72edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
→Music: Add Tolgahan Çoğulu's ''Ne ağlarsın benim zülfü siyahım'' (2026) – transcription by Stephen Weigel
TallKite (talk | contribs)
autopatrolled, Sysops
4,207 edits
→Intervals: added interval names to the main table
 
Line 38: Line 38:
! Cents
! Cents
! Approximate ratios<ref group="note">As a 19-limit temperament, inconsistent intervals in ''italic''. For a table of intervals by prime limit, see [[Table of 72edo intervals]].</ref>
! Approximate ratios<ref group="note">As a 19-limit temperament, inconsistent intervals in ''italic''. For a table of intervals by prime limit, see [[Table of 72edo intervals]].</ref>
! [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="2" |[[Kite's ups and downs notation|Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>6</sup>A1 and d2)
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
|P1
| {{UDnote|step=0}}
| {{UDnote|step=0}}
|-
|-
Line 48: Line 50:
| 16.7
| 16.7
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
|^1
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
Line 53: Line 56:
| 33.3
| 33.3
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
|^^
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
Line 58: Line 62:
| 50.0
| 50.0
| [[33/32]], [[36/35]], [[40/39]]
| [[33/32]], [[36/35]], [[40/39]]
|^<sup>3</sup>1, v<sup>3</sup>m2
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
Line 63: Line 68:
| 66.7
| 66.7
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
|vvm2
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|-
|-
Line 68: Line 74:
| 83.3
| 83.3
| [[20/19]], [[21/20]], [[22/21]]
| [[20/19]], [[21/20]], [[22/21]]
|vm2
| {{UDnote|step=5}}
| {{UDnote|step=5}}
|-
|-
Line 73: Line 80:
| 100.0
| 100.0
| [[17/16]], [[18/17]], [[19/18]]
| [[17/16]], [[18/17]], [[19/18]]
|m2
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
Line 78: Line 86:
| 116.7
| 116.7
| [[15/14]], [[16/15]]
| [[15/14]], [[16/15]]
|^m2
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|-
|-
Line 83: Line 92:
| 133.3
| 133.3
| [[13/12]], [[14/13]], [[27/25]]
| [[13/12]], [[14/13]], [[27/25]]
|^^m2, v~2
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
Line 88: Line 98:
| 150.0
| 150.0
| [[12/11]]
| [[12/11]]
|~2
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
Line 93: Line 104:
| 166.7
| 166.7
| [[11/10]], [[21/19]]
| [[11/10]], [[21/19]]
|^~2, vvM2
| {{UDnote|step=10}}
| {{UDnote|step=10}}
|-
|-
Line 98: Line 110:
| 183.3
| 183.3
| [[10/9]]
| [[10/9]]
|vM2
| {{UDnote|step=11}}
| {{UDnote|step=11}}
|-
|-
Line 103: Line 116:
| 200.0
| 200.0
| [[9/8]], [[19/17]]
| [[9/8]], [[19/17]]
|M2
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|-
|-
Line 108: Line 122:
| 216.7
| 216.7
| [[17/15]], [[25/22]]
| [[17/15]], [[25/22]]
|^M2
| {{UDnote|step=13}}
| {{UDnote|step=13}}
|-
|-
Line 113: Line 128:
| 233.3
| 233.3
| [[8/7]]
| [[8/7]]
