Otones12-24

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

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