72edo: Difference between revisions
m -monotonicity -major 2nd in infobox |
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| Line 51: | Line 51: | ||
| 45/44 | | 45/44 | ||
| ^^ | | ^^ | ||
| | | dup unison | ||
| ^^D | | ^^D | ||
|- | |- | ||
| Line 65: | Line 65: | ||
| 25/24 | | 25/24 | ||
| vvm2 | | vvm2 | ||
| | | dudminor 2nd | ||
| vvEb | | vvEb | ||
|- | |- | ||
| Line 135: | Line 135: | ||
| 8/7 | | 8/7 | ||
| ^^M2 | | ^^M2 | ||
| | | dupmajor 2nd | ||
| ^^E | | ^^E | ||
|- | |- | ||
| Line 149: | Line 149: | ||
| 7/6 | | 7/6 | ||
| vvm3 | | vvm3 | ||
| | | dudminor 3rd | ||
| vvF | | vvF | ||
|- | |- | ||
| Line 219: | Line 219: | ||
| 9/7 | | 9/7 | ||
| ^^M3 | | ^^M3 | ||
| | | dupmajor 3rd | ||
| ^^F# | | ^^F# | ||
|- | |- | ||
| Line 233: | Line 233: | ||
| 21/16, 17/13 | | 21/16, 17/13 | ||
| vv4 | | vv4 | ||
| | | dud 4th | ||
| vvG | | vvG | ||
|- | |- | ||
| Line 345: | Line 345: | ||
| 32/21 | | 32/21 | ||
| ^^5 | | ^^5 | ||
| | | dup 5th | ||
| ^^A | | ^^A | ||
|- | |- | ||
| Line 359: | Line 359: | ||
| 14/9 | | 14/9 | ||
| vvm6 | | vvm6 | ||
| | | dudminor 6th | ||
| vvBb | | vvBb | ||
|- | |- | ||
| Line 429: | Line 429: | ||
| 12/7 | | 12/7 | ||
| ^^M6 | | ^^M6 | ||
| | | dupmajor 6th | ||
| ^^B | | ^^B | ||
|- | |- | ||
| Line 443: | Line 443: | ||
| 7/4 | | 7/4 | ||
| vvm7 | | vvm7 | ||
| | | dudminor 7th | ||
| vvC | | vvC | ||
|- | |- | ||
| Line 513: | Line 513: | ||
| 27/14 | | 27/14 | ||
| ^^M7 | | ^^M7 | ||
| | | dupmajor 7th | ||
| ^^C# | | ^^C# | ||
|- | |- | ||
| Line 527: | Line 527: | ||
| 49/25 | | 49/25 | ||
| vv8 | | vv8 | ||
| | | dud octave | ||
| vvD | | vvD | ||
|- | |- | ||
| Line 556: | Line 556: | ||
! examples | ! examples | ||
|- | |- | ||
| | | dud minor | ||
| zo | | zo | ||
| {a, b, 0, 1} | | {a, b, 0, 1} | ||
| Line 591: | Line 591: | ||
| 9/8, 27/16 | | 9/8, 27/16 | ||
|- | |- | ||
| | | dup major | ||
| ru | | ru | ||
| {a, b, 0, -1} | | {a, b, 0, -1} | ||
| Line 612: | Line 612: | ||
| C vvEb G | | C vvEb G | ||
| Cvvm | | Cvvm | ||
| C | | C dud minor | ||
|- | |- | ||
| gu | | gu | ||
| Line 640: | Line 640: | ||
| C ^^E G | | C ^^E G | ||
| C^^ | | C^^ | ||
| C | | C dupmajor or C dup | ||
|} | |} | ||
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]]. | For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]]. | ||
| Line 1,272: | Line 1,272: | ||
== Music == | == Music == | ||
* [ | * [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] by [[Gene Ward Smith]] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday''] by [[Jake Freivald]] in the [[lazysunday]] scale. | * [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday''] by [[Jake Freivald]] in the [[lazysunday]] scale. | ||
| Line 1,284: | Line 1,284: | ||
* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music | * [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music | ||
* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list | * [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list | ||
* [ | * [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo] | ||
[[Category:72edo| ]] <!-- main article --> | [[Category:72edo| ]] <!-- main article --> | ||
Revision as of 06:29, 17 September 2022
| ← 71edo | 72edo | 73edo → |
The 72 equal divisions of the octave (72edo), or 72-tone equal temperament (72tet, 72et) when viewed from a regular temperament perspective, divides the octave into 72 steps or moria. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12edo. 72edo is also a superset of 24edo, a common and standard tuning of Arabic music, and has itself been used to tune Turkish music.