|^^M2
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|-
|-
Line 118: Line 134:
| 250.0
| 250.0
| [[15/13]], [[22/19]]
| [[15/13]], [[22/19]]
|^<sup>3</sup>M2, v<sup>3</sup>m3
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|-
|-
Line 123: Line 140:
| 266.7
| 266.7
| [[7/6]]
| [[7/6]]
|vvm3
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
Line 128: Line 146:
| 283.3
| 283.3
| [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
|vm3
| {{UDnote|step=17}}
| {{UDnote|step=17}}
|-
|-
Line 133: Line 152:
| 300.0
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| [[19/16]], [[25/21]], [[32/27]]
|m3
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
Line 138: Line 158:
| 316.7
| 316.7
| [[6/5]]
| [[6/5]]
|^m3
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|-
|-
Line 143: Line 164:
| 333.3
| 333.3
| [[17/14]], ''[[39/32]]'', [[40/33]]
| [[17/14]], ''[[39/32]]'', [[40/33]]
|^^m3, v~3
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|-
|-
Line 148: Line 170:
| 350.0
| 350.0
| [[11/9]], [[27/22]]
| [[11/9]], [[27/22]]
|~3
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|-
|-
Line 153: Line 176:
| 366.7
| 366.7
| [[16/13]], [[21/17]], [[26/21]]
| [[16/13]], [[21/17]], [[26/21]]
|^~3, vvM3
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|-
|-
Line 158: Line 182:
| 383.3
| 383.3
| [[5/4]]
| [[5/4]]
|vM3
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
Line 163: Line 188:
| 400.0
| 400.0
| [[24/19]]
| [[24/19]]
|M3
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
Line 168: Line 194:
| 416.7
| 416.7
| [[14/11]], [[19/15]]
| [[14/11]], [[19/15]]
|^M3
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|-
|-
Line 173: Line 200:
| 433.3
| 433.3
| [[9/7]]
| [[9/7]]
|^^M3
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|-
|-
Line 178: Line 206:
| 450.0
| 450.0
| [[13/10]], [[22/17]]
| [[13/10]], [[22/17]]
|^<sup>3</sup>M3, v<sup>3</sup>4
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|-
|-
Line 183: Line 212:
| 466.7
| 466.7
| [[17/13]], [[21/16]]
| [[17/13]], [[21/16]]
|vv4
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|-
|-
Line 188: Line 218:
| 483.3
| 483.3
| [[33/25]]
| [[33/25]]
|v4
| {{UDnote|step=29}}
| {{UDnote|step=29}}
|-
|-
Line 193: Line 224:
| 500.0
| 500.0
| [[4/3]]
| [[4/3]]
|P4
| {{UDnote|step=30}}
| {{UDnote|step=30}}
|-
|-
Line 198: Line 230:
| 516.7
| 516.7
| [[27/20]]
| [[27/20]]
|^4
| {{UDnote|step=31}}
| {{UDnote|step=31}}
|-
|-
Line 203: Line 236:
| 533.3
| 533.3
| [[15/11]], [[19/14]], ''[[26/19]]''
| [[15/11]], [[19/14]], ''[[26/19]]''
|^^4, v~4
| {{UDnote|step=32}}
| {{UDnote|step=32}}
|-
|-
Line 208: Line 242:
| 550.0
| 550.0
| [[11/8]]
| [[11/8]]
|~4
| {{UDnote|step=33}}
| {{UDnote|step=33}}
|-
|-
Line 213: Line 248:
| 566.7
| 566.7
| [[18/13]], [[25/18]]
| [[18/13]], [[25/18]]
|^~4, vvA4
| {{UDnote|step=34}}
| {{UDnote|step=34}}
|-
|-
Line 218: Line 254:
| 583.3
