The PHI(F/8P) comma and the Iph Heptaphone

Hi you Phi-maniacs, and everybody,

I thought this could be of interest in these Phi discussions, if it has not been already discovered and quoted in your exchanges :

One most important Phi-related comma is (9 sqrt of 5 + 20) / 40 = 1.003115295 or 5.384921316 cents,
the ratio between Phi^6 and 8 sqrt of 5, very well approximated by 322/321 or 5,38487 cents (certainly not a hazard, since 322 belongs to the P-series).

Why is it an important comma ?
It allows to link Phi in its two main series F and P, with factor 2.
If you have a look in the two main Phi series, F (Fibonacci) and P (powers of Phi, also named Lucas) :
(F) 1 2 3 5 8 13 21 34 55 89 144 etc. and
(P) 1 3 4 7 11 18 29 47 76 123 etc...,
you find that numbers 18 and 144 are octave-related, more precisely 8 x 18 = 144.
But in real fractal Phi ratios actually, they differ by (9 sqrt of 5 + 20) / 40 = 1.003115295.
,
The F and P whole numbers series dissolve that comma, that could also be dissolved by stretching the 2/1 octave by the cubic root of 1.003115295, or 1.795 c., allowing for a temperament preserving pure Phi and all its differential qualities, or reversely, using pure 2/1 and "unpure" Phi, or both tempered why not, I am not specialist).

Among other applications, compilation of series P, 2P and 4P will link perfectly / or almost perfectly in continuity with series F and following series 2F, etc.
Extending to the five series P, 2P, 4P, F, 2F you will get different scales showing balanced divisions with two types of intervals of either phiaves or octaves :
a 5 tones per phiave (the periodic pattern) or a semi-periodic 7 tones per octave sort of "Arax-type" heptaphone.

Borrowing from P, 2P, 4P, F and 2F series, one of the numerous just intonation models of this scale would be :
55 58 68 72 76 89 94 110
(in which most of difference tones are Fibonaccis or related)
Or, a fractal version can be attained in pure fractal ratios by using powers of " Iph " (= sqrt of 5 - 1, or 2/Phi) = 1.2360679775 six times as generator (F+P illustration here : 76 94 116 144 178 220 272)
Starting from series F frequencies the fractal ratios are :
Iph-Arax heptaphone
1/1
1.055728
1.236068
1.309017
1.381966
1.618034
1.708204

(where 1.618034 and 1.708204 only repeat the first phiave)

This scale can be apparented to a form of a persian medieval "Buzurg" scale I am calling Arax, that follows the same scalar pattern, in its symetrical form, as Surak-Nat-Buzurg, or Mohajira ( s L s L s L s ).
Arax tetrachord is like Buzurg but with 7/6 as large interval instead of 8/7 and smaller semitones - you can have a look at a Surak-Nat-Buzurg decaphonic guitar, containing an heptaphone not far away from the scale Cameron recently described, tuned in
1/1 13/12 8/7 26/21 55/42 39/28 3/2 34/21 12/7 13/7 2/1, at
http://aeh.free.fr
(then going to the page "recherche microtonale")

More dissonnant, Iph-Arax realises the exploit to combine fractal Phi waveforms-compatible intervals- and practically only two sizes of intervals, the chromatic 2(2 - 1/Phi) / Phi^2 ) = 1.055728 = 93.885968 c. (or his cousin (Phi^3) / 4 = 1.059017 = 99.2709 c. ) and the fractal Phi "7/6-like" 8/(Phi^4) = 1.16718427 = 267.6388 c.
With fractal-Phi waveforms it sounds quite persian and dreamy.

Note that those two s & L intervals are almost in 3 to 8 proportions and therefore can be approched roughly in an octave of (3 x 8) + (4 x 3) = 36 equal divisions, already mentionned in recent posts for related scales.

- - - - - - - -
Jacques

  Jacques>"Starting from series F frequencies the fractal ratios are :Iph-Arax heptaphone1/11.0557281.2360681.3090171.3819661.6180341.708204"

     Nice...I tried this but the note 1.309 and 1.708204 sounded "off" to me.  I was just wondering what you'd think of this
1/1
1.055728
1.236068
1.309017
1.381966
1.532 (new tone of 1.618034 / 1.05573)
1.618034
1.8541 (new tone of 1.618034 * 1.14589 where (1/PHI)^5+1 = 1.14589)

     To me it sounds a good deal clearer and yet keeps a lot of the inspirational Persian vibe of the original scale which encouraged me to play around with it in the first place.