Composers that used 72edo include Alois Hába, Ivan Wyschnegradsky, Julián Carrillo (who is better associated with 96edo), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
Theory
72edo approximates 11-limit just intonation exceptionally well, is consistent in the 17-limit, and is the ninth Zeta integral tuning. The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third (5/4) measures 23 steps, not 24, and other 5-limit major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
72edo is an excellent tuning for the miracle temperament, especially the 11-limit version, and the related rank-3 temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -2.98 | -2.16 | -1.32 | -7.19 | -4.96 | +2.49 | +5.06 | +3.76 | +4.96 |
| Relative (%) | +0.0 | -11.7 | -17.9 | -13.0 | -7.9 | -43.2 | -29.7 | +14.9 | +30.4 | +22.5 | +29.8 | |
| Steps (reduced) |
72 (0) |
114 (42) |
167 (23) |
202 (58) |
249 (33) |
266 (50) |
294 (6) |
306 (18) |
326 (38) |
350 (62) |
357 (69) | |
Intervals
| Degrees | Cents | Approximate Ratios * | Ups and Downs Notation | ||
|---|---|---|---|---|---|
| 0 | 0.000 | 1/1 | P1 | perfect unison | D |
| 1 | 16.667 | 81/80 | ^1 | up unison | ^D |
| 2 | 33.333 | 45/44 | ^^ | dup unison | ^^D |
| 3 | 50.000 | 33/32 | ^31, v3m2 |
triple-up unison, triple-down minor 2nd |
^3D, v3Eb |
| 4 | 66.667 | 25/24 | vvm2 | dudminor 2nd | vvEb |
| 5 | 83.333 | 21/20 | vm2 | downminor 2nd | vEb |
| 6 | 100.000 | 35/33, 17/16, 18/17 | m2 | minor 2nd | Eb |
| 7 | 116.667 | 15/14, 16/15 | ^m2 | upminor 2nd | ^Eb |
| 8 | 133.333 | 27/25, 13/12, 14/13 | v~2 | downmid 2nd | ^^Eb |
| 9 | 150.000 | 12/11 | ~2 | mid 2nd | v3E |
| 10 | 166.667 | 11/10 | ^~2 | upmid 2nd | vvE |
| 11 | 183.333 | 10/9 | vM2 | downmajor 2nd | vE |
| 12 | 200.000 | 9/8 | M2 | major 2nd | E |
| 13 | 216.667 | 25/22, 17/15 | ^M2 | upmajor 2nd | ^E |
| 14 | 233.333 | 8/7 | ^^M2 | dupmajor 2nd | ^^E |
| 15 | 250.000 | 81/70, 15/13 | ^3M2, v3m3 |
triple-up major 2nd, triple-down minor 3rd |
^3E, v3F |
| 16 | 266.667 | 7/6 | vvm3 | dudminor 3rd | vvF |
| 17 | 283.333 | 33/28, 13/11, 20/17 | vm3 | downminor 3rd | vF |
| 18 | 300.000 | 25/21 | m3 | minor 3rd | F |
| 19 | 316.667 | 6/5 | ^m3 | upminor 3rd | ^F |
| 20 | 333.333 | 40/33, 17/14 | v~3 | downmid 3rd | ^^F |
| 21 | 350.000 | 11/9 | ~3 | mid 3rd | ^3F |
| 22 | 366.667 | 99/80, 16/13, 21/17 | ^~3 | upmid 3rd | vvF# |
| 23 | 383.333 | 5/4 | vM3 | downmajor 3rd | vF# |
| 24 | 400.000 | 44/35 | M3 | major 3rd | F# |
| 25 | 416.667 | 14/11 | ^M3 | upmajor 3rd | ^F# |
| 26 | 433.333 | 9/7 | ^^M3 | dupmajor 3rd | ^^F# |
| 27 | 450.000 | 35/27, 13/10 | ^3M3, v34 |
triple-up major 3rd, triple-down 4th |
^3F#, v3G |
| 28 | 466.667 | 21/16, 17/13 | vv4 | dud 4th | vvG |
