12afdo
12afdo (arithmetic frequency division of the octave), or 12odo (otonal division of the octave), divides the octave into twelve parts of 1/12 each. 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.
Scales
- Forrest Cahoon’s Sevens tetrachord: 14/12-18/12-21/12-24/12
- Lou Harrison’s “Kyai Gunter Sari pelog”: 13/12-14/12-17/12-18/12-19/12-21/12-24/12
Music
- 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