Full thread (71 messages)
From: Paul Erlich (2003-12-08)
Subject: Question for Manuel, Gene, Kees, or whomever . . .
What is the Kees van Prooijen expressibility-reduced (aka odd-limit
reduced) 72-tone 11-limit periodicity block? In other words, each
interval of 72-equal expressed as the simplest (in odd limit) 11-
limit ratio with which it is epimorphic, or whatever the right way of
saying that is.
George Secor's paper includes a big 72-equal keyboard diagram. It's
marked with ratios, and I don't like them :)
From: Gene Ward Smith (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> What is the Kees van Prooijen expressibility-reduced (aka odd-limit
> reduced) 72-tone 11-limit periodicity block? In other words, each
> interval of 72-equal expressed as the simplest (in odd limit) 11-
> limit ratio with which it is epimorphic, or whatever the right way
of
> saying that is.
I was doing this sort of thing using MT reduction. What is the
criterion for van Prooijen reduction?
From: Paul Erlich (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > What is the Kees van Prooijen expressibility-reduced (aka odd-
limit
> > reduced) 72-tone 11-limit periodicity block? In other words, each
> > interval of 72-equal expressed as the simplest (in odd limit) 11-
> > limit ratio with which it is epimorphic, or whatever the right
way
> of
> > saying that is.
>
> I was doing this sort of thing using MT reduction. What is the
> criterion for van Prooijen reduction?
each ratio is a "ratio of" the smallest possible odd number. see
http://www.sonic-arts.org/dict/ratio-of.htm
http://www.kees.cc/tuning/perbl.html
From: Gene Ward Smith (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > I was doing this sort of thing using MT reduction. What is the
> > criterion for van Prooijen reduction?
>
> each ratio is a "ratio of" the smallest possible odd number. see
>
> http://www.sonic-arts.org/dict/ratio-of.htm
>
> http://www.kees.cc/tuning/perbl.html
Why is this preferable to removing any factors of 2 and taking the
product of numerator and denominator?
From: Paul Erlich (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > I was doing this sort of thing using MT reduction. What is the
> > > criterion for van Prooijen reduction?
> >
> > each ratio is a "ratio of" the smallest possible odd number. see
> >
> > http://www.sonic-arts.org/dict/ratio-of.htm
> >
> > http://www.kees.cc/tuning/perbl.html
>
> Why is this preferable to removing any factors of 2 and taking the
> product of numerator and denominator?
It's *way* preferable. The latter is based on a false view of octave-
reducing the tenney lattice, at best. Do you think 5:3 and 15:8
should count as equally 'distant' octave-equivalence classes from
1:1? What I was asking about is supported by Partch, octave-
equivalent harmonic entropy, and pretty straighforward explanations I
posted for Maximiliano on the tuning list . .
From: Gene Ward Smith (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> It's *way* preferable. The latter is based on a false view of
octave-
> reducing the tenney lattice, at best. Do you think 5:3 and 15:8
> should count as equally 'distant' octave-equivalence classes from
> 1:1? What I was asking about is supported by Partch, octave-
> equivalent harmonic entropy, and pretty straighforward explanations
I
> posted for Maximiliano on the tuning list . .
Is the measure in question one which involves removing all factors of
two, reducing to lowest form p/q, and taking max(p,q)?
From: Paul Erlich (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > It's *way* preferable. The latter is based on a false view of
> octave-
> > reducing the tenney lattice, at best. Do you think 5:3 and 15:8
> > should count as equally 'distant' octave-equivalence classes from
> > 1:1? What I was asking about is supported by Partch, octave-
> > equivalent harmonic entropy, and pretty straighforward
explanations
> I
> > posted for Maximiliano on the tuning list . .
>
> Is the measure in question one which involves removing all factors
of
> two, reducing to lowest form p/q, and taking max(p,q)?
yes.
From: Gene Ward Smith (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > Is the measure in question one which involves removing all
factors
> of
> > two, reducing to lowest form p/q, and taking max(p,q)?
>
> yes.
Another possibility would be a variant on the Euclidean reduced
scales I did once--minimal geometric complexity.
From: Paul Erlich (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > Is the measure in question one which involves removing all
> factors
> > of
> > > two, reducing to lowest form p/q, and taking max(p,q)?
> >
> > yes.
>
> Another possibility would be a variant on the Euclidean reduced
> scales I did once--minimal geometric complexity.
Do you want to try answering the question before changing it?
From: Gene Ward Smith (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > Another possibility would be a variant on the Euclidean reduced
> > scales I did once--minimal geometric complexity.
>
> Do you want to try answering the question before changing it?
Har har har--little do you know, my friend!
I first Tenney reduced in the range -35 to 36 steps, and shifted
anything less than 1 up an octave. I then reduced this scale for
minimum geometric complexity. Finally reduced it your way. The result
in all three cases turns out to be *exactly the same*!
Take that. :)
I think under the circumstances I am justified in calling this the
canonical 11-limit reduced 72-epimorphic JI scale, in whatever order
you prefer those words in.
! red72_11.scl
Canonical 11-limit reduced scale
72
!
81/80
45/44
33/32
25/24
21/20
35/33
77/72
175/162
35/32
54/49
49/44
55/49
198/175
8/7
81/70
64/55
33/28
25/21
6/5
40/33
11/9
99/80
5/4
63/50
14/11
77/60
35/27
21/16
175/132
147/110
66/49
49/36
48/35
242/175
88/63
99/70
63/44
175/121
35/24
72/49
49/33
220/147
264/175
32/21
54/35
120/77
11/7
100/63
8/5
160/99
18/11
33/20
5/3
42/25
56/33
55/32
140/81
7/4
175/99
98/55
88/49
49/27
64/35
324/175
144/77
66/35
40/21
48/25
64/33
88/45
160/81
2/1
From: Gene Ward Smith (2003-12-08)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I think under the circumstances I am justified in calling this the
> canonical 11-limit reduced 72-epimorphic JI scale, in whatever
order
> you prefer those words in.
I see however it doesn't contain either a 9/8 or a 10/9, so I'd
better check to see if I've totally goofed again.
From: Paul Erlich (2003-12-09)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > Another possibility would be a variant on the Euclidean reduced
> > > scales I did once--minimal geometric complexity.
> >
> > Do you want to try answering the question before changing it?
>
> Har har har--little do you know, my friend!
>
> I first Tenney reduced in the range -35 to 36 steps, and shifted
> anything less than 1 up an octave. I then reduced this scale for
> minimum geometric complexity. Finally reduced it your way. The
result
> in all three cases turns out to be *exactly the same*!
>
> Take that. :)
I knew Tenney would agree with "my way" if you shifted to -1/2 to 1/2
octaves. Didn't know about geo. complexity.
>
> I think under the circumstances I am justified in calling this the
> canonical 11-limit reduced 72-epimorphic JI scale, in whatever
order
> you prefer those words in.
>
> ! red72_11.scl
> Canonical 11-limit reduced scale
> 72
> !
