Self Similar E, Phi and Pi Scales

Self Similar E, Phi and Pi Scales

---------------------------------

Here are 3 Self Similar scales which I have found enjoyment
improvising with, based on E, Phi and Pi. These scales are
constructed by iteratively dividing the cents values for the first
constant interval by itself, imposing an octave boundary, then
inverting the intervals to obtain the scale degrees, which forms
scales with inversional symmetry. One will notice myriad of
symmetrical interconnections in each scale relative to its constant
generator.

These unusual but beautiful scales are both sonically and musically
interesting in that the intervals grow smaller at the octave
boundaries, which allows one to play drones with the close pitch
clusters, whilst playing melodic passages with the middle degrees of
the scale.

31 Tone Self Similar E Scale
E= 2.718281828
Cents Consecutive
0.000
1.579 1.579
2.713 1.134
4.291 1.579
7.374 3.082
11.665 4.291
20.044 8.379
31.709 11.665
54.484 22.776
86.193 31.709
148.104 61.911
234.297 86.193
402.588 168.291
636.885 234.297
668.766 31.881
1094.349 425.583
1305.651 211.303
1731.234 425.583
1763.115 31.881
1997.412 234.297
2165.703 168.291
2251.896 86.193
2313.807 61.911
2345.516 31.709
2368.291 22.776
2379.956 11.665
2388.335 8.379
2392.626 4.291
2395.709 3.082
2397.287 1.579
2398.421 1.134
2400.000 1.579

29 Tone Self Similar Phi Scale
Phi= 1.618033989
Cents Consecutive
0.000
1.599 1.599
2.587 0.988
4.186 1.599
6.774 2.587
10.960 4.186
17.733 6.774
28.693 10.960
46.427 17.733
75.120 28.693
121.546 46.427
196.666 75.120
318.212 121.546
514.878 196.666
833.090 318.212
1566.910 733.819
1885.122 318.212
2081.788 196.666
2203.334 121.546
2278.454 75.120
2324.880 46.427
2353.573 28.693
2371.307 17.733
2382.267 10.960
2389.040 6.774
2393.226 4.186
2395.814 2.587
2397.413 1.599
2398.401 0.988
2400.000 1.599

27 Tone Self Similar Pi Scale
Pi= 3.141592654
Cents Consecutive
0.000
2.061 2.061
4.415 2.353
6.476 2.061
13.869 7.393
20.345 6.476
43.571 23.226
63.916 20.345
136.882 72.966
200.798 63.916
418.205 217.407
430.027 11.823
630.825 200.798
1049.030 418.205
1350.970 301.941
1769.175 418.205
1969.973 200.798
1981.795 11.823
2199.202 217.407
2263.118 63.916
2336.084 72.966
2356.429 20.345
2379.655 23.226
2386.131 6.476
2393.524 7.393
2395.585 2.061
2397.939 2.353
2400.000 2.061

Notes:

1. These scales are inspired and constructed by inference from
reading past posts and some communications with Dr. John Chalmers,
and by the paper "Self Similar Pitch Structures, Their Duals, and
Rhythmic Analogues", by Norman Carey and David Clampitt, to whom I am
indebted for revealing this unique type of tuning to me.

2. As John Chalmers pointed out, the ultimate transposition of
this (at least for phi) is to use this kind of tuning with timbres
based on the tuning itself. My preliminary tests with a phi timbre
has revealed a new world of musical sound design and tuning
possibilities by matching timbre to tunings based on constants.

3. For those who may not be aware of it, and may like to
experiment with these scales on a keyboard which supports full
retuning, you can copy the above cents values into a text file and
use the "Load/Cents" command with Manuel Op de Coul's wonderful Scala
program, to create your midi tuning file for your synth. Only use the
first column though, and leave off the "0" - Scala will add this for
you!

4. In designing these scales, I let the iterative operation
terminate at the point where a 768 tuning unit synth would be able to
render the small intervals at the octave boundaries, so that they can
be heard with reasonable accuracy on a TX81Z or the like.

Thanks,

Jacky Ligon

P.S. Feel free to compose music with these if you like, and please
let me hear what you might come up with. I've found them quite
interesting and surprisingly musical to play with harmonic timbres
too.

