Tuning-Math Archive Section 6: 5075

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

- Contents - Hide Contents - Home - Section 6

Previous Next

5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950

5050 - 5075 -

Message: 5075 - Contents - Hide Contents

Date: Wed, 10 Jul 2002 22:53:20

Subject: Re: More on chord geometry

From: Gene W Smith

The symmetrical 7-limit note lattice has "deep holes" which are
geometrically octahedra, and musically hexanies. The dual cubic lattice
of 7-limit tetrads just discussed has eight-tetrad cubes as holes; if we
look at the corresponding notes we have a 14-note stellated octahedron.
Fans of superparticular ratios may be interested to hear that considered
as a scale, this has all of its steps superparticular rations, though of
highly variable size:

1-21/20-15/14-35/32-9/8-5/4-21/16-35/24-3/2-49/32-25/16-105/64-7/4-15/8

Message: 5076 - Contents - Hide Contents

Date: Thu, 11 Jul 2002 20:29:34

Subject: The 7-limit tetrads of Blackjack

From: Gene W Smith

15/14, an approximate secor, translates to [0 0 2] in terms of the cubic
lattice representation. In the place of a chain of secors, we can regard
the chords of Blackjack as a chain of [0 0 1] generators, two of which
make up a 15/14 modulation up. The 7-limit commas of the 72-et are

225/224 [1 1 4]
1029/1024 [3 4 1]
4375/4374 [5 -6 -3]

and redundently but importantly,

2401/2400 [2 3 -3]

Our chain of [0 0 1] generators gives us this:

[0 1 -2] ~ [-2 -2 1] (2401/2400)
[-2 -2 0]
[-2 -2 -1] ~ [-1 -1 3]  (225/224)
[-1 -1 2]
[-1 -1 1]
[-1 -1 0]
[-1 -1 -1]
[-1 -1 -2] ~ [0 0 2] (225/224)
[0 0 1]
[0 0 0]
[0 0 -1]
[0 0 -2]
[0 0 -3]
[1 1 0] ~ [0 0 -4] (225/224)
[1 1 -1]
[-1 -2 1] ~ [1 1 -2] (2401/2400)

Message: 5077 - Contents - Hide Contents

Date: Thu, 11 Jul 2002 20:52:46

Subject: A chord analog to Fokker blocks

From: Gene W Smith

As I reported in a revious article, the 7-limit tetrads form a cubic
lattice. I haven't gotten much feedback on this stuff, and wonder if this
is well known, not known, or somewhere in between.) By identifying the
major tetrad with root q, q a 7-limit octave equivalence class, with q
itself, we may represent the note-lattice of classes as a sublattice of
the cubic lattice of chords. The basis is [0 1 1] representing 3/2, [0 1
0] 5/4, and [1 1 0] 7/4, so that 3^a 5^b 7^c is represented by
[b+c,a+c,a+b]. In this form, the *usual* Euclidean metric applies to the
note lattice.

Using this representation, we may define a block in a way entirely
analogous to note-class blocks. If for instance we take
<9/8, 15/14, 25/24>, the TM-reduced basis for the kernel of h4, we obtain
upon transforming to the cubic lattice coordinates <[0 2 2], [0 0 2], [2
-1 -1]>. Taking the adjoint matrix M of the matrix (of determinant +-8 =
2*4) defined by these as rows, we may construct a corresponding block by
requiring that if [p q r] = [a b c]M, then -4<p<5, -4<q<5, -4<r<5. This
gives us the following set of eight (= 2*4) chords: 

[0, 0, 0], [0, 0, 1], [0, 1, 1], [0, 1, 2], [1, -1, 0], [1, -1, 1], [1,
0, 1], [1, 0, 2]

The notes of these give the following scale:

[1, 25/24, 15/14, 35/32, 9/8, 75/64, 5/4, 21/16, 75/56, 45/32, 35/24,
3/2, 25/16, 45/28, 5/3, 7/4, 25/14, 15/8]



Message: 5078 - Contents - Hide Contents

Date: Fri, 12 Jul 2002 17:30:42

Subject: Re: n-limit chord search

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:

>It looks good to me!


Ok, it's optimised now and can be downloaded.

Manuel



Message: 5079 - Contents - Hide Contents

Date: Fri, 12 Jul 2002 23:51:55

Subject: The chord-generator approach to Miracle and Orwell

From: Gene W Smith

The simplest generators for a chain of chords, in terms of the cubic
lattice picture, are [1 0 0], [0 1 0] and [0 0 1]. If we take a chain of
two of these, and look at the corresponding note, or in other words at
the roots of the chords 
[2 0 0], [0 2 0] and [0 0 2] we obtain 35/24, 21/20 and 15/14
respectively. From (15/14)^2/(8/7) = 225/224 and
(3/2)/(8/7)^3 = 1029/1024 we see that tempering a chain of chords with
generator [0 0 1] via Miracle is a likely plan.
Similarly, from (21/20)^2/(10/9) = (21/20)^3/(7/6) = 4000/3969, it would
seem the planar temperament defined by 4000/3969 works well with the
chords generated by [0 1 0]. One approach to this is to add 245/243 or
2401/2400 to the mix, and obtain the linear temperament with wedgie [8 18
11 -25 5 10], which is covered by 41 and 68, and has a generator of about
21/20. Finally, from (15/7)/(35/24)^2 = 1728/1715 and (28/9)/(35/24)^3 =
6144/6125, it would seem Orwell is a good choice for tempering the chain
of chords defined by [1 0 0].

Below I give scales derived from chains of generators of size 1 to 15, in
72-et (for [0 0 1]), 68-et (for [0 1 0]) and 84-et 
(for [1 0 0]). It should be noted that eventually, these all produce
MOS--Blackjack for 72-et, the 22-note Orwell MOS, and a 27-note 5/68 MOS.

72-et scales

1   [0, 7, 23, 42, 58, 65]   [7, 16, 19, 16, 7, 7]

2   [0, 7, 23, 30, 42, 49, 58, 65]   [7, 16, 7, 12, 7, 9, 7, 7]

3   [0, 7, 14, 23, 30, 42, 49, 58, 65]   [7, 7, 9, 7, 12, 7, 9, 7, 7]

4   [0, 7, 14, 23, 30, 37, 42, 49, 56, 58, 65]   
[7, 7, 9, 7, 7, 5, 7, 7, 2, 7, 7]

5   [0, 7, 14, 21, 23, 30, 37, 42, 49, 56, 58, 65]   
[7, 7, 7, 2, 7, 7, 5, 7, 7, 2, 7, 7]

6   [0, 7, 14, 21, 23, 30, 37, 42, 44, 49, 56, 58, 63, 65]   
[7, 7, 7, 2, 7, 7, 5, 2, 5, 7, 2, 5, 2, 7]

7   [0, 7, 14, 21, 23, 28, 30, 37, 42, 44, 49, 56, 58, 63, 65]   
[7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 7, 2, 5, 2, 7]

8   [0, 7, 14, 14, 21, 23, 28, 30, 37, 42, 44, 49, 51, 56, 58, 63, 65,
70]   
[7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

9   [0, 7, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58, 63, 65,
70]   
[7, 7, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

10   [0, 5, 7, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58, 63,
65, 70]   
[5, 2, 7, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

11   [0, 5, 7, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58, 63,
65, 70]   
[5, 2, 7, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