> 81/80
> 45/44
> 33/32
> 25/24
> 21/20
> 35/33
> 77/72
> 175/162
> 35/32
> 54/49
> 49/44
> 55/49
> 198/175
> 8/7
> 81/70
> 64/55
> 33/28
> 25/21
> 6/5
> 40/33
> 11/9
> 99/80
> 5/4
> 63/50
> 14/11
> 77/60
> 35/27
> 21/16
> 175/132
> 147/110
> 66/49
> 49/36
> 48/35
> 242/175
> 88/63
> 99/70
> 63/44
> 175/121
> 35/24
> 72/49
> 49/33
> 220/147
> 264/175
> 32/21
> 54/35
> 120/77
> 11/7
> 100/63
> 8/5
> 160/99
> 18/11
> 33/20
> 5/3
> 42/25
> 56/33
> 55/32
> 140/81
> 7/4
> 175/99
> 98/55
> 88/49
> 49/27
> 64/35
> 324/175
> 144/77
> 66/35
> 40/21
> 48/25
> 64/33
> 88/45
> 160/81
> 2/1
Thanks, Gene. I appreciate it.
From: Paul Erlich (2003-12-09)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > I think under the circumstances I am justified in calling this
the
> > canonical 11-limit reduced 72-epimorphic JI scale, in whatever
> order
> > you prefer those words in.
>
> I see however it doesn't contain either a 9/8 or a 10/9, so I'd
> better check to see if I've totally goofed again.
Yes, it would seem so . . .
From: Manuel Op de Coul (2003-12-09)
Subject: Re: [tuning-math] Question for Manuel, Gene, Kees, or whomever . . .
What's the 11-limit TM-reduced basis of 72-tET again?
Manuel
From: Manuel Op de Coul (2003-12-09)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
Here's an attempt, but it's possible that Gene finds a
still better one. It's also strictly proper.
0: 1/1 0.000 unison, perfect prime
1: 126/125 13.795 small septimal comma
2: 45/44 38.906 1/5-tone
3: 33/32 53.273 undecimal comma, 33rd harmonic
4: 25/24 70.672 classic chromatic semitone, minor chroma
5: 21/20 84.467 minor semitone
6: 35/33 101.867
7: 77/72 116.234
8: 27/25 133.238 large limma, BP small semitone
9: 49/45 147.428 BP minor semitone
10: 11/10 165.004 4/5-tone, Ptolemy's second
11: 125/112 190.115 classic augmented semitone
12: 9/8 203.910 major whole tone
13: 25/22 221.309
14: 55/48 235.677
15: 81/70 252.680 Al-Hwarizmi's lute middle finger
16: 7/6 266.871 septimal minor third
17: 33/28 284.447 undecimal minor third
18: 25/21 301.847 BP second, quasi-tempered minor third
19: 6/5 315.641 minor third
20: 175/144 337.543
21: 27/22 354.547 neutral third, Zalzal wosta of al-Farabi
22: 99/80 368.914
23: 5/4 386.314 major third
24: 63/50 400.108 quasi-equal major third
25: 14/11 417.508 undecimal diminished fourth or major third
26: 77/60 431.875
27: 35/27 449.275 9/4-tone, septimal semi-diminished fourth
28: 98/75 463.069
29: 175/132 488.180
30: 75/56 505.757
31: 27/20 519.551 acute fourth
32: 15/11 536.951 undecimal augmented fourth
33: 11/8 551.318 undecimal semi-augmented fourth
34: 25/18 568.717 classic augmented fourth
35: 7/5 582.512 septimal or Huygens' tritone, BP fourth
36: 99/70 600.088 2nd quasi-equal tritone
37: 10/7 617.488 Euler's tritone
38: 81/56 638.994
39: 35/24 653.185 septimal semi-diminished fifth
40: 81/55 670.188
41: 49/33 684.379
42: 3/2 701.955 perfect fifth
43: 121/80 716.322
44: 55/36 733.722
45: 77/50 747.516
46: 14/9 764.916 septimal minor sixth
47: 243/154 789.631
48: 35/22 803.822
49: 45/28 821.398
50: 81/50 835.193 acute minor sixth
51: 18/11 852.592 undecimal neutral sixth
52: 33/20 866.959
53: 5/3 884.359 major sixth, BP sixth
54: 42/25 898.153 quasi-tempered major sixth
55: 56/33 915.553
56: 55/32 937.632
57: 210/121 954.459
58: 7/4 968.826 harmonic seventh
59: 99/56 986.402
60: 98/55 1000.020
61: 9/5 1017.596 just minor seventh, BP seventh
62: 49/27 1031.787
63: 11/6 1049.363 21/4-tone, undecimal neutral seventh
64: 231/125 1063.158
65: 15/8 1088.269 classic major seventh
66: 125/66 1105.668
67: 21/11 1119.463
68: 27/14 1137.039 septimal major seventh
69: 35/18 1151.230 septimal semi-diminished octave
70: 55/28 1168.806
71: 99/50 1182.601
72: 2/1 1200.000 octave
Manuel
From: George D. Secor (2003-12-09)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> George Secor's paper includes a big 72-equal keyboard diagram. It's
> marked with ratios, and I don't like them :)
So you don't like ratios, eh? So why did you ever join tuning-
math? ;-)
--George
From: Kees van Prooijen (2003-12-09)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
Preliminary, I think this is it.
Hope layout wont be totally screwed up.
2 : 3 ( 1 0 0 0 )
3 : 4 ( -1 0 0 0 )
4 : 5 ( 0 1 0 0 )
5 : 6 ( 1 -1 0 0 )
6 : 7 ( -1 0 1 0 )
7 : 8 ( 0 0 -1 0 )
8 : 9 ( 2 0 0 0 )
9 : 10 ( -2 1 0 0 )
10 : 11 ( 0 -1 0 1 ) 2
11 : 12 ( 1 0 0 -1 )
14 : 15 ( 1 1 -1 0 )
15 : 16 ( -1 -1 0 0 )
20 : 21 ( 1 -1 1 0 )
21 : 22 ( -1 0 -1 1 )
24 : 25 ( -1 2 0 0 ) 3
27 : 28 ( -3 0 1 0 ) 7
32 : 33 ( 1 0 0 1 ) 10
35 : 36 ( 2 -1 -1 0 )
44 : 45 ( 2 1 0 -1 )
48 : 49 ( -1 0 2 0 )
49 : 50 ( 0 2 -2 0 )
54 : 55 ( -3 1 0 1 )
55 : 56 ( 0 -1 1 -1 ) 10
63 : 64 ( -2 0 -1 0 ) 10
80 : 81 ( 4 -1 0 0 )
98 : 99 ( 2 0 -2 1 )
99 : 100 ( -2 2 0 -1 ) 12
120 : 121 ( -1 -1 0 2 ) 14
125 : 126 ( 2 -3 1 0 )
175 : 176 ( 0 -2 -1 1 ) 15
224 : 225 ( 2 2 -1 0 ) 31
242 : 243 ( 5 0 0 -2 )
384 : 385 ( -1 1 1 1 )
440 : 441 ( 2 -1 2 -1 )
539 : 540 ( 3 1 -2 -1 )
2400 : 2401 ( -1 -2 4 0 )
3024 : 3025 ( -3 2 -1 2 )
4374 : 4375 ( -7 4 1 0 ) 72
9800 : 9801 ( 4 -2 -2 2 )
151250 : 151263 ( 2 -4 5 -2 )
1771470 : 1771561 ( -11 -1 0 6 ) 342
3294172 : 3294225 ( 2 2 -7 4 )
781250000 : 781258401 ( 2 -11 2 6 )
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> What is the Kees van Prooijen expressibility-reduced (aka odd-limit
> reduced) 72-tone 11-limit periodicity block? In other words, each
> interval of 72-equal expressed as the simplest (in odd limit) 11-
> limit ratio with which it is epimorphic, or whatever the right way
of
> saying that is.
