Mailing list post
From: M. Schulter (2001-02-01)
Subject: Re: Fine-tunings and well-untemperaments (for Jacky Ligon)
Hello there to Jacky Ligon, and I'd like to affirm a "prime directive"
of music and intonational approaches to it: diversity, with complex
rational intonations as one road open for the taking.
I was deeply moved to see your rational intonation (RI) tuning
approximating my favored major third at 14:11, as well as Robert
Walker's proposal of an acoustical basis for the kind of weighted
variations on Classic Mediants (e.g. minor thirds at 20:17, 13:11)
which I have described as "walking the gentle plateau," a poetic
phrase borrowed in part from Dave Keenan (without in any way implying that
he necessarily endorses this integer-based technique as one alternative
method of exploring the continuum).
What special "properties" the series of primes may have for music is
an open question, but I would say that tuning generally is a process
of playing with numbers and seeing what interesting musics may
result.
For me, a special feature of complex RI is the way it can produce what
I might term "fine-tunings" of various kinds somewhat analogous to
well-temperaments -- without necessarily the implication of closure,
although the technique lends itself to circulating tunings also.
Since "implementation is the most sincere form of flattery," to quote
one software developer, why don't I give Scala files for two examples
of this technique, an open 16-note "fine-tuning" and what might be
described as a circulating 24-note well-untemperament.
With either RI tuning, we get intervals clustering around some regions
of the continuum, but scattered a bit, in a somewhat freer way than
might occur in a usual well-temperament or the like.
Am I "playing with numbers" -- of course, that's the name of the game
here.
In fact, I'm tempted to call this a kind of "Monte Carlo method" in
the informal rather than strict sense, I'd guess: a kind of aleatory
process in which we pick likely-looking ratios, and see just what
kinds of other ratios they produce, and what ranges of variation. We
can then adjust the defining intervals to taste.
There's fun in using a 7:5 to define a tuning that doesn't have that
much to do with conventional 5-limit or 7-limit harmony, or in picking
other ratios with an historical tradition behind them to come up with
something maybe not quite that traditional.
This only speaks to part of RI, of course, and I find your "mediants
of simplicity," Jacky, fascinating (two complex ratios yielding a
simple ratio as their mediant).
Maybe there's a certain parsimonious viewpoint that says, "Unless this
ratio is audibly discreet, it's just superfluous mathematics."
However, the emblematic use of numbers in music is something that I
treasure.
For example, the Pythagorean ratios of 81:64 and 32:27 for major and
minor thirds are part of a centuries-old tradition which I revere, and
these ratios are a part of my musical life. They may not have the same
significance to others, and there's room on this list for lots of
different approaches.
Some mathematical structures of beauty may have more than one musical
application -- for example, Erv Wilson's Scale Tree, with branching
mediants which can be read either as fractions of an octave or as
rational interval ratios. Let us marvel at the structure and celebrate
the diversity of interpretations, hopefully not only theoretical but
musical.
Here is a 16-note "fine-tuning" designed for a two-manual keyboard,
with one manual in the usual 12-note Eb-G# range, the other sharing
the diatonic notes and Bb while including the other accidentals: Db,
D#, Gb, Ab. Basically this is an RI variation on a 17-note system of
early 15th-century Europe (Gb-A#), but without A#, so that both
keyboards can share the more common Bb, making them easier to play
together with contrasting timbres if this is desired.
In neo-Gothic terms, it's a kind of "11-flavor" fine-tuning, with
minor thirds around 13:11 or 33:28, and major thirds around 14:11.
Here I give a Scala file plus some Scala output showing values in
cents, with Pythagorean-like note names added in brackets:
----------------- Scala file starts on next line of text ------------
! neogp16a.scl
!
Scale from mainly prime-to-prime ratios and octave complements (Gb-D#)
16
!
43/41
6439/5989
53/47
13/11
137/113
47/37
4/3
7/5
1781/1243
3/2
11/7
21/13
22/13
946/533
82/43
2/1
------------- Scala file ended on previous line of text --------
Scale from mainly prime-to-prime ratios and octave complements (Gb-D#)
0: 1/1 0.000000 unison, perfect prime [C]
1: 43/41 82.45533 [Db]
2: 6439/5989 125.4258 [C#]
3: 53/47 207.9980 [D]
4: 13/11 289.2098 [Eb]
5: 137/113 333.4238 [D#]
6: 47/37 414.1627 [E]
7: 4/3 498.0452 perfect fourth [F]
8: 7/5 582.5125 septimal tritone, BP fourth [Gb]
9: 1781/1243 622.6337 [F#]
10: 3/2 701.9553 perfect fifth [G]
11: 11/7 782.4924 undecimal augmented fifth [Ab]
12: 21/13 830.2536 [G#]
13: 22/13 910.7907 [A]
14: 946/533 993.2460 [Bb]
15: 82/43 1117.545 [B]
16: 2/1 1200.000 octave [C]
Here's a Scala file for a 24-note well-untemperament, not specifically
neo-Gothic although it could be so used -- people might find other
interesting applications. Again, I give a Scala file plus Scala's
listing of interval sizes in cents:
------------------ Scala file starts next line of text -------------
! jiri24a.scl
!