12   [0, 5, 7, 12, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58,
63, 65, 70]   
[5, 2, 5, 2, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

13   [0, 5, 7, 12, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58,
63, 65, 70]   
[5, 2, 5, 2, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

14   [0, 5, 7, 12, 14, 19, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56,
58, 63, 65, 70]   
[5, 2, 5, 2, 5, 2, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]

15   [0, 5, 7, 12, 14, 19, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56,
58, 63, 65, 70]   
[5, 2, 5, 2, 5, 2, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]


84-et scales

1   [0, 11, 27, 46, 49, 68]   [11, 16, 19, 3, 19, 16]

2   [0, 11, 27, 30, 46, 49, 68, 73]   [11, 16, 3, 16, 3, 19, 5, 11]

3   [0, 8, 11, 27, 30, 46, 49, 57, 68, 73]   
[8, 3, 16, 3, 16, 3, 8, 11, 5, 11]

4   [0, 8, 11, 27, 30, 35, 46, 49, 57, 68, 73, 76]   
[8, 3, 16, 3, 5, 11, 3, 8, 11, 5, 3, 8]

5   [0, 8, 11, 19, 27, 30, 35, 46, 49, 54, 57, 68, 73, 76]   
[8, 3, 8, 8, 3, 5, 11, 3, 5, 3, 11, 5, 3, 8]

6   [0, 8, 11, 19, 27, 30, 35, 38, 46, 49, 54, 57, 68, 73, 76, 81]   
[8, 3, 8, 8, 3, 5, 3, 8, 3, 5, 3, 11, 5, 3, 5, 3]

7   [0, 8, 11, 16, 19, 27, 30, 35, 38, 46, 49, 54, 57, 65, 68, 73, 76,
81]
   [8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 5, 3]

8   [0, 8, 11, 16, 19, 27, 30, 35, 38, 43, 46, 49, 54, 57, 65, 68, 73,
76, 81]   
[8, 3, 5, 3, 8, 3, 5, 3, 5, 3, 3, 5, 3, 8, 3, 5, 3, 5, 3]

9   [0, 8, 11, 16, 19, 27, 30, 35, 38, 43, 46, 49, 54, 57, 62, 65, 68,
73, 76, 81]   
[8, 3, 5, 3, 8, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3]

10   [0, 5, 8, 11, 16, 19, 27, 30, 35, 38, 
43, 46, 49, 54, 57, 62, 65, 68, 73, 76, 81]   
[5, 3, 3, 5, 3, 8, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3]

11   [0, 5, 8, 11, 16, 19, 24, 27, 30, 35, 
38, 43, 46, 49, 54, 57, 62, 65, 68, 73, 76, 81]   
[5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3]

12   [0, 5, 8, 11, 16, 19, 24, 27, 30, 35, 38, 
43, 46, 49, 51, 54, 57, 62, 65, 68, 73, 76, 81]   
[5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 5, 3, 5, 3]

13   [0, 5, 8, 11, 16, 19, 24, 27, 30, 35, 38,
 43, 46, 49, 51, 54, 57, 62, 65, 68, 70, 73, 76, 81]   
[5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3]

14   [0, 5, 8, 11, 13, 16, 19, 24, 27, 30, 35, 38, 
43, 46, 49, 51, 54, 57, 62, 65, 68, 70, 73, 76, 81]   
[5, 3, 3, 2, 3, 3, 5, 3, 3, 5, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5,
3]

15   [0, 5, 8, 11, 13, 16, 19, 24, 27, 30, 32, 35, 38, 
43, 46, 49, 51, 54, 57, 62, 65, 68, 70, 73, 76, 81]   
[5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3,
5, 3]


68-et scales

1   [0, 5, 22, 27, 40, 55]   [5, 17, 5, 13, 15, 13]

2   [0, 5, 22, 27, 40, 45, 55, 60]   [5, 17, 5, 13, 5, 10, 5, 8]

3   [0, 5, 10, 22, 27, 32, 40, 45, 55, 60]   
[5, 5, 12, 5, 5, 8, 5, 10, 5, 8]

4   [0, 5, 10, 22, 27, 32, 40, 45, 50, 55, 60, 65]   
[5, 5, 12, 5, 5, 8, 5, 5, 5, 5, 5, 3]

5   [0, 5, 10, 15, 22, 27, 32, 37, 40, 45, 50, 55, 60, 65]   
[5, 5, 5, 7, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3]

6   [0, 2, 5, 10, 15, 22, 27, 32, 37, 40, 45, 50, 55, 60, 65]   
[2, 3, 5, 5, 7, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3]

7   [0, 2, 5, 10, 15, 20, 22, 27, 32, 37, 40, 42, 45, 50, 55, 60, 65]   
[2, 3, 5, 5, 5, 2, 5, 5, 5, 3, 2, 3, 5, 5, 5, 5, 3]

8   [0, 2, 5, 7, 10, 15, 20, 22, 27, 32, 37, 40, 42, 45, 50, 55, 60, 65] 
 
[2, 3, 2, 3, 5, 5, 2, 5, 5, 5, 3, 2, 3, 5, 5, 5, 5, 3]

9   [0, 2, 5, 7, 10, 15, 20, 22, 25, 27, 32, 37, 40, 42, 45, 47, 50, 55,
60, 65]   
[2, 3, 2, 3, 5, 5, 2, 3, 2, 5, 5, 3, 2, 3, 2, 3, 5, 5, 5, 3]

10   [0, 2, 5, 7, 10, 12, 15, 20, 22, 25, 27, 32, 37, 40, 42, 45, 47, 50,
55, 60, 65]   
[2, 3, 2, 3, 2, 3, 5, 2, 3, 2, 5, 5, 3, 2, 3, 2, 3, 5, 5, 5, 3]

11   [0, 2, 5, 7, 10, 12, 15, 20, 22, 25, 27, 30, 32, 37, 40, 42, 45, 47,
50, 52, 55, 60, 65]   
[2, 3, 2, 3, 2, 3, 5, 2, 3, 2, 3, 2, 5, 3, 2, 3, 2, 3, 2, 3, 5, 5, 3]

12   [0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27,
 30, 32, 37, 40, 42, 45, 47, 50, 52, 55, 60, 65]   
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 3, 2, 3, 2, 3, 2, 3, 5, 5, 3]

13   [0, 2, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 
30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 65]   
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
5, 3]

14   [0, 2, 2, 5, 7, 7, 10, 12, 15, 17, 20, 22, 22, 25, 27,
 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 65]   
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
5, 3]

15   [0, 2, 2, 5, 7, 7, 10, 12, 15, 17, 20, 22, 22, 25, 27, 
30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65]   
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
2, 3, 3]

Message: 5080 - Contents - Hide Contents

Date: Fri, 12 Jul 2002 17:41:58

Subject: Re: A chord analog to Fokker blocks

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

From a brief look this question arose: are these chord
analogues a subset of real periodicity blocks in some
special way?
Your scale reminded me of the 4:5:6:7 double tied circular
mirroring, but it's bigger and not a superset.