>
> George Secor's paper includes a big 72-equal keyboard diagram. It's
> marked with ratios, and I don't like them :)
From: Kees van Prooijen (2003-12-09)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
oops, I was too hasty. I thought you just wanted the simplest basis
for the block. Needs more work then. No time :-(
--- In tuning-math@yahoogroups.com, "Kees van Prooijen" <lists@k...>
wrote:
> Preliminary, I think this is it.
> Hope layout wont be totally screwed up.
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > What is the Kees van Prooijen expressibility-reduced (aka odd-
limit
> > reduced) 72-tone 11-limit periodicity block? In other words, each
> > interval of 72-equal expressed as the simplest (in odd limit) 11-
> > limit ratio with which it is epimorphic, or whatever the right
way
> of
> > saying that is.
> >
From: Gene Ward Smith (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> What's the 11-limit TM-reduced basis of 72-tET again?
See http://groups.yahoo.com/group/tuning-math/message/7392
for these. Its <225/224, 243/242, 385/384, 4000/3993>
From: Kees van Prooijen (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
Does this look like something?
1
121/120 1.3e-003
2
55/54 9.0e-004
3
36/35 7.1e-004
4
28/27 2.1e-003
80/77 2.9e-004
5
21/20 6.5e-004
6
16/15 6.8e-003
35/33 1.1e-003
7
15/14 1.6e-003
77/72 2.5e-004
8
27/25 5.5e-005
9
12/11 3.7e-004
10
11/10 9.6e-004
54/49 8.9e-004
11
10/9 5.4e-004
12
9/8 2.3e-003
28/25 2.2e-003
55/49 1.2e-005
13
25/22 2.7e-003
14
8/7 1.2e-003
63/55 1.0e-003
15
55/48 8.3e-003
64/55 7.1e-003
81/70 1.5e-003
16
7/6 1.2e-004
17
32/27 6.2e-003
33/28 6.4e-004
18
25/21 1.1e-003
19
6/5 5.9e-004
20
11/9 8.1e-003
40/33 1.7e-004
21
11/9 1.5e-003
49/40 7.7e-004
60/49 3.6e-004
22
27/22 7.0e-003
56/45 6.9e-003
99/80 1.3e-003
100/81 1.1e-003
23
5/4 1.7e-003
96/77 8.8e-004
24
44/35 2.2e-003
63/50 6.3e-005
25
14/11 4.9e-004
26
9/7 1.0e-003
77/60 8.4e-004
27
35/27 4.2e-004
28
21/16 2.4e-003
55/42 1.1e-004
29
21/16 7.3e-003
33/25 1.6e-003
30
4/3 1.1e-003
31
27/20 1.7e-003
66/49 6.0e-004
32
15/11 2.1e-003
49/36 2.4e-004
33
11/8 7.6e-004
34
25/18 1.2e-003
35
7/5 4.7e-004
36
45/32 5.6e-003
64/45 5.6e-003
99/70 5.1e-005
From: Manuel Op de Coul (2003-12-10)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
>Its <225/224, 243/242, 385/384, 4000/3993>
I tried that one indeed, but it's very uneven.
Also the triangular lattice size is larger than the
scale I posted has.
I wouldn't know how to assemble a PB from Kees' list.
Manuel
From: Paul Erlich (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Here's an attempt, but it's possible that Gene finds a
> still better one. It's also strictly proper.
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 126/125 13.795 small septimal comma
Obviously 81/80 is simpler already. Sorry guys I'm behind.
From: Manuel Op de Coul (2003-12-10)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
> 1: 126/125 13.795 small septimal comma
>Obviously 81/80 is simpler already. Sorry guys I'm behind.
Whoa, something is wrong with my algorithm.
Manuel
From: Paul Erlich (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Kees van Prooijen" <lists@k...>
wrote:
> Does this look like something?
>
> 1
> 121/120
Thanks Kees, but obviously 81/80 also maps to 1 degree of 72 and it's
got lower "expressibility" than 121/120.
> 35
> 7/5 4.7e-004
> 36
> 45/32 5.6e-003
> 64/45 5.6e-003
> 99/70 5.1e-005
This is clearly incorrect, 45/32 maps to 35 degrees of 72, not 36. It
has to be the same as 7/5 since 225:224 vanishes in 72, as you know.
I didn't even expect to hear from you, so thanks. I'm still hoping
Gene and/or Manuel can give the solution, I didn't think it would be
this hard for them given similar things they've posted before . . .
From: Manuel Op de Coul (2003-12-10)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
I forgot that the deviation was also weighted in.
So the result is now this, it became more uneven and also
improper but it's much better.