| 29 | 483.333 | 33/25 | v4 | down 4th | vG |
| 30 | 500.000 | 4/3 | P4 | perfect 4th | G |
| 31 | 516.667 | 27/20 | ^4 | up 4th | ^G |
| 32 | 533.333 | 15/11 | v~4 | downmid 4th | ^^G |
| 33 | 550.000 | 11/8 | ~4 | mid 4th | ^3G |
| 34 | 566.667 | 25/18, 18/13 | ^~4 | upmid 4th | vvG# |
| 35 | 583.333 | 7/5 | vA4, vd5 | downaug 4th, updim 5th | vG#, vAb |
| 36 | 600.000 | 99/70, 17/12 | A4, d5 | aug 4th, dim 5th | G#, Ab |
| 37 | 616.667 | 10/7 | ^A4, ^d5 | upaug 4th, downdim 5th | ^G#, ^Ab |
| 38 | 633.333 | 36/25, 13/9 | v~5 | downmid 5th | ^^Ab |
| 39 | 650.000 | 16/11 | ~5 | mid 5th | v3A |
| 40 | 666.667 | 22/15 | ^~5 | upmid 5th | vvA |
| 41 | 683.333 | 40/27 | v5 | down 5th | vA |
| 42 | 700.000 | 3/2 | P5 | perfect 5th | A |
| 43 | 716.667 | 50/33 | ^5 | up 5th | ^A |
| 44 | 733.333 | 32/21 | ^^5 | dup 5th | ^^A |
| 45 | 750.000 | 54/35, 17/11 | ^35, v3m6 |
triple-up 5th, triple-down minor 6th |
^3A, v3Bb |
| 46 | 766.667 | 14/9 | vvm6 | dudminor 6th | vvBb |
| 47 | 783.333 | 11/7 | vm6 | downminor 6th | vBb |
| 48 | 800.000 | 35/22 | m6 | minor 6th | Bb |
| 49 | 816.667 | 8/5 | ^m6 | upminor 6th | ^Bb |
| 50 | 833.333 | 81/50, 13/8 | v~6 | downmid 6th | ^^Bb |
| 51 | 850.000 | 18/11 | ~6 | mid 6th | v3B |
| 52 | 866.667 | 33/20, 28/17 | ^~6 | upmid 6th | vvB |
| 53 | 883.333 | 5/3 | vM6 | downmajor 6th | vB |
| 54 | 900.000 | 27/16 | M6 | major 6th | B |
| 55 | 916.667 | 56/33, 17/10 | ^M6 | upmajor 6th | ^B |
| 56 | 933.333 | 12/7 | ^^M6 | dupmajor 6th | ^^B |
| 57 | 950.000 | 121/70 | ^3M6, v3m7 |
triple-up major 6th, triple-down minor 7th |
^3B, v3C |
| 58 | 966.667 | 7/4 | vvm7 | dudminor 7th | vvC |
| 59 | 983.333 | 44/25 | vm7 | downminor 7th | vC |
| 60 | 1000.000 | 16/9 | m7 | minor 7th | C |
| 61 | 1016.667 | 9/5 | ^m7 | upminor 7th | ^C |
| 62 | 1033.333 | 20/11 | v~7 | downmid 7th | ^^C |
| 63 | 1050.000 | 11/6 | ~7 | mid 7th | ^3C |
| 64 | 1066.667 | 50/27 | ^~7 | upmid 7th | vvC# |
| 65 | 1083.333 | 15/8 | vM7 | downmajor 7th | vC# |
| 66 | 1100.000 | 66/35, 17/9 | M7 | major 7th | C# |
| 67 | 1116.667 | 21/11 | ^M7 | upmajor 7th | ^C# |
| 68 | 1133.333 | 27/14 | ^^M7 | dupmajor 7th | ^^C# |
| 69 | 1150.000 | 35/18 | ^3M7, v38 |
triple-up major 7th, triple-down octave |
^3C#, v3D |
| 70 | 1166.667 | 49/25 | vv8 | dud octave | vvD |
| 71 | 1183.333 | 99/50 | v8 | down octave | vD |
| 72 | 1200.000 | 2/1 | P8 | perfect octave | D |
* based on treating 72edo as a 17-limit temperament
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| dud minor | zo | {a, b, 0, 1} | 7/6, 7/4 |
| minor | fourthward wa | {a, b}, b < -1 | 32/27, 16/9 |
| upminor | gu | {a, b, -1} | 6/5, 9/5 |
| mid | ilo | {a, b, 0, 0, 1} | 11/9, 11/6 |
| " | lu | {a, b, 0, 0, -1} | 12/11, 18/11 |