Manuel



Message: 5081 - Contents - Hide Contents

Date: Fri, 12 Jul 2002 09:12:50

Subject: More chord-block scales

From: Gene W Smith

# h2 scale blocks

cm1 := [9/7, 6/5, 8/7];
c1 := [[-1, 0, 0], [-1, 0, 1], [-1, 1, 1], [0, 0, 0]];
s1 := [1, 15/14, 6/5, 5/4, 9/7, 3/2, 12/7, 7/4];

cm2 := [6/5, 8/7, 9/8];
c2 := [[-1, 0, 0], [-1, 1, 1], [0, 0, 0], [0, 1, 1]];
s2 := [1, 9/8, 6/5, 5/4, 9/7, 21/16, 3/2, 12/7, 7/4, 9/5, 15/8];

# h3 scale block

cm3 := [7/6, 10/9, 16/15];
c3 := [[0, -1, -1], [0, 0, -1], [0, 0, 0], [0, 1, 1], 
[1, -1, -1], [1, 0, 0]];
s3 := [1, 35/32, 9/8, 7/6, 5/4, 21/16, 4/3, 7/5, 35/24, 3/2, 
5/3, 7/4, 15/8, 35/18];

Message: 5082 - Contents - Hide Contents

Date: Fri, 12 Jul 2002 16:39:15

Subject: Re: A chord analog to Fokker blocks

From: genewardsmith

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:

> From a brief look this question arose: are these chord
> analogues a subset of real periodicity blocks in some
> special way?

There's
a close relationship with periodicity blocks. Pick either major or
minor tetrads, and with those, one particular chord element--for
instance the roots of the minor tetrads, or the major third element of
the major tetrads. These will all individually be corresponding Fokker
blocks.

> Your scale reminded me of the 4:5:6:7 double tied circular
> mirroring, but it's bigger and not a superset.

Where is the 4:5:6:7 double tied circular mirroring discussed?



Message: 5083 - Contents - Hide Contents

Date: Fri, 12 Jul 2002 11:17:59

Subject: Glumma

From: Gene W Smith

Inspired by Carl's 12 note, 7-limit JI scale "lumma", I hereby present
"glumma", a modified version which now matches stelhex for number of
complete tetrads (6.)

glumma   [1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 7/4,
96/49]

chords [[-2, -1, 0], [-2, 0, 0], [-1, -1, 0], [-1, 0, 0], [0, -1, 0], [0,
0, 0]]


Compare to

lumma [1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]
[[-2, 0, 0], [-1, -1, 0], [-1, 0, 0], [0, -1, 0]]


stelhex  [1, 21/20, 7/6, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 9/5]
 [[-1, 0, -1], [-1, 1, -1], [-1, 1, 0], [0, 0, -1], [0, 0, 0], [0 1, 0]]


class [1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4,
15/8]
 [[-1, 1, -1], [0, 0, 0], [0, 1, 0], [1, 0, 1]]


prism [1, 16/15, 28/25, 7/6, 5/4, 4/3, 7/5, 112/75, 8/5, 5/3, 7/4, 28/15]
[[-1, -1, -2], [-1, 0, -2], [0, -1, -1], [0, 0, -1]]



Message: 5084 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 02:07:19

Subject: Re: The 7-limit tetrads of Blackjack

From: dkeenanuqnetau

--- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:

> 15/14, an approximate secor, translates to [0 0 2] in terms of the 
cubic
> lattice representation. In the place of a chain of secors, we can 
regard
> the chords of Blackjack as a chain of [0 0 1] generators, two of 
which
> make up a 15/14 modulation up. The 7-limit commas of the 72-et are
> 
> 225/224 [1 1 4]
> 1029/1024 [3 4 1]
> 4375/4374 [5 -6 -3]
> 
> and redundently but importantly,
> 
> 2401/2400 [2 3 -3]
> 
> Our chain of [0 0 1] generators gives us this:
> 
> [0 1 -2] ~ [-2 -2 1] (2401/2400)
> [-2 -2 0]
> [-2 -2 -1] ~ [-1 -1 3]  (225/224)
> [-1 -1 2]
> [-1 -1 1]
> [-1 -1 0]
> [-1 -1 -1]
> [-1 -1 -2] ~ [0 0 2] (225/224)
> [0 0 1]
> [0 0 0]
> [0 0 -1]
> [0 0 -2]
> [0 0 -3]
> [1 1 0] ~ [0 0 -4] (225/224)
> [1 1 -1]
> [-1 -2 1] ~ [1 1 -2] (2401/2400)

Hi Gene,


this is probably fascinating stuff, but I can make neither head nor 
tail of it. As I think I've said before, more headings on tables, more 
diagrams and more textual explanations are always a good idea. I 
assume your intention is to actually be understood, as opposed to say 
merely posting to list as a way of recording your ideas.

Regards,
-- Dave Keenan

Message: 5085 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 13:08:07

Subject: Three generator step scales

From: Gene W Smith

I want to look a little more closely at some of the scales I discussed in
the previous article, focusing on the ones with three generator steps. By
this I mean the scales deriving from [-2 0 0]-[-1 0 0]-[0 0 0]-[1 0 0], 
[0 -2 0]-[0 -1 0]-[0 0 0]-[0 1 0], and [0 0 -2]-[0 0 -1]-[0 0 0]-[0 0 1].
As JI scales, these are isomorphic sets of chords in the lattice, and
hence have isomorphic 7-limit graphs. We have:

From [0 0 1] as generator

[1, 15/14, 7/6, 5/4, 7/5, 3/2, 49/30, 7/4, 28/15, 15/8] 

72-et version [0, 7, 16, 23, 35, 42, 51, 58, 65]

From [0 1 0] as generator

[1, 21/20, 25/21, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4, 40/21]

68-et version [0, 5, 17, 22, 27, 35, 40, 50, 55, 63]

From [1 0 0] as generator

[1, 36/35, 35/32, 6/5, 5/4, 48/35, 35/24, 3/2, 12/7, 7/4]

84-et version [0, 3, 11, 22, 27, 38, 46, 49, 65, 68]

All of the JI scales are graph-isomorphic, each having 21 intervals, 16
triads and 4 tetrads in the 7-limit.
The first scale makes the least sense as a JI scale, having a step of
225/224; two notes are conflated in the 72-et
version, giving an excellent 9-note scale. The second scale makes the
most sense as a JI scale, and the least sense as a tempered scale, since
the tempered version is isomorphic to the non-tempered version. The last
scale is a little irregular so far as step size goes, but does gain from
tempering via Orwell, as the tempered version has four more intervals and
two more triads.

Message: 5086 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 06:22:04

Subject: Re: The 7-limit tetrads of Blackjack

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> this is probably fascinating stuff, but I can make neither head nor 
> tail of it. 

Did
you read the articles previous to this, where I explain (I hope) this
representation of 7-limit chords in terms of a cubic lattice; that is,
vectors [a b c] where a, b and c are integers?

Anyone else following this, or am I talking to myself? What needs
explaining? I don't have the software to draw diagrams, I'm afraid.



Message: 5087 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 15:27:01

Subject: Four generator step scales

From: Gene W Smith

The JI versions of these all have 26 intervals and 20 triads. There are
also minor versions of these.


