The Prime Series as Generator: High Prime Explorations

The Prime Series as Generator:

It has long been a source of musical mystery and fascination to
consider the infinitely rich possibilities of the Prime Series. That
in number theory, the prime series represents points of
irreducibility, also stands as one of the most important concepts of
JI/RI. We must look no further than our old friend "3/2" to see what
has guided the focus to this new path of tuning exploration (3/2 has
the very interesting and unique quality of having both prime over and
under numbers). My interest in this irreducible quality has led me to
explore scales formed from the prime series, where both the numerator
and denominator are prime numbers. Where both are Prime, these truly
represent unique points of irreducibility along the pitch continuum;
which sometimes is mathematically "unique", and many other
times "audibly" unique. One could argue that the human ear will
perceive these as lower number ratios, so it should be pointed out
straight away, that the purpose behind the exploration of these
ratios does not pretend to object to this point; with the primary
goal being to identify and make use of the musical possibilities of
the quality of "irreducibility", by using the Prime Series as a scale
generator. And the truth is; after a point, harmonic concepts
of "limit" become completely irrelevant, and the numbers serve the
sole function of being irreducible scale degree generators. A major
part of the interest in this kind of exploration of high prime
rational intonation, is to see the many audibly "identical"
intervals, along side of many that are either very close to lower
number ratios, or ones which are altogether alien to the language of
3-5 limit Just Intonation.

An initial interest and challenge toward the goal of harnessing the
untapped musical forces of the Prime Series, was to devise a method
by which a huge slice of the lower prime series ratios could be
produced and thereby studied. I found the solution rather easy to
produce. It is as follows:

I found that if one takes a small portion of the Prime Series ("PS"
from here forth), and places it in a first column - treating it as
the numerator series, then takes the same series and places it in the
second column - treating it as the denominator series, then shifts
the second up by one number, then the first infinite prime series of
ratios is found (see below).

For purposes of discussion I will call this the "First Order PS
Ratios". This could best describe the series because unlike the
harmonic series where there is a succession of superparticulars, here
there is a "lumpy" series, where we begin the journey with our
familiar superparticular 3/2, but quickly progress into
superpartients - never to see another superparticular ratio again. As
far as I'm aware, there is never another occurrence of a
superparticular PS Ratio beyond 3/2 (also another noteworthy quality
of this ratio).

2/
3/2 701.955
5/3 884.358
7/5 582.512
11/7 782.492
13/11 289.209
17/13 464.427
19/17 192.557
23/19 330.761
29/23 401.302
31/29 115.458
37/31 306.308
41/37 177.718
43/41 82.455
47/43 153.988
53/47 207.997
59/53 185.667
61/59 57.713
67/61 162.422
71/67 100.389
73/71 48.092
79/73 136.747
83/79 85.510
89/83 120.832
97/89 149.015
101/97 69.958
etc.

It is helpful to imagine this series being extended into infinity, so
as to internalize the fact that we are only looking at an
infinitesimally small slice of the prime universe.

Now this represents a kind of series on the "X" axis, but let's take
a look at the "Y". To produce the second infinite progression of
intervals, we simply move the denominator column up one more place.

This we will call this the "Second Order PS Ratios", which looks like
this:

2/
3/
5/
7/
11/
13/7 1071.702
17/11 753.637
19/13 656.985
23/17 523.319
29/19 732.064
31/23 516.761
37/29 421.767
41/31 484.027
43/37 260.174
47/41 236.444
53/43 361.987
59/47 393.665
61/53 243.380
67/59 220.135
71/61 262.812
73/67 148.482
79/71 184.840
83/73 222.258
89/79 206.343
97/83 269.848
101/89 218.974
etc.

And to obtain the Third Order PS Ratios, we slide the denominator
series up one more place again, giving the below ratios:

2
3
5
7
11
13
17
19/11 946.195
23/13 987.747
29/17 924.622
31/19 847.523
37/23 823.070
41/29 599.485
43/31 566.482
47/37 414.163
53/41 444.442
59/43 547.654
61/47 451.378
67/53 405.802
71/59 320.525
73/61 310.905
79/67 285.230
83/71 270.351
89/73 343.091
97/79 355.359
101/83 339.806
103/89 252.921
107/97 169.865
109/101 131.967
113/103 160.414
127/107 296.661
131/109 318.286
137/113 333.424
etc.