0: 1/1 0.000 unison, perfect prime
1: 81/80 21.506 syntonic comma, Didymus comma
2: 45/44 38.906 1/5-tone
3: 33/32 53.273 undecimal comma, 33rd harmonic
4: 25/24 70.672 classic chromatic semitone, minor chroma
5: 21/20 84.467 minor semitone
6: 35/33 101.867
7: 15/14 119.443 major diatonic semitone
8: 27/25 133.238 large limma, BP small semitone
9: 12/11 150.637 3/4-tone, undecimal neutral second
10: 11/10 165.004 4/5-tone, Ptolemy's second
11: 10/9 182.404 minor whole tone
12: 9/8 203.910 major whole tone
13: 25/22 221.309
14: 8/7 231.174 septimal whole tone
15: 81/70 252.680 Al-Hwarizmi's lute middle finger
16: 7/6 266.871 septimal minor third
17: 33/28 284.447 undecimal minor third
18: 25/21 301.847 BP second, quasi-tempered minor third
19: 6/5 315.641 minor third
20: 40/33 333.041
21: 11/9 347.408 undecimal neutral third
22: 100/81 364.807 grave major third
23: 5/4 386.314 major third
24: 44/35 396.178
25: 14/11 417.508 undecimal diminished fourth or major third
26: 9/7 435.084 septimal major third, BP third
27: 35/27 449.275 9/4-tone, septimal semi-diminished fourth
28: 21/16 470.781 narrow fourth
29: 33/25 480.646 2 pentatones
30: 4/3 498.045 perfect fourth
31: 27/20 519.551 acute fourth
32: 15/11 536.951 undecimal augmented fourth
33: 11/8 551.318 undecimal semi-augmented fourth
34: 25/18 568.717 classic augmented fourth
35: 7/5 582.512 septimal or Huygens' tritone, BP fourth
36: 99/70 600.088 2nd quasi-equal tritone
37: 10/7 617.488 Euler's tritone
38: 36/25 631.283 classic diminished fifth
39: 16/11 648.682 undecimal semi-diminished fifth
40: 22/15 663.049 undecimal diminished fifth
41: 40/27 680.449 grave fifth
42: 3/2 701.955 perfect fifth
43: 50/33 719.354 3 pentatones
44: 32/21 729.219 wide fifth
45: 54/35 750.725 septimal semi-augmented fifth
46: 14/9 764.916 septimal minor sixth
47: 11/7 782.492 undecimal augmented fifth
48: 35/22 803.822
49: 8/5 813.686 minor sixth
50: 81/50 835.193 acute minor sixth
51: 18/11 852.592 undecimal neutral sixth
52: 33/20 866.959
53: 5/3 884.359 major sixth, BP sixth
54: 42/25 898.153 quasi-tempered major sixth
55: 56/33 915.553
56: 12/7 933.129 septimal major sixth
57: 140/81 947.320
58: 7/4 968.826 harmonic seventh
59: 44/25 978.691
60: 16/9 996.090 Pythagorean minor seventh
61: 9/5 1017.596 just minor seventh, BP seventh
62: 20/11 1034.996 large minor seventh
63: 11/6 1049.363 21/4-tone, undecimal neutral seventh
64: 50/27 1066.762 grave major seventh
65: 15/8 1088.269 classic major seventh
66: 66/35 1098.133
67: 21/11 1119.463
68: 48/25 1129.328 classic diminished octave
69: 64/33 1146.727 33rd subharmonic
70: 88/45 1161.094
71: 160/81 1178.494 octave - syntonic comma
72: 2/1 1200.000 octave
Manuel
From: Paul Erlich (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> I forgot that the deviation was also weighted in.
> So the result is now this, it became more uneven and also
> improper but it's much better.
Thanks so very much Manuel. Your other methods may prove useful as
well, I appreciate the time you've taken to code all of them.
I hope George, if it's not too late, will consider using these ratios
for his 72-equal keyboard diagram -- the unevenness is probably not
important for him since he told me he intended the ratios to show how
*intervals* look on the keyboard, not as a representation of the
*pitches* (those of you who have been reading my posts for years know
what i mean -- intervals I notate as a:b, while pitches I notate
a/b) . . .
George, you already have some 3-digit numbers, so the below shouldn't
be a problem, should it?
If this isn't acceptable, maybe a 17-limit version of same?
Or feel free to ignore me.
-Paul
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 81/80 21.506 syntonic comma, Didymus comma
> 2: 45/44 38.906 1/5-tone
> 3: 33/32 53.273 undecimal comma, 33rd harmonic
> 4: 25/24 70.672 classic chromatic semitone,
minor chroma
> 5: 21/20 84.467 minor semitone
> 6: 35/33 101.867
> 7: 15/14 119.443 major diatonic semitone
> 8: 27/25 133.238 large limma, BP small semitone
> 9: 12/11 150.637 3/4-tone, undecimal neutral
second
> 10: 11/10 165.004 4/5-tone, Ptolemy's second
> 11: 10/9 182.404 minor whole tone
> 12: 9/8 203.910 major whole tone
> 13: 25/22 221.309
> 14: 8/7 231.174 septimal whole tone
> 15: 81/70 252.680 Al-Hwarizmi's lute middle
finger
> 16: 7/6 266.871 septimal minor third
> 17: 33/28 284.447 undecimal minor third
> 18: 25/21 301.847 BP second, quasi-tempered
minor third
> 19: 6/5 315.641 minor third
> 20: 40/33 333.041
> 21: 11/9 347.408 undecimal neutral third
> 22: 100/81 364.807 grave major third
> 23: 5/4 386.314 major third
> 24: 44/35 396.178
> 25: 14/11 417.508 undecimal diminished fourth
or major third
> 26: 9/7 435.084 septimal major third, BP third
> 27: 35/27 449.275 9/4-tone, septimal semi-
diminished fourth
> 28: 21/16 470.781 narrow fourth
> 29: 33/25 480.646 2 pentatones
> 30: 4/3 498.045 perfect fourth
> 31: 27/20 519.551 acute fourth
> 32: 15/11 536.951 undecimal augmented fourth
> 33: 11/8 551.318 undecimal semi-augmented
fourth
> 34: 25/18 568.717 classic augmented fourth
> 35: 7/5 582.512 septimal or Huygens' tritone,
BP fourth
> 36: 99/70 600.088 2nd quasi-equal tritone
> 37: 10/7 617.488 Euler's tritone
> 38: 36/25 631.283 classic diminished fifth
> 39: 16/11 648.682 undecimal semi-diminished
fifth
> 40: 22/15 663.049 undecimal diminished fifth
> 41: 40/27 680.449 grave fifth
> 42: 3/2 701.955 perfect fifth
> 43: 50/33 719.354 3 pentatones
> 44: 32/21 729.219 wide fifth
> 45: 54/35 750.725 septimal semi-augmented fifth
> 46: 14/9 764.916 septimal minor sixth
> 47: 11/7 782.492 undecimal augmented fifth
> 48: 35/22 803.822
> 49: 8/5 813.686 minor sixth
> 50: 81/50 835.193 acute minor sixth
> 51: 18/11 852.592 undecimal neutral sixth
> 52: 33/20 866.959
> 53: 5/3 884.359 major sixth, BP sixth
> 54: 42/25 898.153 quasi-tempered major sixth
> 55: 56/33 915.553
> 56: 12/7 933.129 septimal major sixth
> 57: 140/81 947.320
> 58: 7/4 968.826 harmonic seventh
> 59: 44/25 978.691
> 60: 16/9 996.090 Pythagorean minor seventh
> 61: 9/5 1017.596 just minor seventh, BP seventh
> 62: 20/11 1034.996 large minor seventh
> 63: 11/6 1049.363 21/4-tone, undecimal neutral
seventh
> 64: 50/27 1066.762 grave major seventh
> 65: 15/8 1088.269 classic major seventh
> 66: 66/35 1098.133
> 67: 21/11 1119.463
> 68: 48/25 1129.328 classic diminished octave
> 69: 64/33 1146.727 33rd subharmonic
> 70: 88/45 1161.094
> 71: 160/81 1178.494 octave - syntonic comma
> 72: 2/1 1200.000 octave
>
> Manuel
From: Kees van Prooijen (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
Hi Paul,
81/80 maps to 1.29 steps and was rejected in the first pass of the
algorithm, where I look for the simplest ratio of successive values
in the series.