| downmajor | yo | {a, b, 1} | 5/4, 5/3 |
| major | fifthward wa | {a, b}, b > 1 | 9/8, 27/16 |
| dup major | ru | {a, b, 0, -1} | 9/7, 12/7 |
All 72-edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| zo | 6:7:9 | 0-16-42 | C vvEb G | Cvvm | C dud minor |
| gu | 10:12:15 | 0-19-42 | C ^Eb G | C^m | C upminor |
| ilo | 18:22:27 | 0-21-42 | C v3E G | C~ | C mid |
| yo | 4:5:6 | 0-23-42 | C vE G | Cv | C downmajor or C down |
| ru | 14:18:27 | 0-26-42 | C ^^E G | C^^ | C dupmajor or C dup |
For a more complete list, see Ups and Downs Notation #Chord names in other EDOs.
Notations
Sagittal
From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 72edo in the Revo flavor of Sagittal:
JI approximation
Z function
72edo is the ninth zeta integral edo, as well as being a peak and gap edo, and the maximum value of the Z function in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
15-odd-limit interval mappings
The following table shows how 15-odd-limit intervals are represented in 72edo. Prime harmonics are in bold. As 72edo is consistent in the 15-odd-limit, the results by direct approximation and patent val mapping are the same. The following table shows how 15-odd-limit intervals are represented in 72edo. Prime harmonics are in bold.
As 72edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 7/6, 12/7 | 0.204 | 1.2 |
| 11/6, 12/11 | 0.637 | 3.8 |
| 7/5, 10/7 | 0.821 | 4.9 |
| 11/7, 14/11 | 0.841 | 5.0 |
| 9/5, 10/9 | 0.930 | 5.6 |
| 5/3, 6/5 | 1.025 | 6.2 |
| 11/8, 16/11 | 1.318 | 7.9 |
| 11/10, 20/11 | 1.662 | 10.0 |
| 9/7, 14/9 | 1.751 | 10.5 |
| 3/2, 4/3 | 1.955 | 11.7 |
| 7/4, 8/7 | 2.159 | 13.0 |
| 15/13, 26/15 | 2.259 | 13.6 |
| 11/9, 18/11 | 2.592 | 15.6 |
| 15/14, 28/15 | 2.776 | 16.7 |
| 5/4, 8/5 | 2.980 | 17.9 |
| 13/9, 18/13 | 3.284 | 19.7 |
| 15/11, 22/15 | 3.617 | 21.7 |
| 9/8, 16/9 | 3.910 | 23.5 |
| 13/10, 20/13 | 4.214 | 25.3 |
| 15/8, 16/15 | 4.935 | 29.6 |
| 13/7, 14/13 | 5.035 | 30.2 |
| 13/12, 24/13 | 5.239 | 31.4 |
| 13/11, 22/13 | 5.876 | 35.3 |
| 13/8, 16/13 | 7.194 | 43.2 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 15625/15552, 531441/524288 | [⟨72 114 167]] | +0.839 | 0.594 | 3.56 |
| 2.3.5.7 | 225/224, 1029/1024, 4375/4374 | [⟨72 114 167 202]] | +0.822 | 0.515 | 3.09 |
| 2.3.5.7.11 | 225/224, 243/242, 385/384, 4000/3993 | [⟨72 114 167 202 249]] | +0.734 | 0.493 | 2.96 |
| 2.3.5.7.11.13 | 169/168, 225/224, 243/242, 325/324, 385/384 | [⟨72 114 167 202 266 249]] | +0.936 | 0.638 | 3.82 |
| 2.3.5.7.11.13.17 | 169/168, 221/220, 225/224, 243/242, 273/272, 325/324 | [⟨72 114 167 202 249 266 294]] | +0.975 | 0.599 | 3.59 |
72et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next ETs doing better in these subgroups are 99, 270, 224, 494, and 217, respectively.