-Michael

--- On Thu, 5/14/09, Jacques Dudon <fotosonix@...> wrote:

From: Jacques Dudon <fotosonix@...>
Subject: [tuning] The PHI(F/8P) comma and the Iph Heptaphone
To: tuning@yahoogroups.com
Date: Thursday, May 14, 2009, 10:22 AM

Hi you Phi-maniacs, and everybody,
I thought this could be of interest in these Phi discussions, if it has not been already discovered and quoted in your exchanges :
One most important Phi-related comma is (9 sqrt of 5 + 20) / 40   = 1.003115295 or 5.384921316 cents,  the ratio between Phi^6 and 8 sqrt of 5, very well approximated by 322/321 or 5,38487 cents (certainly not a hazard, since 322 belongs to the P-series). 
Why is it an important comma ?It allows to link Phi in its two main series F and P, with factor 2. If you have a look in the two main Phi series, F (Fibonacci) and P (powers of Phi, also named Lucas) :(F) 1 2 3 5 8 13 21 34 55 89 144 etc. and(P) 1 3 4 7 11 18 29 47 76 123 etc..., you find
that  numbers 18 and 144 are octave-related, more precisely   8 x 18 = 144.But in real fractal Phi ratios actually,  they differ by (9 sqrt of 5 + 20) / 40 = 1.003115295.,The F and P whole numbers series dissolve that comma, that could also be dissolved by stretching the 2/1 octave by the cubic root of  1.003115295, or 1.795 c., allowing for a temperament preserving pure Phi and all its differential qualities, or  reversely, using pure 2/1 and  "unpure" Phi, or both tempered why not, I am not specialist). 
Among other applications, compilation of series P, 2P and 4P will link perfectly / or almost perfectly in continuity with series F and following series 2F, etc.Extending to the five series P, 2P, 4P, F, 2F  you will get different scales showing balanced divisions with two types of
intervals of either phiaves or octaves :a 5 tones per phiave (the periodic pattern) or a semi-periodic 7 tones per octave sort of "Arax-type" heptaphone.
Borrowing from P, 2P, 4P, F and 2F series, one of the numerous just intonation models of this scale would be :55 58 68 72 76 89 94 110(in which most of difference tones are Fibonaccis or related)Or, a fractal version can be attained  in pure fractal ratios by using powers of " Iph " (= sqrt of 5  - 1,   or 2/Phi)  = 1.2360679775    six times as generator (F+P illustration here : 76 94 116 144 178 220 272)Starting from series F frequencies the fractal ratios are :Iph-Arax heptaphone1/11.0557281.2360681.3090171.3819661.6180341.708204
(where 1.618034 and 1.708204 only repeat the first phiave)
This scale can be apparented to a form of a persian medieval "Buzurg" scale I am calling Arax, that follows the same scalar pattern, in its symetrical form, as Surak-Nat-Buzurg, or Mohajira  ( s L s L s L s ).Arax tetrachord is like Buzurg but with 7/6 as large interval instead of 8/7 and smaller semitones - you can have a look at a Surak-Nat-Buzurg decaphonic guitar, containing an heptaphone not far away from the scale Cameron recently described, tuned in 1/1  13/12  8/7  26/21  55/42  39/28  3/2  34/21  12/7  13/7  2/1,  athttp://aeh.free. fr(then going to the page "recherche microtonale"
)
More dissonnant, Iph-Arax realises the exploit to combine fractal Phi waveforms-compatibl e intervals- and practically only two sizes of intervals, the chromatic 2(2 - 1/Phi) / Phi^2 ) = 1.055728 = 93.885968 c. (or his cousin (Phi^3) / 4 = 1.059017 = 99.2709 c. ) and the fractal Phi "7/6-like"   8/(Phi^4) = 1.16718427 = 267.6388 c.With fractal-Phi waveforms it sounds quite persian and dreamy.
Note that those two s & L intervals are almost in 3 to 8 proportions and therefore can be approched roughly in an octave of (3 x 8) + (4 x 3) = 36 equal divisions, already mentionned in recent posts for related scales.  - - - - - - - -Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Starting from series F frequencies the fractal ratios are :
> Iph-Arax heptaphone
> 1/1
> 1.055728
> 1.236068
> 1.309017
> 1.381966
> 1.618034
> 1.708204
>
> (where 1.618034 and 1.708204 only repeat the first phiave)

This sounds great- using the "phiave", it has the interesting property of "opening out" in sound as you go higher or lower than the original ambitus, getting brighter and more "major" sounding. In my experience this doesn't happen so often in non-octave tunings, as you tend to get either into murky or hard-sounding intervals as you move more than say 2.5* the frequency up (or .4* down) from the original mid-rangey ambitus.

>
> Hi you Phi-maniacs, and everybody,
>
> I thought this could be of interest in these Phi discussions, if >it
> has not been already discovered and quoted in your exchanges :
>
> One most important Phi-related comma is (9 sqrt of 5 + 20) / 40 =
> 1.003115295 or 5.384921316 cents,
> the ratio between Phi^6 and 8 sqrt of 5, very well approximated by
> 322/321 or 5,38487 cents (certainly not a hazard, since 322 >belongs
> to the P-series).
>
> Why is it an important comma ?
> It allows to link Phi in its two main series F and P, with factor 2.
> If you have a look in the two main Phi series, F (Fibonacci) and P
> (powers of Phi, also named Lucas) :
> (F) 1 2 3 5 8 13 21 34 55 89 144 etc. and
> (P) 1 3 4 7 11 18 29 47 76 123 etc...,
> you find that numbers 18 and 144 are octave-related, more
> precisely 8 x 18 = 144.
> But in real fractal Phi ratios actually, they differ by (9 sqrt of >5
> + 20) / 40 = 1.003115295.