45/32 was accepted in the second pass. I agree this causes an
unevenness in acceptance. Still, these are best ratios for the
complexity. Obviously, 99/70 should be the representative.
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Kees van Prooijen"
<lists@k...>
> wrote:
> > Does this look like something?
> >
> > 1
> > 121/120
>
> Thanks Kees, but obviously 81/80 also maps to 1 degree of 72 and
it's
> got lower "expressibility" than 121/120.
>
> > 35
> > 7/5 4.7e-004
> > 36
> > 45/32 5.6e-003
> > 64/45 5.6e-003
> > 99/70 5.1e-005
>
> This is clearly incorrect, 45/32 maps to 35 degrees of 72, not 36.
It
> has to be the same as 7/5 since 225:224 vanishes in 72, as you know.
>
> I didn't even expect to hear from you, so thanks. I'm still hoping
> Gene and/or Manuel can give the solution, I didn't think it would
be
> this hard for them given similar things they've posted before . . .
From: Paul Erlich (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Kees van Prooijen" <lists@k...>
wrote:
> Hi Paul,
>
> 81/80 maps to 1.29 steps and was rejected in the first pass of the
> algorithm, where I look for the simplest ratio of successive values
> in the series.
I'm unclear on what that means.
> 45/32 was accepted in the second pass. I agree this causes an
> unevenness in acceptance. Still, these are best ratios for the
> complexity.
45/32 should *only* map to 35 steps of 72, never to 36 steps of 72,
if you are constructing your periodicity block correctly.
From: Kees van Prooijen (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
I totally agree Paul. I just threw an algorithm together and gave the
raw results. I didn't even consider periodicity blocks. I just tried
to find relatively best rationals for the steps. That, of course,
doesn't have to result in consistent mapping.
Sorry if I only caused confusion.
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Kees van Prooijen"
<lists@k...>
> wrote:
>
> > Hi Paul,
> >
> > 81/80 maps to 1.29 steps and was rejected in the first pass of
the
> > algorithm, where I look for the simplest ratio of successive
values
> > in the series.
>
> I'm unclear on what that means.
>
> > 45/32 was accepted in the second pass. I agree this causes an
> > unevenness in acceptance. Still, these are best ratios for the
> > complexity.
>
> 45/32 should *only* map to 35 steps of 72, never to 36 steps of 72,
> if you are constructing your periodicity block correctly.
From: Paul Erlich (2003-12-10)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
No problem Kees -- my question did concern periodicity blocks, but I
shouldn't have assumed that you'd have read every post. No apology
necessary from your end.
--- In tuning-math@yahoogroups.com, "Kees van Prooijen" <lists@k...>
wrote:
> I totally agree Paul. I just threw an algorithm together and gave
the
> raw results. I didn't even consider periodicity blocks. I just
tried
> to find relatively best rationals for the steps. That, of course,
> doesn't have to result in consistent mapping.
> Sorry if I only caused confusion.
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Kees van Prooijen"
> <lists@k...>
> > wrote:
> >
> > > Hi Paul,
> > >
> > > 81/80 maps to 1.29 steps and was rejected in the first pass of
> the
> > > algorithm, where I look for the simplest ratio of successive
> values
> > > in the series.
> >
> > I'm unclear on what that means.
> >
> > > 45/32 was accepted in the second pass. I agree this causes an
> > > unevenness in acceptance. Still, these are best ratios for the
> > > complexity.
> >
> > 45/32 should *only* map to 35 steps of 72, never to 36 steps of
72,
> > if you are constructing your periodicity block correctly.
From: George D. Secor (2003-12-11)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> ...
> I hope George, if it's not too late, will consider using these
ratios
> for his 72-equal keyboard diagram -- the unevenness is probably not
> important for him since he told me he intended the ratios to show
how
> *intervals* look on the keyboard, not as a representation of the
> *pitches*
The ratios on the keys illustrate pitches, but my intention was that
interval vectors may be deduced from these.
> (those of you who have been reading my posts for years know
> what i mean -- intervals I notate as a:b, while pitches I notate
> a/b) . . .
Yes, I am in total agreement with that, except that I agree with Dave
Keenan that (unless context dictates otherwise) the small number
should precede the colon to reflect the practice of building
intervals (and chords) from the bottom up, and also to be consistent
with the manner in which we indicate ratios for chords, e.g., 4:5:6.
I believe that Helmholtz and Ellis (and most other pre-20th-century)
writers followed this practice. Could it be that putting the larger
number first in a ratio is an *American* convention?
> George, you already have some 3-digit numbers, so the below
shouldn't
> be a problem, should it?
These are ones I have I had to squeeze into a limited amount of space
by redoing the characters pixel-by-pixel, as was also the case for
ratios having 2 digits in both numerator and denominator. It was
very time-consuming and I'm sorry, but there just isn't enough time
now to change this many ratios. (For example, why are 64/63 and
63/32 being replaced by 45/44 and 88/45?)
Besides, something that I wanted to illustrate with the ratios that I
have is that the keyboard does not limit a JI tuning to 72 pitches
per octave.
> If this isn't acceptable, maybe a 17-limit version of same?
>
> Or feel free to ignore me.
I'm not trying to ignore you. If I've been slow to respond, it's
because there has been so much to do lately (including microtonal
projects) that I have hardly had time to read the postings. Several
days ago I had nearly a week's worth of digests unread, and I had to
get the oldest ones out of the way by just scanning the tables of
contents, searching for occurrences of my last name, and then
deleting them. (I now notice that I probably would have missed
replying to this one if I had not been reading the latest ones more
carefully (once I noticed that the presence of a new member on the
main list caused quite a bit of controversy.)
--George
From: Manuel Op de Coul (2003-12-12)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
George wrote:
>(For example, why are 64/63 and
>63/32 being replaced by 45/44 and 88/45?)
Because they have a lower van Prooijen harmonic distance
value. Also a lower Erlich complexity, which is easier:
log2( max( num and den without factors 2 ) ).
Manuel
From: George D. Secor (2003-12-12)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> George wrote:
> >(For example, why are 64/63 and
> >63/32 being replaced by 45/44 and 88/45?)
>
> Because they have a lower van Prooijen harmonic distance
> value. Also a lower Erlich complexity, which is easier:
> log2( max( num and den without factors 2 ) ).
>
> Manuel
However useful those criteria may be, I consider 64/63 and 63/32
simpler because:
1) The prime numbers in the factors are lower; and
2) The range of numbers in the ratios (32 to 64) is lower (than 44 to
88).
Paul, if you're objecting to my use of ratios of 19 in my diagram
because 72-ET is not 19-limit consistent, may I point out that the
only 19-limit consonances that participate in the inconsistency are
19/13 and 26/19, and neither of those appear in the diagram.
