Otones12-24

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A dedecatonic (12-note) scale borrowed from the overtone series, also known as harm24:

12/12, 13/12, 14/12, 15/12, 16/12, 17/12, 18/12, 19/12, 20/12, 21/12, 22/12, 23/12, 24/12

which reduces to:

1/1, 13/12, 7/6, 5/4, 4/3, 17/12, 3/2, 19/12, 5/3, 7/4, 11/6, 23/12, 2/1

in cents:

0, 139, 267, 386, 498, 603, 702, 796, 884, 969, 1049, 1126, 1200

Intervals between adjacent steps gradually get smaller, as is expected for overtone chunks like this:

13:12, 14:13, 15:14, 16:15, 17:16, 18:17, 19:18, 21:20, 22:21, 23:22, 24:23

in cents:

139, 128, 119, 112, 105, 99, 94, 89, 84, 81, 77, 74

modes

For your tuning pleasure, all 12 modes, arranged in a handy-dandy table. The following matrix uses a keyboard mapping that starts the scale on C. Thus, C=1/1, C#=13/12, etc. To find an interval, say the interval from C to F#, first find the lower pitch on the left, C, & follow it across the row to the column of the higher pitch, F# to find 702 cents. To find the name of the interval in just intonation, use the number of the higher pitch as the numerator (18) and the number of the lower pitch (12) as the denominator, then reduce (3/2).

C-12 C#-13 D-14 D#-15 E-16 F-17 F#-18 G-19 G#-20 A-21 A#-22 B-23
C-12 0 139 267 386 498 603 702 796 884 969 1049 1126
C#-13 1061 0 128 248 359 464 563 657 746 830 911 988
D-14 933 1072 0 119 231 336 435 529 617 702 782 859
D#-15 814 952 1081 0 112 217 316 409 498 583 663 740
E-16 702 841 969 1088 0 105 204 298 386 471 551 628
F-17 597 736 864 983 1095 0 99 193 281 366 446 523
F#-18 498 637 765 884 996 1101 0 94 182 267 347 424
G-19 404 543 671 791 902 1007 1106 0 89 173 254 331
G#-20 316 454 583 702 814 919 1018 1111 0 84 165 242
A-21 231 370 498 617 729 834 933 1027 1116 0 81 157
A#-22 151 289 418 537 649 754 853 946 1035 1119 0 77
B-23 74 212 341 460 572 677 776 869 958 1043 1123 0

You can see that, due to the varying step sizes, this relatively small scale contains a large number of unique rational intervals up to the 23-limit.

Compositions

ant lizard dragon man, original song by Threshold of Pain, words by Scott Marshall (2006). This recording is an arrangement for otonal organ, otonal dulcimer, hand claps, and voice by Andrew Heathwaite in otones 12-24 (arranged & recorded 2010).

Inventory of intervals from 0 to 1200 cents:

0 - 1/1 - ...

74 - 24/23 - B-C

77 - 23/22 - A#-B

81 - 22/21 - A-A#

84 - 21/20 - G#-A

89 - 20/19 - G-G#

94 - 19/18 - F#-G

99 - 18/17 - F-F#

105 - 17/16 - E-F

112 - 16/15 - D#-E

119 - 15/14 - D-D#

128 - 14/13 - C#-D

139 - 13/12 - C-C#

151 - 12/11 - A#-C

157 - 23/21 - A-B

165 - 11/10 - A#-B#

173 - 21/19 - G-A

182 - 10/9 - F#-G#

193 - 19/17 - F-G

204 - 9/8 - E-F#

212 - 26/23 - B-C#

217 - 17/15 - D#-F

231 - 8/7 - D-E ; A-C

242 - 23/20 - G#-B

248 - 15/13 - C#-D#

254 - 22/19 - G-A#

267 - 7/6 - C-D ; F#-A

281 - 20/17 - F-G#

289 - 13/11 - A#-C#

316 - 6/5 - D#-F# ; G#-C

331 - 23/19 - G-B

336 - 17/14 - D-F

347 - 11/9 - F#-A#

359 - 16/13 - C#-E

366 - 21/17 - F-A

370 - 26/21 - A-C#

386 - 5/4 - C-D# ; E-G#

404 - 24/19 - G-C

409 - 19/15 - D#-G

418 - 14/11 - A#-D

424 - 23/18 - F#-B

435 - 9/7 - D-F#

446 - 22/17 - F-A#

454 - 13/10 - G#-C#

460 - 30/23 - B-D#

464 - 17/13 - C#-F

471 - 21/16 - E-A

498 - 4/3 - C-E ; D#-G# ; F#-C ; A-D

523 - 23/17 - F-B

529 - 19/14 - D-G

537 - 15/11 - A#-D#

543 - 26/19 - G-C#

551 - 11/8 - E-A

563 - 18/13 - C#-F#

572 - 32/23 - B-E

583 - 7/5 - D#-A ; G#-D

597 - 24/17 - F-C

603 - 17/12 - C-F

617 - 10/7 - D-G# ; A-D#

628 - 23/16 - E-B

637 - 13/9 - F#-C#

649 - 16/11 - A-E

657 - 19/13 - C#-G

663 - 22/15 - D#-A#

671 - 28/19 - G-D

677 - 34/23 - B-F

...