Harmonics 12–24: Difference between revisions

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{{Infobox ADO|steps=12}}
{{Infobox harmonics|12}}
'''12ado''' is the [[ADO|arithmetic equal division of the octave]] into twelve parts of 1/12 each.  12ado is a subset of the esoteric [[Factor 9 grid]] temperament connected to "A = 432 Hz" conspiracy.
 
{{Harmonics intro|12}}


== Intervals ==
== Intervals ==
Line 85: Line 86:
| [[11/6]]
| [[11/6]]
| 1.83333
| 1.83333
| undecimal neutral sixth
| undecimal neutral seventh
| [[File:Jid_11_6_pluck_adu_dr220.mp3]]
| [[File:Jid_11_6_pluck_adu_dr220.mp3]]
|-
|-
Line 93: Line 94:
| 1.91667
| 1.91667
| vicesimotertial major seventh
| vicesimotertial major seventh
| [[File:Jid_23_12_pluck_adu_dr220.mp3]]
| [[File:Ji-23-12-csound-foscil-220hz.mp3]]
|-
|-
| 12
| 12
Line 103: Line 104:
|}
|}


[[Category:ADO]]
== 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).
 
{| class="wikitable right-all"
|-
! Note
! 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.
<pre>
! 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
</pre>
 
== Scales ==
* 12:13:14:17:18:19:21:24 [[Lou Harrison]]'s "Kyai Gunter Sari [[pelog]]"
* 12:14:16:18:21:24 Septimal minor pentatonic
* 12:14:18:21:24 Forrest Cahoon's Sevens [[tetrachord]]
 
== Music ==
; [[Forrest Cahoon]]
* [https://soundcloud.com/fcahoon/the-sevens ''The Sevens''] (2012) – uses the Sevens tetrachord
 
; [[Francium]]
* [https://www.youtube.com/watch?v=9x66m4ddNok ''Dark Hard, Dream Big''] (2026)
 
; [[Andrew Heathwaite]]
* [https://soundclick.com/share?songid=8839059 ''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.
 
{{Todo| cleanup }}
 
{{Navbox harmonics}}
 
[[Category:Pages with Scala files]]

Latest revision as of 00:07, 11 October 2026

Harmonics 12–24
Prime factorization 22 × 3
Fifth 18/12 (701.955c)

The harmonic segment 12::24 (also harmonics 12–24) consists of harmonics 12 through 24 (12:13:…:24) and spans one octave above the root. Used as a scale, it is also called mode 12 of the harmonic series.

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 seventh
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).

Note 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

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

Music

Forrest Cahoon
Francium
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.
View • Talk • EditOvertone scales 
Small modes 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24
Larger modes 30 • 32 • 36 • 48 • 60 • 128
Families /2: 2 • 4 • 8 • 16 • 32 • 128
/3: 3 • 6 • 9 • 12 • 15 • 18 • 21 • 24
/5: 5 • 10 • 15 • 20 • 25 • 30 • 35 • 60 • 80
/7: 7 • 14 • 21 • 28 • 35 • 56
/11: 11 • 22 • 33
/13: 13 • 26
Related Harmonic series • Subharmonic series • Carlos harmonic scale • Ringer scale • Primodality
Retrieved from "https://en.xen.wiki/index.php?title=Harmonics_12–24&oldid=239213"