Besides, if we're just mapping JI tones to an octave division, I
don't see any problem with a minor inconsistency such as this, as
long as constancy is maintained for every ratio.
--George
From: Manuel Op de Coul (2003-12-12)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
>However useful those criteria may be, I consider 64/63 and 63/32
>simpler because:
>1) The prime numbers in the factors are lower; and
>2) The range of numbers in the ratios (32 to 64) is lower (than 44 to
>88).
Still there are more consonant chords in the scale with the original
pitches.
Manuel
From: Gene Ward Smith (2003-12-12)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
After fixing my program, here is what I am getting for Prooijen and
geometric 11-limit reductions:
! red72_11pro.scl
Prooijen 11-limit reduced scale
72
!
81/80
64/63
33/32
25/24
21/20
128/121
16/15
27/25
12/11
11/10
10/9
9/8
25/22
8/7
297/256
7/6
33/28
32/27
6/5
40/33
11/9
99/80
5/4
81/64
14/11
32/25
128/99
21/16
160/121
4/3
27/20
15/11
11/8
25/18
7/5
512/363
10/7
36/25
16/11
22/15
40/27
3/2
121/80
32/21
99/64
25/16
11/7
128/81
8/5
160/99
18/11
33/20
5/3
27/16
56/33
12/7
512/297
7/4
44/25
16/9
9/5
20/11
11/6
50/27
15/8
121/64
40/21
48/25
64/33
63/32
160/81
2
! red72_11geo.scl
Geometric 11-limit reduced scale
72
!
100/99
56/55
33/32
25/24
21/20
35/33
15/14
27/25
12/11
11/10
10/9
9/8
112/99
8/7
231/200
7/6
33/28
25/21
6/5
40/33
11/9
99/80
5/4
44/35
14/11
9/7
35/27
21/16
33/25
4/3
27/20
15/11
11/8
25/18
7/5
140/99
10/7
36/25
16/11
22/15
40/27
3/2
50/33
32/21
54/35
14/9
11/7
35/22
8/5
160/99
18/11
33/20
5/3
42/25
56/33
12/7
400/231
7/4
99/56
16/9
9/5
20/11
11/6
50/27
28/15
66/35
40/21
48/25
64/33
55/28
99/50
2
From: George D. Secor (2003-12-12)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> >However useful those criteria may be, I consider 64/63 and 63/32
> >simpler because:
> >1) The prime numbers in the factors are lower; and
> >2) The range of numbers in the ratios (32 to 64) is lower (than 44
to
> >88).
>
> Still there are more consonant chords in the scale with the original
> pitches.
>
> Manuel
The next thing that I found was that I would have 28/27 and 27/14
instead of 25/24 and 48/25 (for which I would imagine that your reply
would be the same).
Another question is: why 15/14 and 15/8 (when 16/15 would have been
the inversion of 15/8)?
I may have so many questions regarding what the other pitches in the
scale should be, that to choose ratios on the basis of consonant
chords being produced with them could have us going around in circles.
--George
From: Carl Lumma (2003-12-12)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees
>After fixing my program, here is what I am getting for Prooijen and
>geometric 11-limit reductions:
Thanks for the follow-up, Gene. I wonder what you and Manuel are
doing differently?
-Carl
From: Manuel Op de Coul (2003-12-12)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
George wrote:
>Another question is: why 15/14 and 15/8 (when 16/15 would have been
>the inversion of 15/8)?
Then it wouldn't be epimorphic anymore, nor a constant structure.
The alternatives are limited to changes by the unison vectors of
the PB.
Manuel
From: Manuel Op de Coul (2003-12-12)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
Gene, your geometric reduced scale isn't epimorphic. Is that
a mistake?
Manuel
From: Manuel Op de Coul (2003-12-12)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees
Carl wrote:
>Thanks for the follow-up, Gene. I wonder what you and Manuel are
>doing differently?
We used different periodicity blocks to optimise.
At least that's what I think.
Manuel
From: George D. Secor (2003-12-15)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> George wrote:
> >Another question is: why 15/14 and 15/8 (when 16/15 would have been
> >the inversion of 15/8)?
>
> Then it wouldn't be epimorphic anymore, nor a constant structure.
> The alternatives are limited to changes by the unison vectors of
> the PB.
>
> Manuel
If 15/14 were changed to 16/15, the tuning would still be a constant
structure. But I still haven't worked my way through all of the
intricacies involved in figuring out exactly what epimorphism is
supposed to mean. Is there now a definition that does not require a
degree in mathematics to comprehend?
I don't mean to be giving you guys a hard time, but I can't even
begin to consider changing the ratios on the decimal keyboard diagram
unless I can get the number of required changes reduced to something
that isn't going to eat up a lot of time for other (increasingly
urgent) projects. (I'm presently trying to finish up the rest of the
sagittal graphics for Scala.)
--George
From: Manuel Op de Coul (2003-12-15)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
George wrote:
>If 15/14 were changed to 16/15, the tuning would still be a constant
>structure.
Sorry, yes. I must have made a typo when I tried it.
Those changes are ok indeed, since 225/224 is one of the unison vectors,
the others are 3025/3024, 1375/1372 and 4375/4374.
I'll change it in the archive too.
>Is there now a definition that does not require a
>degree in mathematics to comprehend?
See http://sonic-arts.org/dict/epimorphic.htm
Considering all the higher than 11-limit ratios, I can imagine
it would take a lot of time to change the diagram.
>(I'm presently trying to finish up the rest of the
>sagittal graphics for Scala.)
Splendid, by the way I now don't use xpm files anymore, but png,
but that doesn't matter to you.
Manuel
From: Gene Ward Smith (2003-12-15)
Subject: Re: Question for Manuel, Gene, Kees
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Carl wrote:
> >Thanks for the follow-up, Gene. I wonder what you and Manuel are
> >doing differently?
>
> We used different periodicity blocks to optimise.
> At least that's what I think.
That should make no difference.
From: Gene Ward Smith (2003-12-15)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Gene, your geometric reduced scale isn't epimorphic. Is that
> a mistake?
I just checked this, and I get that it is epimorphic. Can you tell me
where you think there is a problem?
From: Manuel Op de Coul (2003-12-15)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
> Can you tell me
>where you think there is a problem?
I checked and there's a numerical problem in Scala which was silently
ignored. Thanks, I'll see if I can fix it.
Manuel
From: Carl Lumma (2003-12-15)
Subject: epimorphism
http://sonic-arts.org/dict/epimorphic.htm
By the way, the definition on monz's site is woefully
inadequate. For it to work, we need to know Gene's
definition of "scale" which isn't there or on his own
site, so we know what type of value we're plugging in
to h(). We also need to know what the hell kind of
operation is () here.
Furthermore, if CS = epimorphic it should say so. If
there's some weaker relationship it should also say so.
Gene, what do you think about putting all your definitions
in one place, say on Wikipedia, where the interlinks will
happen automagically?