[0 0 1]

[1, 15/14, 7/6, 5/4, 75/56, 7/5, 3/2, 45/28, 49/30, 7/4, 28/15, 15/8]

72-et [0, 7, 16, 23, 30, 35, 42, 49, 51, 58, 65]
30 intervals, 25 triads


[0 1 0]

[1, 21/20, 25/21, 5/4, 21/16, 10/7, 3/2, 63/40, 5/3, 7/4, 147/80, 40/21]

68-et [0, 5, 17, 22, 27, 35, 40, 45, 50, 55, 60, 63]
28 intervals, 21 triads


[1 0 0]

[1, 36/35, 35/32, 6/5, 5/4, 245/192, 48/35, 35/24, 3/2, 12/7, 7/4,
175/96]

84-et [0, 3, 11, 22, 27, 30, 38, 46, 49, 65, 68, 73]
34 intervals, 27 triads

Despite having one less note, the 72-et scale has more 7-limit intervals
and triads than the 68-et scale.

Message: 5088 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 13:25:56

Subject: The two lists

From: Gene W Smith

On Sat, 13 Jul 2002 11:49:12 -0700 Carl Lumma <carl@xxxxx.xxx> writes:

> I'm wondering if anybody is seeing my replies over here?  If nobody 
> is
> going to use this list, it should be deleted, or at least its mailer
> turned off and the forwarding from yahoo turned off.  The situation
> is causing fragmentation of the message archives.

Sorry about that; I meant to switch over, but there seemed to me to be no
reason to hurry. You asked about archiving, and who would want to
moderate the list. I would be happy to co-moderate if you need someone; I
think there should always be more than one list moderator anyway. Is
Robert Walker doing the archiving? I forget. Anyone want to moderate?



Message: 5089 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 18:12:49

Subject: Re: The chord-generator approach to Miracle and Orwell

From: genewardsmith

--- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:


> 72-et scales
> 
> 1   [0, 7, 23, 42, 58, 65]   [7, 16, 19, 16, 7, 7]
> 
> 2   [0, 7, 23, 30, 42, 49, 58, 65]   [7, 16, 7, 12, 7, 9, 7, 7]


Compare to Qm(2): [7, 16, 7, 12, 7, 16, 7]


> 3   [0, 7, 14, 23, 30, 42, 49, 58, 65]   [7, 7, 9, 7, 12, 7, 9, 7, 7]
> 
> 4   [0, 7, 14, 23, 30, 37, 42, 49, 56, 58, 65]   
> [7, 7, 9, 7, 7, 5, 7, 7, 2, 7, 7]


Compare to Qm(3): [7, 7, 9, 7, 7, 5, 7, 7, 9, 7]


> 5   [0, 7, 14, 21, 23, 30, 37, 42, 49, 56, 58, 65]   
> [7, 7, 7, 2, 7, 7, 5, 7, 7, 2, 7, 7]
> 
> 6   [0, 7, 14, 21, 23, 30, 37, 42, 44, 49, 56, 58, 63, 65]   
> [7, 7, 7, 2, 7, 7, 5, 2, 5, 7, 2, 5, 2, 7]


Compare to Qm(4): [7, 7, 7, 2, 7, 7, 5, 2, 5, 7, 7, 2, 7]


> 7   [0, 7, 14, 21, 23, 28, 30, 37, 42, 44, 49, 56, 58, 63, 65]   
> [7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 7, 2, 5, 2, 7]
> 
> 8   [0, 7, 14, 14, 21, 23, 28, 30, 37, 42, 44, 49, 51, 56, 58, 63, 65,
> 70]   
> [7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]


Compare to Qm(5): [7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 2, 5, 7, 2, 5, 2]



Message: 5090 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 21:11:58

Subject: Transformations of glumma

From: Gene W Smith

Here are the 12 versions of glumma; the first being a mode of the
original. "recanbm" means in the cubic lattice description of the scale,
the "n" values range from -1 to 1, and the "m" values from 0 to 1;
"recancm" means instead the "m" values range from -1 to 0 instead of 0 to
1. Hence "reca1b2", which is glumma, has chords [i,j,0] where 
-1<=i<=1 and 0<=j<=1. However glumma does not seem the most interesting
of these scales; recta3c1 not only has nice, low-ratio note values, it is
the most regular in terms of the variation in step size. The presence of
both 15/14 and 16/15 among the step sizes suggests tempering by
225/224~1.


reca1b2 := [1, 21/20, 35/32, 6/5, 5/4, 21/16, 35/24, 3/2, 49/32, 12/7,
7/4, 
9/5];
reca1b3 := [1, 15/14, 35/32, 6/5, 5/4, 9/7, 35/24, 3/2, 25/16, 12/7, 7/4,
15
/8];
reca2b1 := [1, 25/24, 21/20, 35/32, 5/4, 21/16, 10/7, 35/24, 3/2, 49/32,
5/3
, 7/4];
reca2b3 := [1, 21/20, 15/14, 9/8, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4, 25/14,
15
/8];
reca3b1 := [1, 49/48, 15/14, 35/32, 7/6, 5/4, 7/5, 35/24, 3/2, 25/16,
7/4, 
15/8];
reca3b2 := [1, 21/20, 15/14, 9/8, 7/6, 49/40, 5/4, 21/16, 7/5, 3/2, 7/4,
15/
8];
reca1c2 := [1, 25/24, 35/32, 8/7, 6/5, 5/4, 10/7, 35/24, 3/2, 5/3, 12/7,
7/4
];
reca1c3 := [1, 49/48, 35/32, 7/6, 6/5, 5/4, 7/5, 35/24, 3/2, 8/5, 12/7,
7/4]
;
reca2c1 := [1, 21/20, 8/7, 6/5, 5/4, 21/16, 10/7, 3/2, 5/3, 12/7, 7/4,
9/5];
reca2c3 := [1, 21/20, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7, 3/2, 5/3,
7/4]
;
reca3c1 := [1, 15/14, 7/6, 6/5, 5/4, 9/7, 7/5, 3/2, 8/5, 12/7, 7/4,
15/8];
reca3c2 := [1, 15/14, 7/6, 5/4, 4/3, 7/5, 10/7, 3/2, 5/3, 7/4, 25/14,
15/8];



Message: 5091 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 22:30:20

Subject: Re: Transformations of glumma

From: Gene W Smith

I checked each of these 12 scales for intervals which were close to a
7-limit consonance by a comma less than 15 cents. All of them had at
least one such comma, and the 31-et covers all of the commas.



> reca1b2 := [1, 21/20, 35/32, 6/5, 5/4, 21/16, 35/24, 3/2, 49/32, 
> 12/7,
> 7/4, 
> 9/5];


126/125 ^ 1728/1715 = [10, 9, 7, -9, 17, -9]

"Small diesic" temperament covered by 27, 31 and 58


> reca1b3 := [1, 15/14, 35/32, 6/5, 5/4, 9/7, 35/24, 3/2, 25/16, 12/7, 
> 7/4,
> 15
> /8];

1728/1715

> reca2b1 := [1, 25/24, 21/20, 35/32, 5/4, 21/16, 10/7, 35/24, 3/2, 
> 49/32,
> 5/3
> , 7/4];

126/125

> reca2b3 := [1, 21/20, 15/14, 9/8, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4, 
> 25/14,
> 15
> /8];

126/125

> reca3b1 := [1, 49/48, 15/14, 35/32, 7/6, 5/4, 7/5, 35/24, 3/2, 25/16,
> 7/4, 
> 15/8];

225/224

> reca3b2 := [1, 21/20, 15/14, 9/8, 7/6, 49/40, 5/4, 21/16, 7/5, 3/2, 
> 7/4,
> 15/
> 8];


