beating already exists in the standard ET. In fact, the difference in the PHI value gives 0.059017 x 32 = 1.88854, which seems 8ve equiv to its inverse 2^(11/12) = 1.887748625. But the standard semi gives around 1.9 and besides, I can't think of a single (tonal) chord where we would have three semitones together, so I don't think its very useful musically.

-Rick

> ...Not to say the scales are "mine" any more, it's up to public opinion how much they borrow from Lorne Temes' Golden Section scales (which actually are generated in a very similar fashion).
> But, more importantly, this guy (Lorne) actually explains the phenomena I call(ed) "proportionate beating", although he does not mention a formal term for it (we really need one...)
>
>
> Here are my thoughts about his very useful PHI scales paper on
> http://www.anaphoria.com/temes.PDF
>
>
> Temes>"There are, in fact, though, intervals which sound relatively consonant, but for which none of the harmonics coincide"
> This is quite true for the scale I "ran into" as well...every harmonic either coincides or is just far enough away to avoid beating.
> Although I don't make a point of going for >only< avoiding overlap...I go for the goal of either overlapping almost exactly (which produces a chorus effect) or being at least a 1.05 ratio away from the nearest overtone (which makes controlled beating).
>
> Temes>"This number (PHI) we shall denote by 1 + 1/PHI..."
> Indeed, he (unlike anyone in the other paper's I've read) uses the same convention I do: of deriving the golden section from the
> inverse of PHI rather than PHI itself.
>
> Temes> "Now sums and differences ought to have an implicit role in any phenomena involving the way frequencies combine...feed any two frequencies into a non-linear device (like the ear) and out comes, along with the original tone, both their sum and difference!"
>
> If I understood this right...this also holds for where x = 1/x + 2 IE the silver ratio. So indeed, the extra sum and difference tones seem to be a universal coincidence that can be applied to any noble/fractal number that solves an equation in the form x = 1/x + b. This lends a lot of understanding to why the Silver Ratio scale seems to work so well also...and could serve as a "professional" explanation for the phenomena of "proportionate beating" (Carl, do you hear this...) :-)
>
> Temes> "(trying to) get a scale which was nicely distributed, it's degrees neither too plentiful nor crowded"
> In other words Temes, like me, was trying to fit in as many tones as possible while obeying the critical band. This is further proof that this type of scale >does< comply with Helmholtz and Plomp and Llevelt's psycho-acoustic theories for root tones along with harmonics (the relationship with harmonics/overtones was discussed above).
>
> Temes> "Observe that...nor does anyone know for certain what either the fundamental interval or the proposed scale sound like"
> This might explain how...why I came up with several different notes in the scale I found then Temes did, although they are both built on the same formula.
>
>
>
> Here is the scale of Temes' that turned out closest to my own:
> What I came up with
> 833 = PHI (I often round this to 1.625 to preserve consonance)
> 733.86 = 1.527934 octave inverse (just as with his scale!)
> 560 = 1.38
> 366.89 = 1.23606
> 235.76 = 1.14589
> 149.35 = 1.0901
> 93.839903905 = 1.0557
>
> Temes-mix (closest match)
> 833.09030
> 733.82000
> 639.93400
> 560.06700
> 466.18100
> 366.90970
> 273.02400
>
> Notes in common between the two:
> 833
> 733
> 560
> 366
> (that's 4 of 7 notes in common with what I came up with using the (1 + 1/PHI)^2...more than any scale I have run into so far and close enough to say Temes could have easily run into the scale I did using his methods).
>
> Considering Temes never had the technology to actually try these scales with (and regardless of how much people think Temes scale matches with mine)...I think it's fair to say the rest of us can work with our ears and advanced tuning software, VSTs, etc. ...and finish what he started.
>
> Finally, though, a paper that seriously makes sense with the noble/fractal scale generation system I have made (thank you Cameron for showing me Temes' quite relevant work with PHI scales)! :-)
>
> -Michael
>

>"Would have liked some more reasoning behind the choice of those numbers (have you tried them? How do they sound?). "
Simply put, Temes uses the same generation style/theory as mine, but toward a different goal (he even makes a point in his paper that he purposefully tries to keep harmonics as far as possible from aligning...which means they are not tonal but not close enough to each other to cause terrible beating either...hence a result which is assonant: neither consonant nor dissonant)..
Thus, due to this goal/restriction, he ends up with tons of assonant (neither dissonant nor consonant) values yet very few consonant ones while mine includes both assonant and consonant ones (and takes consonant values over assonant ones where possible). So, in short, my theories are
pretty much "just" an extension of his, but so far I've found mine sound better to a fairly significant degree (since I prefer a combination of consonance and assonance over mostly assonance..

>"it is at least mathematically true that they have no harmonics in common. "
But you see this is what trips people up into thinking PHI >has< to be dissonant. What happens when to overtones of two different notes neither collide nor are close enough to beat heavily? Assonance...>not< dissonance! Hence the non-colliding tones, if overtones are far enough from colliding (to my ear, a ratio 1.05+ away from the nearest partial is good enough to avoid dissonance)...form assonance. Which is not a bad thing. The problem is a few partials in the possible scales from the golden section do fall within a 1.015-1.049 ratio and create dissonance. My PHI scales, unlike most I've seen,
make a deliberate huge point of avoiding such ratios...

>"PHI^3/4 = 1.059017, very close to 1.05946 (Is this common knowledge cause I didn't know?)."
Hmm...that beating rate may have something to do with the 1.055 value obtained from Temes theory (which is virtually equivalent to my own).
Good point, though...maybe 12TET itself could have PHI or near-phi roots. Although of course, we don't really use 12TET...only 7-8 tone subsets of it due to overtone clashing problems/issues.
So the problem then becomes how do we enforce either A) overtone matching (create consonance) or B) making overtones far enough from each other not to collide to avoid dissonance (creates assonance)? The obvious solution seems to be to make the timbre of the instrument 12TET (and not the harmonic series) and make the scale the same way...and then check to see of PHI-style difference tones appear.

Also I >have< found out that 10TET does share several values in virtually common with the gold section, including
1.1487 (very near (1/PHI)^4 + 1)
1.23114 (very near (1/PHI)^3 + 1)
1.51572 (not too far from 2 - (1/PHI)^5 + 1)
1.6245 (virtually dead on the 13/8 approximation of PHI)
and one from the silver section, including
1.07177 (very near (1/2.414)^3 + 1)

-Michael