-Carl
From: Manuel Op de Coul (2003-12-15)
Subject: Re: [tuning-math] epimorphism
I found the bug, it will be fixed in the next version.
>Furthermore, if CS = epimorphic it should say so. If
>there's some weaker relationship it should also say so.
Epimorphism implies CS, but not v.v. so it's not the same.
I also explained it in a few sentences in tips.par.
Suggestions for improvement are welcome.
Manuel
From: Manuel Op de Coul (2003-12-15)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees
>> We used different periodicity blocks to optimise.
>> At least that's what I think.
Gene wrote:
>That should make no difference.
But it means the results will not be the same,
doesn't it?
Manuel
From: George D. Secor (2003-12-15)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> George wrote:
> >If 15/14 were changed to 16/15, the tuning would still be a
constant
> >structure.
>
> Sorry, yes. I must have made a typo when I tried it.
> Those changes are ok indeed, since 225/224 is one of the unison
vectors,
> the others are 3025/3024, 1375/1372 and 4375/4374.
> I'll change it in the archive too.
>
> >Is there now a definition that does not require a
> >degree in mathematics to comprehend?
>
> See http://sonic-arts.org/dict/epimorphic.htm
This was the same definition that I saw before, but now that I have
read this again (along with the one for vals), it makes a lot more
sense now, and I now see how this property is more stringent than
CS. I guess it just needed some time to to sink in. :-)
> Considering all the higher than 11-limit ratios, I can imagine
> it would take a lot of time to change the diagram.
It depends mainly on how many of the ratios I have to change. ^
> >(I'm presently trying to finish up the rest of the
> >sagittal graphics for Scala.)
>
> Splendid, by the way I now don't use xpm files anymore, but png,
> but that doesn't matter to you.
Yes, I see that now. BTW, thanks for implementing mid-seq
conversions. I haven't had a chance to try these out yet, but once
things settle down a bit ...
Has anyone ever requested Scala capability to make the computer
keyboard a polyphonic keyboard? With what you now have, it is
necessary to press the key again to stop a tone; instead, releasing a
key would stop a tone. I can think of several possibilities for
arranging tones on the keyboard, and the cursor keys could be
employed to scroll the pitches to avoid running out of keys.
--George
From: Carl Lumma (2003-12-15)
Subject: Re: [tuning-math] epimorphism
>I found the bug, it will be fixed in the next version.
>
>>Furthermore, if CS = epimorphic it should say so. If
>>there's some weaker relationship it should also say so.
>
>Epimorphism implies CS, but not v.v. so it's not the same.
>
>I also explained it in a few sentences in tips.par.
>Suggestions for improvement are welcome.
I didn't know this file existed. I find it generally
obnoxious. Why don't you fold it into the context-sensitive
help? If you really must have a tip-of-the-day, you could
then take it from that unified source.
I read the file for the string "epi". I thought the plain-
English version of Gene's def. on monz's site good, though
I'd still like to get the formal version cleaned up. Note:
() I'm not clear whether "epimorphic" and "JI-epimorphic"
refer to separate things. I don't see why epimorphism would
apply only to JI, but if they really are separate then a
discussion of the non-JI usage is missing.
Also it seems implied that non-torsion = epimorphic. Is
that true?
-Carl
From: Manuel Op de Coul (2003-12-16)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
George wrote:
>Has anyone ever requested Scala capability to make the computer
>keyboard a polyphonic keyboard? With what you now have, it is
>necessary to press the key again to stop a tone; instead, releasing a
>key would stop a tone. I can think of several possibilities for
>arranging tones on the keyboard, and the cursor keys could be
>employed to scroll the pitches to avoid running out of keys.
Yes, Robert Walker has. I also tried to implement it, but it
didn't work under Windows. Under Linux it worked fine (as so often).
So I reversed the change, because under Windows the tone would go
off immediately and I don't like to maintain different versions.
There's a problem with the key-release event, which I hope will be
fixed someday.
Robert was kind enough to show how I could descend into the
Windows depths and might try to work around it, but I avoid
writing platform-specific code like the plague.
Manuel
From: Carl Lumma (2003-12-16)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
>Robert was kind enough to show how I could descend into the
>Windows depths and might try to work around it, but I avoid
>writing platform-specific code like the plague.
Surely a 'typematic' sort of thing would work on all platforms?
-Carl
From: Carl Lumma (2003-12-16)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
>>Surely a 'typematic' sort of thing would work on all platforms?
>
>No idea what you mean by that.
Well if I hold down the "b" key, I'll get a string of bs until
I let it off. That works on all platforms I've ever used. So
one could simply have a buffer which would signal note-off if
it ever emptied. Not that it's necessary to duplicate Robert's
work...
-Carl
From: Manuel Op de Coul (2003-12-16)
Subject: Re: [tuning-math] epimorphism
Carl asked:
>I didn't know this file existed. I find it generally
>obnoxious. Why don't you fold it into the context-sensitive
>help?
Because a large part applies to the gui-elements, and the
help file is about the core part of the program, the command
functions.
> I'm not clear whether "epimorphic" and "JI-epimorphic"
>refer to separate things.
No, but because epimorphism is such a broad term, I called
it "JI-epimorphic" to indicate that it applies to the interval
ratios. (Now Dave probably says then it should be RI-epimorphic
and he would be right).
>Also it seems implied that non-torsion = epimorphic. Is
>that true?
I don't know, but I suspect it's not true.
Manuel
From: Manuel Op de Coul (2003-12-16)
Subject: Re: [tuning-math] epimorphism
>Also it seems implied that non-torsion = epimorphic. Is
>that true?
It's not true because I found a counterexample. The
[225/224, 1029/1024, 25/24] block is not a Constant Structure
and it has no torsion.
Manuel
From: Carl Lumma (2003-12-16)
Subject: Re: [tuning-math] epimorphism
>Carl asked:
>>I didn't know this file existed. I find it generally
>>obnoxious. Why don't you fold it into the context-sensitive
>>help?
>
>Because a large part applies to the gui-elements, and the
>help file is about the core part of the program, the command
>functions.
But now the user has to check two separate sources of
information!
>>Also it seems implied that non-torsion = epimorphic. Is
>>that true?
>
>I don't know, but I suspect it's not true.
The bit I was referring to is here:
>Smith's definition: "Torsion describes a condition wherein an
>independent set of n unison vectors {u1, u2, ..., un} defines a
>non-epimorphic periodicity block, because of the existence of
>a torsion element, meaning an interval which is not the product
>u1^e1 u2^e2 ... un^en of the set of unison vectors raised to
>(positive, negative or zero) integral powers, but some integer
>power of which is. An example would be a block defined by 648/625
>and 2048/2025; here 81/80 is not a product of these commas, but
>(81/80)^2 = (648/625) (2048/2025)^(-1)."
-Carl
From: Manuel Op de Coul (2003-12-16)
Subject: Re: [tuning-math] epimorphism
Carl wrote:
>But now the user has to check two separate sources of
>information!