It becomes immediately obvious that one could go on to extend both on
the x and y as far as desired (I carried it out to the 12th Order PS
Ratios, and up to the first 1000 primes for my study of the PS). With
this much charted above though, one can see that with each successive
slide upward of the denominator column the granularity of the series
becomes much greater.

The next question emanating from the reader must be: "How may I form
a scale from this?" And the answer is that one may either employ a
systematic approach to subsetting from the series, or else use one's
personal taste and knowledge of tuning flavors to "pick by hand" from
the PS web.

First, let's look at a selection of the ratios, sorted by prime limit:

3/2 701.955
5/3 884.359
7/5 582.512
11/7 782.492
13/11 289.210
13/7 1071.702
17/13 464.428
17/11 753.637
19/17 192.558
19/13 656.985
19/11 946.195
23/19 330.761
23/17 523.319
23/13 987.747
29/23 401.303
29/19 732.064
29/17 924.622
31/29 115.458
31/23 516.761
31/19 847.523
31/17 1040.080
37/31 306.308
37/29 421.767
37/23 823.070
37/19 1153.831
41/37 177.718
41/31 484.027
41/29 599.485
41/23 1000.788
43/41 82.455
43/37 260.174
43/31 566.482
43/29 681.941
43/23 1083.243
47/43 153.989
47/41 236.444
47/37 414.163
47/31 720.471
47/29 835.929

Below are 3 scales I created by order of prime limits (a more
systematic method).

Scale #1 (Scala file: "LIGON.SCL". Thanks for including this scale
Manuel!)
1/1 0.000
31/29 115.458
19/17 192.558
13/11 289.210
29/23 401.303
17/13 464.428
7/5 582.512
3/2 701.955
31/19 847.523
11/7 782.492
5/3 884.359
13/7 1071.702
2/1 1200.000

Scale #2
1/1 0.000
43/41 82.455
41/37 177.718
23/19 330.761
37/29 421.767
23/17 523.319
41/29 599.485
19/13 656.985
17/11 753.637
19/11 946.195
23/13 987.747
37/19 1153.831
2/1 1200.000

Scale #3
1/1 0.000
61/59 57.713
47/43 153.989
37/31 306.308
47/37 414.163
31/23 516.761
43/31 566.482
29/19 732.064
37/23 823.070
29/17 924.622
31/17 1040.080
43/23 1083.243
2/1 1200.000

Now these scales represent a very simple way of subsetting from the
series, but one could just as well create scales with many more
pitches to the 2/1, or else map the portions of the series across the
entire midi range. The sky's the limit (although here, "Primes" are
the limit).

Many times my approach to performing on such systematically
constructed scales as above, comes from the pattern creating joy of
reacting to the sound under one's fingertips, on instruments of
choice, where the musical focus is attuned to "How I may make music
with the unique qualities and moods of a particular tuning system,
rather than being repelled by the numbers I see before hearing"?
Commas aren't a problem - they become powerful melodic elements. Wide
5ths - love them! (Perhaps this comes from many years of being an
improvising musician - I'm always "reacting".) There is something
undeniably gorgeous about intervals like 29/19; an interval right at
home on a metallophone timbre. There are myriad of creative
compositional techniques which one can call upon to get the most out
of complex tunings such as these. Things like using proper registers,
and mixed timbres (stay away from the "full fisted" keyboard
approach!), hocketing, intervallic partitioning, klangfarbenmelodie-
like passing of melodic notes between timbres, singing, and extended
melodies. These scales open a new door for me (and like any other
tuning system one can name, it implies its own innate style); a kind
of bridge between worlds of simplicity and points of complexity where
ideas of limits - odd or prime - lose their meaning, and prime
numbers granulate the pitch continuum in ways which are mysterious to
the mind.

To conclude, I have tried these both harmonically (playing chords)
and melodically as well, and they shine the most in a melodic setting
to me, although on the right timbres (e.g. strings), I'm able to
write compelling thematic harmonies with any of the above three
scales. And upon reflection from appreciating them in these different
ways, I must add that they have an overall sound that is eerily
beautiful and uniquely unlike any tunings I've ever explored
previously. I can only convey some vague sense of this quality
through the medium of text, but suffice it to say, that it is
analogous to the ineffable sensation of laying on one's back as a
child, and gazing at the stars.

Jacky Ligon