Just/rational intonation system -- with circulating 24-note set
24
!
36/35
18/17
12/11
9/8
15/13
19/16
11/9
29/23
13/10
4/3
11/8
17/12
16/11
3/2
20/13
46/29
18/11
32/19
26/15
16/9
11/6
17/9
35/18
2/1
------------- Scala file ended on previous line of text --------
Just/rational intonation system -- with circulating 24-note set
0: 1/1 0.000000 unison, perfect prime
1: 36/35 48.77040 1/4-tone, septimal diesis
2: 18/17 98.95463 Arabic lute index finger
3: 12/11 150.6371 3/4-tone, undecimal neutral second
4: 9/8 203.9100 major whole tone
5: 15/13 247.7411
6: 19/16 297.5131 19th harmonic
7: 11/9 347.4080 undecimal neutral third
8: 29/23 401.3030
9: 13/10 454.2141
10: 4/3 498.0452 perfect fourth
11: 11/8 551.3181 undecimal semi-augmented fourth
12: 17/12 603.0007 2nd septendecimal tritone
13: 16/11 648.6823 undecimal semi-diminished fifth
14: 3/2 701.9553 perfect fifth
15: 20/13 745.7864
16: 46/29 798.6975
17: 18/11 852.5924 undecimal neutral sixth
18: 32/19 902.4874 19th subharmonic
19: 26/15 952.2593
20: 16/9 996.0905 Pythagorean minor seventh
21: 11/6 1049.363 21/4-tone, undecimal neutral seventh
22: 17/9 1101.045
23: 35/18 1151.230 septimal semi-diminished octave
24: 2/1 1200.000 octave
Most appreciatively,
Margo Schulter
[email protected]
Full thread (2 messages)
From: M. Schulter (2001-02-01)
Subject: Re: Fine-tunings and well-untemperaments (for Jacky Ligon)
Hello there to Jacky Ligon, and I'd like to affirm a "prime directive"
of music and intonational approaches to it: diversity, with complex
rational intonations as one road open for the taking.
I was deeply moved to see your rational intonation (RI) tuning
approximating my favored major third at 14:11, as well as Robert
Walker's proposal of an acoustical basis for the kind of weighted
variations on Classic Mediants (e.g. minor thirds at 20:17, 13:11)
which I have described as "walking the gentle plateau," a poetic
phrase borrowed in part from Dave Keenan (without in any way implying that
he necessarily endorses this integer-based technique as one alternative
method of exploring the continuum).
What special "properties" the series of primes may have for music is
an open question, but I would say that tuning generally is a process
of playing with numbers and seeing what interesting musics may
result.
For me, a special feature of complex RI is the way it can produce what
I might term "fine-tunings" of various kinds somewhat analogous to
well-temperaments -- without necessarily the implication of closure,
although the technique lends itself to circulating tunings also.
Since "implementation is the most sincere form of flattery," to quote
one software developer, why don't I give Scala files for two examples
of this technique, an open 16-note "fine-tuning" and what might be
described as a circulating 24-note well-untemperament.
With either RI tuning, we get intervals clustering around some regions
of the continuum, but scattered a bit, in a somewhat freer way than
might occur in a usual well-temperament or the like.
Am I "playing with numbers" -- of course, that's the name of the game
here.
In fact, I'm tempted to call this a kind of "Monte Carlo method" in
the informal rather than strict sense, I'd guess: a kind of aleatory
process in which we pick likely-looking ratios, and see just what
kinds of other ratios they produce, and what ranges of variation. We
can then adjust the defining intervals to taste.
There's fun in using a 7:5 to define a tuning that doesn't have that
much to do with conventional 5-limit or 7-limit harmony, or in picking
other ratios with an historical tradition behind them to come up with
something maybe not quite that traditional.
This only speaks to part of RI, of course, and I find your "mediants
of simplicity," Jacky, fascinating (two complex ratios yielding a
simple ratio as their mediant).
Maybe there's a certain parsimonious viewpoint that says, "Unless this
ratio is audibly discreet, it's just superfluous mathematics."
However, the emblematic use of numbers in music is something that I
treasure.
For example, the Pythagorean ratios of 81:64 and 32:27 for major and
minor thirds are part of a centuries-old tradition which I revere, and
these ratios are a part of my musical life. They may not have the same
significance to others, and there's room on this list for lots of
different approaches.
Some mathematical structures of beauty may have more than one musical
application -- for example, Erv Wilson's Scale Tree, with branching
mediants which can be read either as fractions of an octave or as
rational interval ratios. Let us marvel at the structure and celebrate
the diversity of interpretations, hopefully not only theoretical but
musical.