--- In tuning@y..., ligonj@n... wrote:
> Self Similar E, Phi and Pi Scales
>
> ---------------------------------
>

Original post:

/tuning/topicId_18578.html#18578

By popular request, it has been asked that I give a more detailed
explanation of the construction of these scales. So here goes, with
Phi as the object of desire!

If we take the ratio Phi, which is (SQRT(5)+1)/2 = 1.618033989, and
we convert it to cents we get, LOG(R) * (1200/LOG(2)) = 833.090.

Next we take the cents value 833.090, and divide it by Phi, obtaining
= 514.878

Then we take the cents value 514.878, and divide it by Phi, obtaining
= 318.212

etc...

To appreciate the Phi related connections of this scale, we can
easily see that if we take our first cents value and divide it by the
next one in the sequence, it will give the ratio Phi.

833.090/514.878 = 1.618

514.878/318.212 = 1.618

So we can now see how the first descending series is created, and if
we carry this out to the point where we must stop for those using a
768 tuning unit synth (steps of 1.56 cents) we see the below sequence:

833.090
514.878
318.212
196.666
121.546
75.120
46.427
28.693
17.733
10.960
6.774
4.186
2.587
1.599

Now we want to invert the above series, so we will impose our octave
boundary. Please note that here we will use a 2 octave boundary (2400
cents), but one could just as easily use a 1200 cents octave too -
there are indeed many variations on the self similar theme!

So if we take the above series and subtract each member from our
octave value, we get the following ascending series:

2400.000
1566.910
1885.122
2081.788
2203.334
2278.454
2324.880
2353.573
2371.307
2382.267
2389.040
2393.226
2395.814
2397.413
2398.401

Next we combine and sort the two above sequences and add the "0",
obtaining our final scale:

> 29 Tone Self Similar Phi Scale
> Phi= 1.618033989
> Cents Consecutive
> 0.000
> 1.599 1.599
> 2.587 0.988
> 4.186 1.599
> 6.774 2.587
> 10.960 4.186
> 17.733 6.774
> 28.693 10.960
> 46.427 17.733
> 75.120 28.693
> 121.546 46.427
> 196.666 75.120
> 318.212 121.546
> 514.878 196.666
> 833.090 318.212
> 1566.910 733.819
> 1885.122 318.212
> 2081.788 196.666
> 2203.334 121.546
> 2278.454 75.120
> 2324.880 46.427
> 2353.573 28.693
> 2371.307 17.733
> 2382.267 10.960
> 2389.040 6.774
> 2393.226 4.186
> 2395.814 2.587
> 2397.413 1.599
> 2398.401 0.988
> 2400.000 1.599

Here is a scale formed in the above manner, but with a 1200 cents
octave rather than 2400 cents:

29 Tone Self Similar Phi Scale with 1200 cents octave
Phi= 1.618033989
Cents Consecutive
0.000
1.599 1.599
2.587 0.988
4.186 1.599
6.774 2.587
10.960 4.186
17.733 6.774
28.693 10.960
46.427 17.733
75.120 28.693
121.546 46.427
196.666 75.120
318.212 121.546
366.910 48.698
514.878 147.968
685.122 170.244
833.090 147.968
881.788 48.698
1003.334 121.546
1078.454 75.120
1124.880 46.427
1153.573 28.693
1171.307 17.733
1182.267 10.960
1189.040 6.774
1193.226 4.186
1195.814 2.587
1197.413 1.599
1198.401 0.988
1200.000 1.599

Quite a different scale (!), but I actually prefer the sound of the
scale degrees are distributed over a 2 octave range.

One appealing aspect of this kind of scale for me, is that it is in
keeping with my musical and melodic interest in JI scales with
inversional symmetry. I enjoy the sound of symmetry, where the scale
degrees are mirrored ascending and descending at the 1/1 and 2/1. It
can lend a bit more predictability for melodic writing, under certain
conditions, than totally irregular spacing found in many JI scales.
Honestly though, I find them both of equal value musically.

Thanks to all my list friends who have shown interest, and happy
music making!

Jacky Ligon