I've mentioned them both in the readme file. Lots of
programs have separate tips and help file. Some info
might be moved, I agree.
>>I don't know, but I suspect it's not true.
>The bit I was referring to is here:
It doesn't say about the opposite. Anyway it's not true,
and I'll add it to the tip.
Manuel
From: Paul Erlich (2003-12-19)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:
> >
> > >However useful those criteria may be, I consider 64/63 and 63/32
> > >simpler because:
> > >1) The prime numbers in the factors are lower; and
> > >2) The range of numbers in the ratios (32 to 64) is lower (than
44
> to
> > >88).
> >
> > Still there are more consonant chords in the scale with the
original
> > pitches.
> >
> > Manuel
>
> The next thing that I found was that I would have 28/27 and 27/14
> instead of 25/24 and 48/25 (for which I would imagine that your
reply
> would be the same).
George, the reason for choices like these become clearer if you
extend the scale slightly beyond one octave, by octave transposition.
> Another question is: why 15/14 and 15/8 (when 16/15 would have been
> the inversion of 15/8)?
Aha -- looks like Manuel was making an arbitrary choice in the case
of a tie, perhaps letting Tenney complexity break the tie.
From: Paul Erlich (2003-12-19)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> George wrote:
> >Another question is: why 15/14 and 15/8 (when 16/15 would have been
> >the inversion of 15/8)?
>
> Then it wouldn't be epimorphic anymore, nor a constant structure.
Manuel, that can't be right.
> The alternatives are limited to changes by the unison vectors of
> the PB.
Correct, and 225:224 is indeed one of the unison vectors!
From: Paul Erlich (2003-12-19)
Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> After fixing my program, here is what I am getting for Prooijen and
> geometric 11-limit reductions:
>
> ! red72_11pro.scl
> Prooijen 11-limit reduced scale
> 72
> !
> 81/80
> 64/63
Gene -- why isn't this 45/44?
From: Paul Erlich (2003-12-19)
Subject: Re: epimorphism
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> >Also it seems implied that non-torsion = epimorphic. Is
> >that true?
>
> It's not true because I found a counterexample. The
> [225/224, 1029/1024, 25/24] block is not a Constant Structure
> and it has no torsion.
>
> Manuel
ugh! Is this, Gene, one of the cases where the notes are "in the
wrong order"?
From: Manuel Op de Coul (2003-12-20)
Subject: Re: [tuning-math] Re: Question for Manuel, Gene, Kees, or whomever . . .
Paul wrote:
>Aha -- looks like Manuel was making an arbitrary choice in the case
>of a tie, perhaps letting Tenney complexity break the tie.
Yes it's arbitrary, and that latter would be a useful addition.
Manuel
From: Paul Erlich (2003-12-23)
Subject: Re: epimorphism
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:
> >
> > >Also it seems implied that non-torsion = epimorphic. Is
> > >that true?
> >
> > It's not true because I found a counterexample. The
> > [225/224, 1029/1024, 25/24] block is not a Constant Structure
> > and it has no torsion.
> >
> > Manuel
>
> ugh! Is this, Gene, one of the cases where the notes are "in the
> wrong order"?
Manuel, you are wrong. This is indeed a torsional block. The four
determinants are 20, 32, 46, and 56 -- obviously these are all
multiples of 2, so we have torsion!
Is everyone asleep on this list? :)
:)
From: Paul Erlich (2003-12-23)
Subject: Attention Gene
http://groups.yahoo.com/group/tuning-math/message/8269
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > After fixing my program, here is what I am getting for Prooijen
and
> > geometric 11-limit reductions:
> >
> > ! red72_11pro.scl
> > Prooijen 11-limit reduced scale
> > 72
> > !
> > 81/80
> > 64/63
>
> Gene -- why isn't this 45/44?
From: Manuel Op de Coul (2003-12-23)
Subject: Re: [tuning-math] Re: epimorphism
>Manuel, you are wrong. This is indeed a torsional block. The four
>determinants are 20, 32, 46, and 56 -- obviously these are all
>multiples of 2, so we have torsion!
Drag, you're right. Why is it that when you know there's a bug in
the code you can spot it immediately, when otherwise it remains unnoticed.
Manuel
From: Paul Erlich (2003-12-23)
Subject: Re: epimorphism
--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> >Manuel, you are wrong. This is indeed a torsional block. The four
> >determinants are 20, 32, 46, and 56 -- obviously these are all
> >multiples of 2, so we have torsion!
>
> Drag, you're right. Why is it that when you know there's a bug in
> the code you can spot it immediately, when otherwise it remains
unnoticed.
>
> Manuel
so now can you find a *real* counterexample?
From: Manuel Op de Coul (2003-12-24)
Subject: Re: [tuning-math] Re: epimorphism
Paul wrote:
>so now can you find a *real* counterexample?
I guess not...
I've uploaded a new release with the torsion and epimorphism
bugs fixed, updated Sagittal symbols, improved periodicity
block dialog, and smaller improvements.
http://www.xs4all.nl/~huygensf/software/Scala_Setup.exe
Manuel
From: Gene Ward Smith (2003-12-26)
Subject: Re: Attention Gene
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> http://groups.yahoo.com/group/tuning-math/message/8269
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > After fixing my program, here is what I am getting for Prooijen
> and
> > > geometric 11-limit reductions:
> > >
> > > ! red72_11pro.scl
> > > Prooijen 11-limit reduced scale
> > > 72
> > > !
> > > 81/80
> > > 64/63
> >
> > Gene -- why isn't this 45/44?
I guess I'm using the wrong definition of Prooijen complexity.
From: Paul Erlich (2003-12-26)
Subject: Re: Attention Gene
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > http://groups.yahoo.com/group/tuning-math/message/8269
> >
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <gwsmith@s...>
> > > wrote:
> > > > After fixing my program, here is what I am getting for
Prooijen
> > and
> > > > geometric 11-limit reductions:
> > > >
> > > > ! red72_11pro.scl
> > > > Prooijen 11-limit reduced scale
> > > > 72
> > > > !
> > > > 81/80
> > > > 64/63
> > >
> > > Gene -- why isn't this 45/44?
>
> I guess I'm using the wrong definition of Prooijen complexity.
It's called 'expressibility', and it's simply (the log of) the "ratio
of" (or, imprecisely speaking, "odd-limit") measure of the ratio.
http://tonalsoft.com/enc/ratio-of.htm
http://www.kees.cc/tuning/perbl.html
Since log(45)<log(63), you must indeed have the wrong definition.
From: Gene Ward Smith (2003-12-26)
Subject: Re: Attention Gene
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Since log(45)<log(63), you must indeed have the wrong definition.
log(11*45)>log(63), which is what I think I used.
From: Paul Erlich (2003-12-26)
Subject: Re: Attention Gene
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Since log(45)<log(63), you must indeed have the wrong definition.
>
> log(11*45)>log(63), which is what I think I used.
I thought I had already straightened you out on that particular
misunderstanding.