This is going to take me some time to digest- I'm aware of the Lucas series of course but I haven't done more than tinkered with it. Working with intervals, the digits pile up really fast due to the rapid introduction of primes: this means about zero to me as far as sound, but it's purely a practical matter of keeping track where the heck I am in all the digits, hahaha! Whereas with Fibonacci I can do a great deal just in my head, walking in the park.

> ,
> The F and P whole numbers series dissolve that comma, that could >also
> be dissolved by stretching the 2/1 octave by the cubic root of
> 1.003115295, or 1.795 c., allowing for a temperament preserving >pure
> Phi and all its differential qualities, or reversely, using pure 2/>1
> and "unpure" Phi, or both tempered why not, I am not specialist).
>
> Among other applications, compilation of series P, 2P and 4P will
> link perfectly / or almost perfectly in continuity with series F >and
> following series 2F, etc.
> Extending to the five series P, 2P, 4P, F, 2F you will get >different
> scales showing balanced divisions with two types of intervals of
> either phiaves or octaves :
> a 5 tones per phiave (the periodic pattern) or a semi-periodic 7
> tones per octave sort of "Arax-type" heptaphone.
>
> Borrowing from P, 2P, 4P, F and 2F series, one of the numerous >just
> intonation models of this scale would be :
> 55 58 68 72 76 89 94 110
> (in which most of difference tones are Fibonaccis or related)
> Or, a fractal version can be attained in pure fractal ratios by
> using powers of " Iph " (= sqrt of 5 - 1, or 2/Phi) =
> 1.2360679775 six times as generator (F+P illustration here : 76 >94
> 116 144 178 220 272)
> Starting from series F frequencies the fractal ratios are :
> Iph-Arax heptaphone
> 1/1
> 1.055728
> 1.236068
> 1.309017
> 1.381966
> 1.618034
> 1.708204
>
> (where 1.618034 and 1.708204 only repeat the first phiave)
>
> This scale can be apparented to a form of a persian medieval >"Buzurg"
> scale I am calling Arax, that follows the same scalar pattern, in >its
> symetrical form, as Surak-Nat-Buzurg, or Mohajira ( s L s L s L s ).
> Arax tetrachord is like Buzurg but with 7/6 as large interval >instead
> of 8/7 and smaller semitones - you can have a look at a Surak-Nat-
> Buzurg decaphonic guitar, containing an heptaphone not far away >from
> the scale Cameron recently described, tuned in
> 1/1 13/12 8/7 26/21 55/42 39/28 3/2 34/21 12/7 13/7 2/1, at
> http://aeh.free.fr
> (then going to the page "recherche microtonale")
>
> More dissonnant, Iph-Arax realises the exploit to combine fractal Phi
> waveforms-compatible intervals- and practically only two sizes of
> intervals, the chromatic 2(2 - 1/Phi) / Phi^2 ) = 1.055728 =
> 93.885968 c. (or his cousin (Phi^3) / 4 = 1.059017 = 99.2709 c. ) >and
> the fractal Phi "7/6-like" 8/(Phi^4) = 1.16718427 = 267.6388 c.

The 7/6-like interval is practically identical to an augmented second, 9/8*25/24. This would be a slick way to modulate from 5-limit JI to your tuning and back, hm! All you'd have to do is go to
H minor, do a Picardy third cadence, then sustaining the M3 bring C back in at the original pitch, then drop 18/19 from C for the new H, now the tonic of Iph-Arax. This would create a comma shift of about
96/95, depending on how the microtuning of the intervals involved went- I don't know in this case whether it would better to temper it out or let it be in this case but I intend to find out. :-D

It is also possible to link this 7/6 or aug. 2nd-like interval as it appears between 560 and 833 cents (as it does between golden section and golden section of golden section), and phi intervals in general, going from 4/3 Fa in 7-limit, and using traditional tetrachordal methods, as there is the identity of 560 as 28/27 above 4/3.

Like this:

1/1
28/27
68/63
4/3
112/81
272/189
3/2
14/9
34/21
16/9
2/1

the lower, the conjunct, and the disjunct tetrachord all have the same "phi characteristic interval" chromatic tetrachord of Aristoxenos (right on the border of chromatic and enharmonic), in the interpretation with Archytus' 28/27, and the tuning has the golden section, and the golden section of the golden section.

(This is the way I've been doing it.)

> With fractal-Phi waveforms it sounds quite persian and dreamy.

I think these tunings sound very dreamy with harmonic spectra, too.

-Cameron Bobro