126/125 ^ 225/224 ^ 2401/2400 = 31-et


> reca1c2 := [1, 25/24, 35/32, 8/7, 6/5, 5/4, 10/7, 35/24, 3/2, 5/3, 
> 12/7,
> 7/4
> ];


126/125 ^ 1728/1715, small diesic


> reca1c3 := [1, 49/48, 35/32, 7/6, 6/5, 5/4, 7/5, 35/24, 3/2, 8/5, 
> 12/7,
> 7/4]
> ;

1728/1715

> reca2c1 := [1, 21/20, 8/7, 6/5, 5/4, 21/16, 10/7, 3/2, 5/3, 12/7, 
> 7/4,
> 9/5];


1029/1024^126/125 = [9, 5, -3, -21, 30, -13]

"quartaminorthirds" covered by 31, 46 and 77


> reca2c3 := [1, 21/20, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7, 3/2, 
> 5/3,
> 7/4]
> ;


2401/2400^126/125 = 126/125^1728/1715 = small diesic


> reca3c1 := [1, 15/14, 7/6, 6/5, 5/4, 9/7, 7/5, 3/2, 8/5, 12/7, 7/4,
> 15/8];

225/224

> reca3c2 := [1, 15/14, 7/6, 5/4, 4/3, 7/5, 10/7, 3/2, 5/3, 7/4, 25/14,
> 15/8];


126/125^225/224 = [1, 4, 10, 12 -13, 4]

meantone, covered by 19, 31, 43, 50, 74



Message: 5092 - Contents - Hide Contents

Date: Sat, 13 Jul 2002 22:53:41

Subject: Carl's commas

From: Gene W Smith

Here is the same analysis applied to Carl's 12-note, 7-limit JI scales.
Meantone rules, it seems!

lumma:=[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]

126/125

lester:=[1, 21/20, 9/8, 7/6, 5/4, 4/3, 7/5, 3/2, 14/9, 5/3, 7/4, 15/8]

126/125 ^ 225/224, meantone

prism:=[1, 16/15, 28/25, 7/6, 5/4, 4/3, 7/5, 112/75, 8/5, 5/3, 7/4,
28/15]

126/125 ^ 225/224, meantone

stelhex:=[1, 21/20, 7/6, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 9/5]

126/125

class:=[1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4,
15/8]

126/125 ^ 225/224, meantone

Message: 5093 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 10:16:01

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 01:03 18/06/02 -0000, you wrote:

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>>> I would therefore recommend going back to the rational 
>>> complementation system and doing the ET's that way as well.
>> 

>> Agreed. Provided we _always_ use rational complements, whether this 
>> results in matching half-apotomes or not.
>

>In other words, you would prefer having this:
>
>152 (76 ss.):  

)|   |~   /|   |\   ~|)   /|)   /|\  (|)  
(|\  ||~  /||  ||\  ~||)  (||~  /||\

>
>instead of this:
>
>152 (76 ss.):  

)|   |~   /|   |\   /|~   /|)   /|\  (|)  
(|\  ||~  /||  ||\  /||~  /||)  /||\

>
>even if it isn't as easy to remember.

OK.
I think you've got me there. :-) Remember I said I thought we
shouldn't let complements cause us to choose an inferior set of
single-shaft symbols, because some people won't use the purely
saggital complements. I think we both agree that /|~ is a better
choice for 5deg152 than ~|) since it introduces fewer new flags and
puts the ET value closer to the rational value. 

I don't think we have defined a rational complement for /|~ because it
isn't needed for rational tunings. But if we look at complements
consistent with 494-ET (as all the rational complements are) the only
complement for /|~ is ~||(. So we end up with

152 (76 ss.):  
)|   |~    /|   |\   /|~   /|)   /|\  (|)  
(|\  ~||(  /||  ||\  ~||)  /||)  /||\

But this is bad because the flag sequence is different in the two
half-apotomes _and_ ~||( = 10deg152 is inconsistent _and_ too many
flag types. So you're right. I don't want to use strict rational
complements for this, particularly with its importance in representing
1/3 commas. I'd rather have

>152 (76 ss.):  

)|   |~   /|   |\   /|~   /|)   /|\  (|)  
(|\  ||~  /||  ||\  /||~  /||)  /||\

I note that 76-ET can also be notated using its native fifth, as you
give (and I agree) below.

>I suggest that you try some more ET's before insisting on rational 
>complements across the board.  In addition to less memorable symbol 
>sequences, strict rational complementation will also result in some 
>bad symbol arithmetic in instances where the complement symbols are 
>not consistent in some ET's.  I will accept some symbol arithmetic 
>inconsistency (e.g., with ||) in 72-ET), if it isn't too 
>disorienting, but I think that users will need all the help they can 
>get to keep the symbols straight in the larger ET's, and too many 
>flags and bad symbol arithmetic aren't going to help.

Point taken.

>>> I would be agreeable to doing all of the ET's (with the rational 
>>> complementation scheme) using the symbols that we agreed on in 
>>> message #4443.
>> 
>> OK.
>

>I erroneously stated that everything that we last agreed on (using 
>what I would call "inverse complements") would stay the same.  
>However, there is one exception.  This:
>
>32:  )|  /|\  (|)  (||\  /||\  (DK - inverse complements
>
>would become this:
>
>32:  )|  /|\  (|)  (||~  /||\  (rational complements)
>
>To this I am agreeable.

That's ok with me too.

Now to start on the others with 6 or less steps per apotome.

I won't necessarily include the double-shaft symbols from here on. You
should assume they correspond to the rational complements.

We are really having problems with 1deg48 aren't we?

You wrote:

>I think that 48 and 55 have sufficiently different properties that 
>there would be no reason to insist on doing them alike.  Since I 
>would do 96 this way:
>
>96:  /|  |)  /|)  /|\  (|\  ||)  ||\  /||\
>
>I wouldn't see any problem with doing 48 as a subset of 96, 
>particularly since 7 and 11 are among the best factors represented in

>48:
>
>48:  |)  /|\  ||)  /||\

We agree 48 should be every second step of 96, but we haven't agreed
on 96 yet.

I agree 48 doesn't _need_ to be the same as either 41 or 55, but it
would be good to minimise the number of different notations for all
the scales with 4 steps to the apotome.

Both ~|) and ~|\ are consistently 1 degree of 48, 55 and 62-ET, but of
these only ~|) is also 2 degrees of 96-ET.

That's one reason why I favour ~|).

But lets forget 55 and 62 for now. You propose to use |) which is
certainly correct as the 7-comma for both 48 and 96-ET. Why would I
want to add the ~| 17-flag to it when this is zero steps?

One problem is that we're already using |) as one degree of 36-ET and
2 degrees of 72-ET. People will naturally attach the meaning of 1/3
semitone to it in this application, and may find it confusing if 48
and 96-ET use it for 1/4 semitone.

This opens a whole other can of worms regarding notation relative to
12-ET. Lots of people would like to notate their tunings (even those
which are not n*12-ETs) as deviations from 12-ET, rather than as
deviations from a chain of the tuning's own native fifths (or it may
have none).

Since people are going to try to do it anyway, shouldn't we look at
standardising a consistent way of doing it? Some time ago I
investigated this in depth and I now offer a first pass at a
spreadsheet that does it semi-automatically. And, you guessed it, it
requires 1deg48 and 2deg96 to be ~|).