Commas
Commas tempered out by 72edo include…
| Prime Limit |
Ratio[1] | Monzo | Cents | Name(s) |
|---|---|---|---|---|
| 3 | (12 digits) | [-19 12⟩ | 23.46 | Pythagorean comma |
| 5 | 15625/15552 | [-6 -5 6⟩ | 8.11 | Kleisma |
| 5 | (16 digits) | [-25 7 6⟩ | 31.57 | Ampersand |
| 5 | (18 digits) | [-13 17 -6⟩ | 15.35 | Graviton |
| 5 | (26 digits) | [1 -27 18⟩ | 0.86 | Ennealimma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Septimal kleisma, Marvel comma |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.43 | Gamelisma |
| 7 | 2401/2400 | [-5 -1 -2 4⟩ | 0.72 | Breedsma |
| 7 | 4375/4374 | [-1 -7 4 1⟩ | 0.40 | Ragisma |
| 7 | 16875/16807 | [0 3 4 -5⟩ | 6.99 | Mirkwai |
| 7 | 19683/19600 | [-4 9 -2 -2⟩ | 7.32 | Cataharry |
| 7 | (12 digits) | [-6 -8 2 5⟩ | 1.12 | Wizma |
| 7 | (12 digits) | [-4 6 -6 3⟩ | 0.33 | Landscape comma |
| 11 | 243/242 | [-1 5 0 0 -2⟩ | 7.14 | Rastma |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Keenanisma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Werckisma |
| 11 | 540/539 | [2 3 1 -2 -1⟩ | 3.21 | Swetisma |
| 11 | 1375/1372 | [-2 0 3 -3 1⟩ | 3.78 | Moctdel |
| 11 | 3025/3024 | [-4 -3 2 -1 2⟩ | 0.57 | Lehmerisma |
| 11 | 4000/3993 | [5 -1 3 0 -3⟩ | 3.03 | Wizardharry |
| 11 | 6250/6237 | [1 -4 5 -1 -1⟩ | 3.60 | Liganellus comma |
| 11 | 9801/9800 | [-3 4 -2 -2 2⟩ | 0.18 | Kalisma, Gauss' comma |
| 11 | (14 digits) | [16 -3 0 0 6⟩ | 2.04 | Nexus comma |
| 13 | 169/168 | [-3 -1 0 -1 0 2⟩ | 10.27 | Buzurgisma, Dhanvantarisma |
| 13 | 325/324 | [-2 -4 2 0 0 1⟩ | 5.34 | Marveltwin comma |
| 13 | 351/350 | [-1 3 -2 -1 0 1⟩ | 4.94 | Ratwolfsma |
| 13 | 364/363 | [2 -1 0 1 -2 1⟩ | 4.76 | Gentle comma |
| 13 | 625/624 | [-4 -1 4 0 0 -1⟩ | 2.77 | Tunbarsma |
| 13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.56 | Island comma, Parizeksma |
| 13 | 729/728 | [-3 6 0 -1 0 -1⟩ | 2.38 | Squbema |
| 13 | 1001/1000 | [-3 0 -3 1 1 1⟩ | 1.73 | Sinbadma |
| 13 | 1575/1573 | [2 2 1 -2 -1⟩ | 2.20 | Nicola |
| 13 | 1716/1715 | [2 1 -1 -3 1 1⟩ | 1.01 | Lummic comma |
| 13 | 2080/2079 | [5 -3 1 -1 -1 1⟩ | 0.83 | Ibnsinma |
| 13 | 6656/6655 | [9 0 -1 0 -3 1⟩ | 0.26012 | Jacobin comma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Rank-2 temperaments
72edo provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit.
