Masses of asses

I took all 13-limit odd integers up to 99, and then all four-note
chords derivable from them. I then checked for the ones which were
outonal, or asses, and for which the smallest interval was greater
than or equal to 14/13. Below I list these for prime limits from 3 to
13. The chords are in a Tenney reduced form, with minimax Tenney
height. I give this minimax height for each chord; they are sorted in
terms of this but not in terms of another relevant metric, the
smallest interval appearing.

3
[9/8, 4/3, 3/2, 2] 72
[9/8, 4/3, 27/16, 2] 432

5
[5/4, 3/2, 5/3, 2] 20
[6/5, 4/3, 9/5, 2] 45
[3/2, 5/3, 9/5, 2] 45
[9/8, 5/4, 9/5, 2] 72
[9/8, 8/5, 9/5, 2] 72
[27/25, 4/3, 36/25, 2] 900

7
[5/4, 7/5, 7/4, 2] 35
[7/5, 8/5, 7/4, 2] 40
[7/6, 7/5, 5/3, 2] 42
[7/6, 3/2, 7/4, 2] 42
[7/6, 4/3, 7/4, 2] 42
[8/7, 7/5, 8/5, 2] 56
[7/6, 9/7, 3/2, 2] 63
[9/7, 7/5, 9/5, 2] 63
[7/6, 10/7, 5/3, 2] 70
[9/7, 10/7, 9/5, 2] 70
[9/8, 9/7, 7/4, 2] 72
[10/9, 9/7, 10/7, 2] 90
[9/8, 14/9, 7/4, 2] 126
[25/21, 3/2, 25/14, 2] 525
[25/21, 4/3, 25/14, 2] 525
[9/8, 10/7, 45/28, 2] 1260

11
[7/6, 11/7, 11/6, 2] 77
[11/8, 11/7, 7/4, 2] 88
[11/8, 3/2, 11/6, 2] 88
[8/7, 11/8, 11/7, 2] 88
[11/9, 3/2, 11/6, 2] 99
[9/8, 11/9, 11/8, 2] 99
[11/10, 5/3, 11/6, 2] 110
[11/10, 5/4, 11/8, 2] 110
[11/10, 7/5, 11/7, 2] 110
[11/10, 11/9, 9/5, 2] 110
[11/10, 11/8, 8/5, 2] 110
[11/10, 10/7, 11/7, 2] 110
[11/10, 6/5, 11/6, 2] 110
[11/9, 11/8, 16/9, 2] 144
[13/12, 7/6, 13/7, 2] 156
[12/11, 15/11, 8/5, 2] 165
[12/11, 5/4, 15/11, 2] 165
[6/5, 15/11, 18/11, 2] 198
[15/11, 3/2, 20/11, 2] 220
[27/22, 4/3, 18/11, 2] 594
[6/5, 11/8, 33/20, 2] 660
[11/10, 4/3, 33/20, 2] 660
[33/28, 4/3, 11/7, 2] 924
[33/28, 11/8, 12/7, 2] 924
[33/28, 9/7, 11/6, 2] 924
[33/25, 3/2, 44/25, 2] 1100
[10/9, 11/8, 55/36, 2] 1980
[55/42, 10/7, 11/6, 2] 2310
[9/8, 14/11, 63/44, 2] 2772
[63/55, 14/11, 9/5, 2] 3465
[11/10, 14/9, 77/45, 2] 3465
[11/9, 7/5, 77/45, 2] 3465

13
[8/7, 13/8, 13/7, 2] 104
[13/10, 7/5, 13/7, 2] 130
[13/10, 10/7, 13/7, 2] 130
[13/12, 12/7, 13/7, 2] 156
[13/12, 13/10, 5/3, 2] 156
[13/12, 3/2, 13/8, 2] 156
[13/12, 4/3, 13/8, 2] 156
[13/12, 6/5, 13/10, 2] 156
[15/13, 18/13, 5/3, 2] 234
[15/13, 4/3, 20/13, 2] 260
[14/13, 4/3, 21/13, 2] 273
[7/6, 18/13, 21/13, 2] 273
[14/13, 3/2, 21/13, 2] 273
[15/13, 5/4, 24/13, 2] 312
[15/13, 8/5, 24/13, 2] 312
[13/11, 3/2, 39/22, 2] 858
[14/13, 5/4, 35/26, 2] 910
[35/26, 20/13, 7/4, 2] 910
[5/4, 18/13, 45/26, 2] 1170
[39/35, 6/5, 13/7, 2] 1365
[39/35, 13/10, 12/7, 2] 1365
[13/12, 10/7, 65/42, 2] 2730
[63/52, 18/13, 7/4, 2] 3276
[65/56, 10/7, 13/8, 2] 3640
[13/11, 7/5, 91/55, 2] 5005

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> I took all 13-limit odd integers up to 99, and then all four-note
> chords derivable from them. I then checked for the ones which were
> outonal, or asses, and for which the smallest interval was greater
> than or equal to 14/13. Below I list these for prime limits from 3
to
> 13. The chords are in a Tenney reduced form, with minimax Tenney
> height. I give this minimax height for each chord; they are sorted
in
> terms of this but not in terms of another relevant metric, the
> smallest interval appearing.

Could you define outonal? Is that some combination of utonal and
otonal? Thanks.

