12afdo

← 11afdo

12afdo

13afdo →

Prime factorization	22 × 3
Fifth	18/12 (701.955c)

12afdo (arithmetic frequency division of the octave), or 12odo (otonal division of the octave), divides the octave into twelve parts of 1/12 each. It is a superset of 11afdo and a subset of 13afdo. As a scale it may be known as mode 12 of the harmonic series or the Over-12 scale.

Intervals

#	Cents	Ratio	Decimal	Interval name	Audio
0	0	1/1	1.0000	perfect unison
1	138.6	13/12	1.0833	tridecimal neutral second
2	266.9	7/6	1.1667	subminor third
3	386.3	5/4	1.2500	just major third
4	498.0	4/3	1.3333	just perfect fourth
5	603.0	17/12	1.41667	larger septendecimal tritone
6	702.0	3/2	1.50000	just perfect fifth
7	792.6	19/12	1.58333	large undevicesimal minor sixth
8	884.4	5/3	1.66667	just major sixth
9	968.8	7/4	1.75000	harmonic seventh
10	1049.3	11/6	1.83333	undecimal neutral sixth
11	1126.3	23/12	1.91667	vicesimotertial major seventh
12	1200.0	2/1	2.0000	perfect octave

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.

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

…

Scala file

Useable in Scala and any software/hardware supporting it.

! harm24.scl
!
Harmonics 12 to 24 
! Also known as otones12-24                                                             
12
!
13/12
7/6
5/4
4/3
17/12
3/2
19/12
5/3
7/4
11/6
23/12
2/1

Scales

• Forrest Cahoon's Sevens tetrachord: 14/12–18/12–21/12–24/12
• Septimal minor pentatonic: 14/12-16/12-18/12-21/12-24/12
• Lou Harrison's "Kyai Gunter Sari pelog": 13/12–14/12–17/12–18/12–19/12–21/12–24/12

Music

Andrew Heathwaite

• ant lizard dragon man (arranged & recorded 2010) – 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.

Forrest Cahoon

• The Sevens (2012) – uses the Sevens tetrachord