Here is a 16-note "fine-tuning" designed for a two-manual keyboard,
with one manual in the usual 12-note Eb-G# range, the other sharing
the diatonic notes and Bb while including the other accidentals: Db,
D#, Gb, Ab. Basically this is an RI variation on a 17-note system of
early 15th-century Europe (Gb-A#), but without A#, so that both
keyboards can share the more common Bb, making them easier to play
together with contrasting timbres if this is desired.
In neo-Gothic terms, it's a kind of "11-flavor" fine-tuning, with
minor thirds around 13:11 or 33:28, and major thirds around 14:11.
Here I give a Scala file plus some Scala output showing values in
cents, with Pythagorean-like note names added in brackets:
----------------- Scala file starts on next line of text ------------
! neogp16a.scl
!
Scale from mainly prime-to-prime ratios and octave complements (Gb-D#)
16
!
43/41
6439/5989
53/47
13/11
137/113
47/37
4/3
7/5
1781/1243
3/2
11/7
21/13
22/13
946/533
82/43
2/1
------------- Scala file ended on previous line of text --------
Scale from mainly prime-to-prime ratios and octave complements (Gb-D#)
0: 1/1 0.000000 unison, perfect prime [C]
1: 43/41 82.45533 [Db]
2: 6439/5989 125.4258 [C#]
3: 53/47 207.9980 [D]
4: 13/11 289.2098 [Eb]
5: 137/113 333.4238 [D#]
6: 47/37 414.1627 [E]
7: 4/3 498.0452 perfect fourth [F]
8: 7/5 582.5125 septimal tritone, BP fourth [Gb]
9: 1781/1243 622.6337 [F#]
10: 3/2 701.9553 perfect fifth [G]
11: 11/7 782.4924 undecimal augmented fifth [Ab]
12: 21/13 830.2536 [G#]
13: 22/13 910.7907 [A]
14: 946/533 993.2460 [Bb]
15: 82/43 1117.545 [B]
16: 2/1 1200.000 octave [C]
Here's a Scala file for a 24-note well-untemperament, not specifically
neo-Gothic although it could be so used -- people might find other
interesting applications. Again, I give a Scala file plus Scala's
listing of interval sizes in cents:
------------------ Scala file starts next line of text -------------
! jiri24a.scl
!
Just/rational intonation system -- with circulating 24-note set
24
!
36/35
18/17
12/11
9/8
15/13
19/16
11/9
29/23
13/10
4/3
11/8
17/12
16/11
3/2
20/13
46/29
18/11
32/19
26/15
16/9
11/6
17/9
35/18
2/1
------------- Scala file ended on previous line of text --------
Just/rational intonation system -- with circulating 24-note set
0: 1/1 0.000000 unison, perfect prime
1: 36/35 48.77040 1/4-tone, septimal diesis
2: 18/17 98.95463 Arabic lute index finger
3: 12/11 150.6371 3/4-tone, undecimal neutral second
4: 9/8 203.9100 major whole tone
5: 15/13 247.7411
6: 19/16 297.5131 19th harmonic
7: 11/9 347.4080 undecimal neutral third
8: 29/23 401.3030
9: 13/10 454.2141
10: 4/3 498.0452 perfect fourth
11: 11/8 551.3181 undecimal semi-augmented fourth
12: 17/12 603.0007 2nd septendecimal tritone
13: 16/11 648.6823 undecimal semi-diminished fifth
14: 3/2 701.9553 perfect fifth
15: 20/13 745.7864
16: 46/29 798.6975
17: 18/11 852.5924 undecimal neutral sixth
18: 32/19 902.4874 19th subharmonic
19: 26/15 952.2593
20: 16/9 996.0905 Pythagorean minor seventh
21: 11/6 1049.363 21/4-tone, undecimal neutral seventh
22: 17/9 1101.045
23: 35/18 1151.230 septimal semi-diminished octave
24: 2/1 1200.000 octave
Most appreciatively,
Margo Schulter
[email protected]
From: [email protected] (2001-02-02)
Subject: Re: Fine-tunings and well-untemperaments (for Jacky Ligon)
--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> For example, the Pythagorean ratios of 81:64 and 32:27 for major and
> minor thirds are part of a centuries-old tradition which I revere,
and
> these ratios are a part of my musical life. They may not have the
same
> significance to others, and there's room on this list for lots of
> different approaches.
Sure. But it should be noted that 64:81 and 27:32 arise naturally from
chains of just intervals or their octave inversions while ratios of
primes can never arise in this manner. Readers are bound to conclude
(in the absence of accompanying words to the contrary) that the author
who uses large prime ratios feels that they have some special
psychoacoustic significance (i.e. "audibly discrete", as you say). So
far, I am not aware of any evidence of this.
Regards,
-- Dave Keenan