>
> 3
> [9/8, 4/3, 3/2, 2] 72
> [9/8, 4/3, 27/16, 2] 432
>
> 5
> [5/4, 3/2, 5/3, 2] 20
> [6/5, 4/3, 9/5, 2] 45
> [3/2, 5/3, 9/5, 2] 45
> [9/8, 5/4, 9/5, 2] 72
> [9/8, 8/5, 9/5, 2] 72
> [27/25, 4/3, 36/25, 2] 900
>
> 7
> [5/4, 7/5, 7/4, 2] 35
> [7/5, 8/5, 7/4, 2] 40
> [7/6, 7/5, 5/3, 2] 42
> [7/6, 3/2, 7/4, 2] 42
> [7/6, 4/3, 7/4, 2] 42
> [8/7, 7/5, 8/5, 2] 56
> [7/6, 9/7, 3/2, 2] 63
> [9/7, 7/5, 9/5, 2] 63
> [7/6, 10/7, 5/3, 2] 70
> [9/7, 10/7, 9/5, 2] 70
> [9/8, 9/7, 7/4, 2] 72
> [10/9, 9/7, 10/7, 2] 90
> [9/8, 14/9, 7/4, 2] 126
> [25/21, 3/2, 25/14, 2] 525
> [25/21, 4/3, 25/14, 2] 525
> [9/8, 10/7, 45/28, 2] 1260
>
> 11
> [7/6, 11/7, 11/6, 2] 77
> [11/8, 11/7, 7/4, 2] 88
> [11/8, 3/2, 11/6, 2] 88
> [8/7, 11/8, 11/7, 2] 88
> [11/9, 3/2, 11/6, 2] 99
> [9/8, 11/9, 11/8, 2] 99
> [11/10, 5/3, 11/6, 2] 110
> [11/10, 5/4, 11/8, 2] 110
> [11/10, 7/5, 11/7, 2] 110
> [11/10, 11/9, 9/5, 2] 110
> [11/10, 11/8, 8/5, 2] 110
> [11/10, 10/7, 11/7, 2] 110
> [11/10, 6/5, 11/6, 2] 110
> [11/9, 11/8, 16/9, 2] 144
> [13/12, 7/6, 13/7, 2] 156
> [12/11, 15/11, 8/5, 2] 165
> [12/11, 5/4, 15/11, 2] 165
> [6/5, 15/11, 18/11, 2] 198
> [15/11, 3/2, 20/11, 2] 220
> [27/22, 4/3, 18/11, 2] 594
> [6/5, 11/8, 33/20, 2] 660
> [11/10, 4/3, 33/20, 2] 660
> [33/28, 4/3, 11/7, 2] 924
> [33/28, 11/8, 12/7, 2] 924
> [33/28, 9/7, 11/6, 2] 924
> [33/25, 3/2, 44/25, 2] 1100
> [10/9, 11/8, 55/36, 2] 1980
> [55/42, 10/7, 11/6, 2] 2310
> [9/8, 14/11, 63/44, 2] 2772
> [63/55, 14/11, 9/5, 2] 3465
> [11/10, 14/9, 77/45, 2] 3465
> [11/9, 7/5, 77/45, 2] 3465
>
> 13
> [8/7, 13/8, 13/7, 2] 104
> [13/10, 7/5, 13/7, 2] 130
> [13/10, 10/7, 13/7, 2] 130
> [13/12, 12/7, 13/7, 2] 156
> [13/12, 13/10, 5/3, 2] 156
> [13/12, 3/2, 13/8, 2] 156
> [13/12, 4/3, 13/8, 2] 156
> [13/12, 6/5, 13/10, 2] 156
> [15/13, 18/13, 5/3, 2] 234
> [15/13, 4/3, 20/13, 2] 260
> [14/13, 4/3, 21/13, 2] 273
> [7/6, 18/13, 21/13, 2] 273
> [14/13, 3/2, 21/13, 2] 273
> [15/13, 5/4, 24/13, 2] 312
> [15/13, 8/5, 24/13, 2] 312
> [13/11, 3/2, 39/22, 2] 858
> [14/13, 5/4, 35/26, 2] 910
> [35/26, 20/13, 7/4, 2] 910
> [5/4, 18/13, 45/26, 2] 1170
> [39/35, 6/5, 13/7, 2] 1365
> [39/35, 13/10, 12/7, 2] 1365
> [13/12, 10/7, 65/42, 2] 2730
> [63/52, 18/13, 7/4, 2] 3276
> [65/56, 10/7, 13/8, 2] 3640
> [13/11, 7/5, 91/55, 2] 5005
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Could you define outonal? Is that some combination of utonal and
> otonal? Thanks.

A chord is "outonal" if it is an inversely symmetrical chord in
rational intonation. Inversely symmetrical means an octave-equivalent
chord which, on inversion and transposition, gives the same chord
again. "Outonal" suggests that it is equally otonal and utonal.

For example, taking [5/4, 3/2, 5/3, 2] and inverting gives
[4/5, 2/3, 3/5, 1/2]. Multiplying that by 5/2 gives [5/4, 3/2, 5/3, 2]
again.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Could you define outonal? Is that some combination of utonal and
> > otonal? Thanks.
>
> A chord is "outonal" if it is an inversely symmetrical chord in
> rational intonation. Inversely symmetrical means an octave-equivalent
> chord which, on inversion and transposition, gives the same chord
> again. "Outonal" suggests that it is equally otonal and utonal.
>
> For example, taking [5/4, 3/2, 5/3, 2] and inverting gives
> [4/5, 2/3, 3/5, 1/2]. Multiplying that by 5/2 gives [5/4, 3/2, 5/3, 2]
> again.
>
Thanks. I checked your posts on tuning too. Perhaps we should mention
when we are cross-posting...of course I should have checked myself.