Yahoo groups: /tuning-math/files/Dave/Notating... * [with cont.] 

If you examine the formulae in this spreadsheet you will see that the
principle is that each symbol is given, in a lookup table, a range of
cents deviations that it covers. In general the ranges overlap, but
there is a strict order of precedence to resolve the cases where more
than one symbol could notate the same degree. Determining the ranges
was quite tedious, but the main requirement is to ensure that the
symbols actually agree with their comma values, given 12-ET fifths.
e.g. the changover between one symbol and the next, at the same
precedence level, occurs at the point equidistant from their two comma
values relative to a chain of 12-ET fifths.

But how did I choose which symbols to use in the first place? It's so
long ago I've almost forgotten, but the basic idea was for example, to
look at all the n*12-ETs that contained a 25c step and find which
symbol corresponded to 25 cents in all of them, and so on. 

Here's what it gives for all the n*12-ETs whose best fifth is the
12-ET fifth. The dots indicate degrees that cannot be notated.

12:
24:                                                            /|\
36:                                  |) 
48:                            ~|)                             /|\
60:            /|                              |\
72:            /|                    |)                        /|\
84:            /|                    |)             /|)
96:            /|              ~|)             |\              /|\
108:           /|         /|(        |)             /|)
120:           /|    (|              |)        |\              /|\
132: ~|(       /|                    |)        |\   /|)
144: ~|(       /|              ~|)   |)             /|)        /|\
156: ~|(       /|              ~|)   |)        |\   /|)
168: ~|(       /|         /|(        |)        |\   /|)        /|\
180: ~|(       /|    (|        ~|)   |)        |\   /|)
192: ~|(       /|    (|        ~|)   |)        |\   /|)        /|\
204: ~|(       /|    (|        ~|)   |)        |\         (|)  /|\
216: ~|(       /|    (|   /|(  ~|)   |)        |\   /|)        /|\
228: ~|(   |(  /|         /|(  ~|)   |)        |\   /|)   (|)  /|\
240: ~|(   |(  /|    (|        ~|)   |)  ~|\   |\   /|)        /|\
252: ~|(   |(  /|    (|        ~|)   |)  ~|\   |\   /|)   (|)  /|\
264: ~|(   |(  /|    (|   /|(  ~|)   |)        |\ . /|)        /|\
276: ~|(   |(  /|    (|   /|(  ~|)   |)  ~|\   |\   /|)   (|)  /|\
288: ~|(   |(  /|  . (|        ~|)   |)  ~|\   |\   /|) . (|)
300: ~|(   |(  /|  . (|        ~|)   |)  ~|\   |\   /|) . (|)  /|\

Notice that this scheme only uses 6 types of flag since it doesn't go
beyond 17-limit. Of course one has to get used to the fact that ~| is
negative (-5.0 cents).

Notice that 276-ET is the largest that can be fully notated, and that
12,24,36,72 are as previously agreed. We haven't agreed on 60-ET yet,
but the proposal above is different from what either of us suggested
recently.

Notice that 144-ET has bad flag arithmetic, since /| and |) [7 flag]
are 2 and 4 steps respectively and thereby agree with 72-ET, but /|)
is 5 steps and must be interpreted as the 13 flag. If we are not
willing to do this, then we must accept that 144-ET cannot be fully
notated in a manner consistent with 72-ET, simply because we don't
have a separate symbol for the 13-comma, and the 13-schisma doesn't
vanish.

>Now 55 is a real problem, because nothing is really very good for 
>1deg.  The only single flags that will work are |( (17'-17) or (| (as

>the 29 comma), and the only primes that are 1,3,5,n-consistent are 
>17, 23, and 29.
>
>If I wanted to minimize the number of flags, I could do it by 
>introducing only one new flag:
>
>55:  ~|\  /|\  ~||  /||\
>
>so that 1deg55 is represented by the larger version of the 23' comma 
>symbol.  Or doing it another way would introduce only two new flags:
>
>55:  ~|~  /|\  ~||~  /||\
>
>The latter has for 1deg the 17+23 symbol, and its actual size (~25.3 
>cents) is fairly close to 1deg55 (~21.8 cents).  Besides, the symbols

>are very easy to remember.  So this would be my choice.

I would not use a 23 comma to notate this when it can be done in
17-limit. Luckily ~|\ works for 1 step as the 17+(11-5) comma (which
also agrees with 2 steps of 110-ET). So I go for your first (min
flags) suggestion:

55:  ~|\  /|\

>What was your reason for choosing ~|)?

Probably only because I could make it agree with 48-ET.

>> 62:  |)  /|\  (|\  /||\     [13-commas]
>

>Considering that 7 is so well represented in this division, I would 
>hesitate to use |) in the notation if it isn't being used as the 7 
>comma.  In fact, I don't think I would want to use |) for a symbol 
>unless it *did* represent the 7 comma (lest the notation be 
>misleading), although I would allow its use it in combination with 
>other flags.

Good point.

>  So I would prefer this:
>
>62:  /|)  /|\  (|\  /||\     [13-commas]

Agreed.

>> 69,76:  |)  ??  (|\  /||\     [13-comma]
>

>Again, I wouldn't use |) by itself defined as a 13-comma symbol, but 
>would choose /|) instead:
>
>69,76:  /|)  )|\  (|\  /||\     [13-commas]
>
>For 2deg of either 69 or 76, )|\ is about the right size.

Agreed.

I note that 62, 69 and 76 are all 1,3,9-inconsistent and might also be
notated as subsets of 2x or 3x ETs.

We should take a look at the n*19-ET family now that it is complete.

19:                 /||\
38:       /|\       /||\
57:    /|)   (|\    /||\     [13-commas]
76:  /|)  )|\  (|\  /||\     [13-commas]

>> 60:  /|  |)  ||)  ||\  /||\
>

>I notice that 13 is much better represented than 7, so I would prefer

>this (in which the JI symbols also more closely approximate the ET 
>intervals):
>
>60:  /|  /|)  (|\  ||\  /||\

As described above, this would not work in with the other n*12-ETs. My
current proposal uses neither 7 nor 13 comma symbols.

60:  /|  |\  /||  ||\  /||\

>> 67,74: ~|)  /|)  (|\  ~||(  /||\
>

>I'm certainly in agreement with the 2deg and 3deg symbols, and if you

>must do both ET's alike, then what you have for 1deg would be the 
>only choice (apart from (| as the 29 comma).  We both previously 
>chose )|) for 1deg74 (see message #4412), presumably because it's the

>smallest symbol that will work, and I chose |( for 1deg67 (in #4346),

>which would give this:
>
>67:  |(  /|)  (|\  /||)  /||\
>74:  )|)  /|)  (|\  (||(  /||\
>
>So what do you prefer?

I prefer yours, but I'm uncertain about the complement used for 4
steps of 74.

>> 81,88: )|)  /|)  (|\  (||(  /||\         [13-commas]
>

>This is exactly what I have for 74, above.  Should we do 67 as I did 
>it above and do 74, 81, and 88 alike?

Yes.

>On the other hand, why wouldn't 88 be done as a subset of 176?

I have a reason to do both 81 and 88 as subsets, apart from the fact
that they are 1,3,9-inconsistent. When using their native fifths they
need a single shaft symbol for 4 steps and none is available.