| Periods per octave |
Generator | Names |
|---|---|---|
| 1 | 1\72 | Quincy |
| 1 | 5\72 | Marvolo |
| 1 | 7\72 | Miracle/benediction/manna |
| 1 | 11\72 | |
| 1 | 13\72 | |
| 1 | 17\72 | Neominor |
| 1 | 19\72 | Catakleismic |
| 1 | 23\72 | |
| 1 | 25\72 | Sqrtphi |
| 1 | 29\72 | |
| 1 | 31\72 | Marvo/zarvo |
| 1 | 35\72 | Cotritone |
| 2 | 1\72 | |
| 2 | 5\72 | Harry |
| 2 | 7\72 | Semimiracle |
| 2 | 11\72 | Unidec/hendec |
| 2 | 13\72 | Wizard/lizard/gizzard |
| 2 | 19\72 | Bikleismic |
| 3 | 1\72 | |
| 3 | 5\72 | Tritikleismic |
| 3 | 7\72 | |
| 3 | 11\72 | Mirkat |
| 4 | 1\72 | Quadritikleismic |
| 4 | 5\72 | |
| 4 | 7\72 | |
| 6 | 1\72 | |
| 6 | 5\72 | |
| 8 | 1\72 | Octoid |
| 8 | 2\72 | Octowerck |
| 8 | 4\72 | |
| 9 | 1\72 | |
| 9 | 3\72 | Ennealimmal/ennealimmic |
| 12 | 1\72 | Compton |
| 18 | 1\72 | Hemiennealimmal |
| 24 | 1\72 | Hours |
| 36 | 1\72 |
Scales
- smithgw72a, smithgw72b, smithgw72c, smithgw72d, smithgw72e, smithgw72f, smithgw72g, smithgw72h, smithgw72i, smithgw72j
- blackjack, miracle_8, miracle_10, miracle_12, miracle_12a, miracle_24hi, miracle_24lo
- keenanmarvel, xenakis_chrome, xenakis_diat, xenakis_schrome
- Euler(24255) genus in 72 equal
- JuneGloom
Harmonic scale
Mode 8 of the harmonic series – harmonics 8 through 16, octave repeating – is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
| Harmonics in "Mode 8": | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | ||||||||
| …as JI Ratio from 1/1: | 1/1 | 9/8 | 5/4 | 11/8 | 3/2 | 13/8 | 7/4 | 15/8 | 2/1 | ||||||||
| …in cents: | 0 | 203.9 | 386.3 | 551.3 | 702.0 | 840.5 | 968.8 | 1088.3 | 1200.0 | ||||||||
| Nearest degree of 72edo: | 0 | 12 | 23 | 33 | 42 | 50 | 58 | 65 | 72 | ||||||||
| …in cents: | 0 | 200.0 | 383.3 | 550.0 | 700.0 | 833.3 | 966.7 | 1083.3 | 1200.0 | ||||||||
| Steps as Freq. Ratio: | 9:8 | 10:9 | 11:10 | 12:11 | 13:12 | 14:13 | 15:14 | 16:15 | |||||||||
| …in cents: | 203.9 | 182.4 | 165.0 | 150.6 | 138.6 | 128.3 | 119.4 | 111.7 | |||||||||
| Nearest degree of 72edo: | 12 | 11 | 10 | 9 | 8 | 8 | 7 | 7 | |||||||||
| …in cents: | 200.0 | 183.3 | 166.7 | 150.0 | 133.3 | 133.3 | 116.7 | 116.7 |
Music
- Kotekant play by Gene Ward Smith
- Twinkle canon – 72 edo by Claudi Meneghin
- Lazy Sunday by Jake Freivald in the lazysunday scale.
- June Gloom #9 by Prent Rodgers
See also
External links
- OrthodoxWiki Article on Byzantine chant, which uses 72edo
- Ekmelic Music Society/Gesellschaft für Ekmelische Musik, a group of composers and researchers dedicated to 72edo music
- Rick Tagawa's 72edo site, including theory and composers' list
- Danny Wier, composer and musician who specializes in 72-edo