>It is with some surprise that I find that |( is 1deg in both 67 and 
>81, so 81 could also be done the same way as I have for 67, above.

Better to do it the same as 74 and 88 (or as a subset).

>> 6 steps per apotome
>> 37,44,51: )|  /|  /|)  ||\  (||\  /||\         [13-commas]
>> or
>> 37,44,51: |)  )|)  /|)  (||(  ||)  /||\         [13-commas]
>

>For 51 I had something a bit simpler (using lower primes):

So are you agreeing to one of these for 37 and 44? Presumably not the
second one because of |) not being the 7-comma. And with rational
complements?

>51:  |)  /|  /|)  ||\  ||)  /||\

OK.

>> 58: /|  |\  /|\  /||  ||\  /||\
>> or
>> 58: /|  |)  /|\  ||)  ||\  /||\      [13-comma]
>

>I think I would avoid your version2 -- this is another instance where

>it's too easy to be misled into thinking that |) is the 7 comma.  If 
>we wanted to avoid the confusability of all straight flags, we could 
>try:
>
>58: /|  /|)  /|\  (|\  ||\  /||\
>
>Here |) would be kept as the 7 comma and (| would be the 11'-7 comma 
>of 2deg58.  However, I think that it would be too easy to forget 
>that /|) and (|\ aren't representing ratios of 13.  So I think that 
>the safest choice is version 1 -- all straight flags.

Agreed:

58: /|  |\  /|\  /||  ||\  /||\

>> 86,93,100:  )|)  |)  )|\  (|\  (||(  /||\   [13-commas]
>> or
>> 86,100:  )|(  |)  )|\  (|\  (||)  /||\   [13-commas]
>> 93:  |(  |)  )|\  (|\  /||)  /||\   [13-commas]
>

>I would do 93-ET and 100-ET as subsets of 186-ET and 200-ET, 
>respectively.

I can agree to that for 100-ET since there is no single-shaft symbol
for 5 steps, but it is of course 2*50, and 93 is 3*31, so the fifth
sizes are quite acceptable.

>For 86, I wouldn't use |) by itself as anything other than the 7 
>comma, as explained above,

I totally agree we should avoid this in all cases.

> but would use convex flags for symbols 
>that are actual ratios of 13.  So this is how I would do it:
>
>86:  ~|~  /|)  (|~  (|\  ~||~  /||\   [13-commas and 23-comma]
>
>The two best primes are 13 and 23, so there is some basis for 
>defining |~ as the 23 flag.  In any event, I believe that (|~ can be 
>a strong candidate for half an apotome if neither /|\ nor /|) nor (|\

>can be used.

I have no argument about the even steps (they agree with 43 and
50-ET). But again I don't see the need to use a 23-comma. We have
already used )|\ for a half-apotome in the case of 69 and 76-ETs. It
works here too. 86-ET is 1,3,7,13,19-consistent. So why not:

86,93,100:  )|)  /|)  )|\  (|\  ??  /||\   [13-commas]

We can now consider the 31-ET family.

31:            /|\           /||\
62:     /|)    /|\    (|\    /||\     [13-commas]
93:  )|)  /|)  )|\  (|\  ??  /||\     [13-commas]

and compare it to the 19-ET family

19:                          /||\
38:            /|\           /||\
57:       /|)       (|\      /||\     [13-commas]
76:     /|)    )|\    (|\    /||\     [13-commas]

Whew!

With that I must sadly inform you that I will not be able to
contribute to this discussion again for quite some time. I need to get
seriously involved in an electronic design project for some months
now. The trouble is I'm a tuning theory addict. I can't have just a
little.

George, I strongly encourage you to present what we've agreed upon so
far,  to the wider community for comment.

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 5094 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 15:31:52

Subject: Geometric complexity

From: Gene W Smith

The complexity measures we have been using, based on linear temperaments,
give rise to problems when we try to generalize beyond the linear case.
Moreover, as can be seen in the examples of commas associated to 7-limit
JI scales I recently posted, the commas which seem to be the most
important for tempering JI scales are, reasonably enough, the ones with
small Euclidean length in terms of the lattice of octave classes.

I propose we scrap the linear approach and define complexity in terms of
lattice geometry. We can define a Euclidean metric by requiring that for
odd primes p, q and Euclidean distance from the unison L, that we have
the following:

L(p) = ln(p)
if p<q then L(p/q) = L(q/p) = ln(q)

This uniquely determines a Euclidean metric, with quadratic form

L(n)^2 = sum_{i,j} ln(p)^2 x_p x_q,

where x_p is the exponent of p n the factorization of n, and x_q the
exponent of q.

Choosing an orthonormal basis in this space, we define the geometric
complexity as the length of the wedge product of a set of
octave-equivalent commas (commas stripped of 2, a la Graham). Because of
the wedge product, this is independent of the choice of comma basis, and
depends only on the temperament.

Here are some examples; showing first Keenan-style weighted complexity,
geometric complexity as defined above, and 
geometric complexity using a symmetric metric, where 3, 5, and 7 all have
the same length.

[50/49, 64/63]   [2, -4, -4, 2, 12, -11] pajara
3.93867776085766   11.9251094548197   10.3923048454133

[81/80, 126/125]   [1, 4, 10, 12, -13, 4] meantone
5.32244723964455   15.1018056341299   15.5884572681199

[225/224, 1029/1024]   [6, -7, -2, 15, 20, -25] miracle
7.60914796670902   24.9266291754661   18.6279360101972

[225/224, 1728/1715]   [7, -3, 8, 27, 7, -21] orwell
7.42151179771811   25.4206296354264   18.5472369909914

[2401/2400, 4375/4374]   [18, 27, 18, -34, 22, 1] ennealimmal
16.9575882920421   58.8407776477707   39.2300904918661



Message: 5095 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 18:24:27

Subject: Temperaments sorted by "geometric badness"

From: Gene W Smith

Here is a list of 45 7-limit linear temperaments I've given before, this
time sorted according to a badness measure computed using geometric
complexity.

[18, 27, 18, -34, 22, 1]
comp   58.84077764   rms   .1304491741   bad   451.6459719

[1, 4, 10, 12, -13, 4]
comp   15.10180563   rms   3.665035228   bad   835.8645488

[6, -7, -2, 15, 20, -25]
comp   24.92662917   rms   1.637405196   bad   1017.380173

[2, 25, 13, -40, -15, 35]
comp   46.45156501   rms   .5851564738   bad   1262.620148

[2, -4, -4, 2, 12, -11]
comp   11.92510945   rms   10.90317755   bad   1550.521640

[3, 0, -6, -14, 18, -7]
comp   14.16874336   rms   8.100678834   bad   1626.237914

[0, 5, 0, -14, 0, 8]
comp   10.25428060   rms   15.81535241   bad   1662.988586

[7, -3, 8, 27, 7, -21]
comp   25.42062964   rms   2.589237496   bad   1673.187049

[16, 2, 5, 6, 37, -34]
comp   43.84212122   rms   .8753631224   bad   1682.563113

[1, -8, -14, -10, 25, -15]
comp   24.41447354   rms   2.859336356   bad   1704.354666

[5, 13, -17, -76, 41, 9]
comp   46.68750453   rms   .8458796028   bad   1843.783292

[6, 5, 22, 37, -18, -6]
comp   34.26986563   rms   1.610555448   bad   1891.474472

[5, 1, 12, 25, -5, -10]
comp   21.62473825   rms   4.139050792   bad   1935.541443

[1, 4, -2, -16, 6, 4]
comp   9.836559603   rms   20.16328150   bad   1950.956872

[3, 12, -1, -36, 10, 12]
comp   24.63368765   rms   3.579262150   bad   2171.962729

[9, 5, -3, -21, 30, -13]
comp   27.04575319   rms   3.065961726   bad   2242.667503

[0, 12, 24, 22, -38, 19]
comp   38.80790985   rms   1.496892545   bad   2254.400806

[3, 5, -6, -28, 18, 1]
comp   18.24110330   rms   6.808961862   bad   2265.599328

[3, 0, 6, 14, -1, -7]
comp   12.01994256   rms   16.59867843   bad   2398.160778

[8, 18, 11, -25, 5, 10]
comp   34.23414359   rms   2.064339812   bad   2419.357927

[1, 9, -2, -30, 6, 12]
comp   19.47032028   rms   6.410458352   bad   2430.162271

[2, -9, -4, 16, 12, -19]
comp   19.94265308   rms   6.245315858   bad   2483.820897

[4, 4, 4, -2, 5, -3]
comp   11.40589690   rms   19.13699259   bad   2489.617178

[10, 9, 7, -9, 17, -9]
comp   27.53173943   rms   3.320167332   bad   2516.675801

[4, -3, 2, 13, 8, -14]
comp   14.72969739   rms   12.18857055   bad   2644.480840

[4, 2, 2, -1, 8, -6]
comp   10.57420044   rms   23.94525150   bad   2677.407524

[2, 6, 6, -3, -4, 5]
comp   11.92510945   rms   18.86388876   bad   2682.600333

[4, -8, 14, 55, -11, -22]
comp   34.89878325   rms   2.250483424   bad   2740.920186

[2, -4, -16, -26, 31, -11]
comp   26.97297092   rms   3.821630536   bad   2780.393514

[2, 8, 8, -4, -7, 8]
comp   15.87113260   rms   11.21894132   bad   2825.971103

[1, -3, 5, 20, -5, -7]
comp   12.33750942   rms   18.58450012   bad   2828.823679

[5, -11, -12, 3, 33, -29]
comp   32.44371031   rms   2.697384486   bad   2839.251640

[1, 4, -9, -32, 17, 4]
comp   19.68579597   rms   7.652394368   bad   2965.536698

[2, 8, 1, -20, 4, 8]
comp   15.29862604   rms   12.69007837   bad   2970.086938

[7, 9, 13, 5, -1, -2]
comp   24.36497795   rms   5.052931030   bad   2999.683372

[2, 8, -11, -48, 23, 8]
comp   28.86573677   rms   3.732363180   bad   3109.919806

[3, 17, -1, -50, 10, 20]
comp   34.40312184   rms   2.729116326   bad   3230.113288

[6, 5, 3, -7, 12, -6]
comp   16.38306753   rms   12.27380956   bad   3294.350648

[5, 1, -7, -19, 25, -10]
comp   19.98216004   rms   8.727168682   bad   3484.642557

[12, 10, -9, -49, 48, -12]
comp   42.88340322   rms   1.896512488   bad   3487.660430

[15, -2, -5, -6, 50, -38]
comp   45.81266906   rms   1.731229740   bad   3633.506097

[12, -2, 20, 52, 2, -31]
comp   45.66691576   rms   1.753213789   bad   3656.269843

[9, 10, -3, -35, 30, -5]
comp   30.78274747   rms   4.052704060   bad   3840.251351

[13, -10, 6, 42, 27, -46]
comp   48.03151023   rms   1.678518039   bad   3872.384715

[8, 6, 6, -3, 13, -9]
comp   21.57627467   rms   10.13226624   bad   4716.930933

Message: 5096 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 22:37:17

Subject: Philippe de Vitry and Levi ben Gerson

From: Gene W Smith

The finiteness of the 3-limit superparticulars is a very old result, and
arose from musical considerations:

Science News Online: Ivars Peterson's MathTrek... * [with cont.]  (Wayb.)



Message: 5097 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 22:42:32

Subject: Hendrik talks, but says who knows what?

From: Gene W Smith

Can anyone read this?

http://msri.org/publications/ln/msri/1998/mandm/lenstra/1/ * [with cont.]  (Wayb.)



Message: 5098 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 00:02:04

Subject: Square scales

From: Gene W Smith

There are three scales deriving from the three orientations of a 3x3
square in the cubic lattice with a major tetrad in the center, and three
more with a minor tetrad. These I give below, along with the
corresponding commas, which turn out to be the same for the major and
minor form of each square.

maj12 := [1, 25/24, 21/20, 35/32, 8/7, 6/5, 5/4, 21/16, 10/7, 35/24, 3/2,
49/32, 5/3, 12/7, 7/4, 9/5]

min12 := [1, 36/35, 21/20, 8/7, 6/5, 5/4, 21/16, 48/35, 10/7, 36/25, 3/2,
5/3, 12/7, 7/4, 9/5, 96/49]

commas 126/125, 1029/1024, 1728/1715

126/125^1029/1024^1728/1715 = 31-et
1029/1024^126/125 = [6, 3, -3, -5, 6, -4], a linear temperament with no
name
126/125^1728/1715 = [10, 9, 7, -9, 17, -9], small diesic
1029/1024^1728/1715 = [3, 12, -1, -36, 10, 12], supermajor seconds


maj13 := [1, 49/48, 15/14, 35/32, 7/6, 6/5, 5/4, 9/7, 7/5, 35/24, 3/2,
25/16, 8/5, 12/7, 7/4, 15/8]

min13 := [1, 36/35, 15/14, 7/6, 6/5, 5/4, 9/7, 48/35, 7/5, 72/49, 3/2,
8/5, 12/7, 7/4, 15/8, 48/25]

commas 225/224, 1728/1715
1728/1715^225/224 = orwell

maj23:=[1, 21/20, 15/14, 9/8, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7,
3/2, 5/3, 7/4, 25/14, 15/8]

min23 := [1, 21/20, 15/14, 9/8, 8/7, 6/5, 60/49, 9/7, 4/3, 7/5, 10/7,
3/2, 8/5, 42/25, 12/7, 9/5]

commas 126/125, 225/224, 2401/2400
126/125^225/224^2401/2400 = 31-et
126/125^225/224 = meantone
126/125^2401/2400 = small diesic
225/224^2401/2400 = miracle

Message: 5099 - Contents - Hide Contents

Date: Sun, 14 Jul 2002 14:05:49

Subject: Re: A chord analog to Fokker blocks

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:

>There's a close relationship with periodicity blocks. Pick either major or

minor tetrads, and with those, one >particular chord element--for instance
the roots of the minor tetrads, or the major third element of the major

>tetrads. These will all individually be corresponding Fokker blocks.

I see, sort of a Carthesian product of a Fokker block with a chord then?

>Where is the 4:5:6:7 double tied circular mirroring discussed?

I wrote a posting to the Tuning List several years ago, I can't find
the specific date at the moment. In the scala archive they are the
scales *kring*.scl.
In a double tied circular mirroring a chord is inverted repeatedly
with two tones in common each time with the next inversion, until
coming back to the original.
They are closely related to Partch diamonds.

Manuel

Previous Next

5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950

5050 - 5075 -