| 583.3
| [[7/5]]
| [[7/5]]
|vA4, vd5
| {{UDnote|step=35}}
| {{UDnote|step=35}}
|-
|-
Line 223: Line 260:
| 600.0
| 600.0
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
|A4, d5
| {{UDnote|step=36}}
| {{UDnote|step=36}}
|-
|-
Line 228: Line 266:
| 616.7
| 616.7
| [[10/7]]
| [[10/7]]
|^A4, ^d5
| {{UDnote|step=37}}
| {{UDnote|step=37}}
|-
|-
Line 233: Line 272:
| 633.3
| 633.3
| [[13/9]], [[36/25]]
| [[13/9]], [[36/25]]
|v~5, ^^d5
| {{UDnote|step=38}}
| {{UDnote|step=38}}
|-
|-
Line 238: Line 278:
| 650.0
| 650.0
| [[16/11]]
| [[16/11]]
|~5
| {{UDnote|step=39}}
| {{UDnote|step=39}}
|-
|-
Line 243: Line 284:
| 666.7
| 666.7
| ''[[19/13]]'', [[22/15]], [[28/19]]
| ''[[19/13]]'', [[22/15]], [[28/19]]
|vv5, ^~5
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|-
|-
Line 248: Line 290:
| 683.3
| 683.3
| [[40/27]]
| [[40/27]]
|v5
| {{UDnote|step=41}}
| {{UDnote|step=41}}
|-
|-
Line 253: Line 296:
| 700.0
| 700.0
| [[3/2]]
| [[3/2]]
|P5
| {{UDnote|step=42}}
| {{UDnote|step=42}}
|-
|-
Line 258: Line 302:
| 716.7
| 716.7
| [[50/33]]
| [[50/33]]
|^5
| {{UDnote|step=43}}
| {{UDnote|step=43}}
|-
|-
Line 263: Line 308:
| 733.3
| 733.3
| [[26/17]], [[32/21]]
| [[26/17]], [[32/21]]
|^^5
| {{UDnote|step=44}}
| {{UDnote|step=44}}
|-
|-
Line 268: Line 314:
| 750.0
| 750.0
| [[17/11]], [[20/13]]
| [[17/11]], [[20/13]]
|^<sup>3</sup>5, v<sup>3</sup>m6
| {{UDnote|step=45}}
| {{UDnote|step=45}}
|-
|-
Line 273: Line 320:
| 766.7
| 766.7
| [[14/9]]
| [[14/9]]
|vvm6
| {{UDnote|step=46}}
| {{UDnote|step=46}}
|-
|-
Line 278: Line 326:
| 783.3
| 783.3
| [[11/7]], [[30/19]]
| [[11/7]], [[30/19]]
|vm6
| {{UDnote|step=47}}
| {{UDnote|step=47}}
|-
|-
Line 283: Line 332:
| 800.0
| 800.0
| [[19/12]]
| [[19/12]]
|m6
| {{UDnote|step=48}}
| {{UDnote|step=48}}
|-
|-
Line 288: Line 338:
| 816.7
| 816.7
| [[8/5]]
| [[8/5]]
|^m6
| {{UDnote|step=49}}
| {{UDnote|step=49}}
|-
|-
Line 293: Line 344:
| 833.3
| 833.3
| [[13/8]], [[21/13]], [[34/21]]
| [[13/8]], [[21/13]], [[34/21]]
|^^m6, v~6
| {{UDnote|step=50}}
| {{UDnote|step=50}}
|-
|-
Line 298: Line 350:
| 850.0
| 850.0
| [[18/11]], [[44/27]]
| [[18/11]], [[44/27]]
|~6
| {{UDnote|step=51}}
| {{UDnote|step=51}}
|-
|-
Line 303: Line 356:
| 866.7
| 866.7
| [[28/17]], [[33/20]], ''[[64/39]]''
| [[28/17]], [[33/20]], ''[[64/39]]''
|^~6, vvM6
| {{UDnote|step=52}}
| {{UDnote|step=52}}
|-
|-
Line 308: Line 362:
| 883.3
| 883.3
| [[5/3]]
| [[5/3]]
|vM6
| {{UDnote|step=53}}
| {{UDnote|step=53}}
|-
|-
Line 313: Line 368:
| 900.0
| 900.0
| [[27/16]], [[32/19]], [[42/25]]
| [[27/16]], [[32/19]], [[42/25]]
|M6
| {{UDnote|step=54}}
| {{UDnote|step=54}}
|-
|-
Line 318: Line 374:
| 916.7
| 916.7
| [[17/10]], [[22/13]]
| [[17/10]], [[22/13]]
|^M6
| {{UDnote|step=55}}
| {{UDnote|step=55}}
|-
|-
Line 323: Line 380:
| 933.3
| 933.3
| [[12/7]]
| [[12/7]]
|^^M6
| {{UDnote|step=56}}
| {{UDnote|step=56}}
|-
|-
Line 328: Line 386:
| 950.0
| 950.0
| [[19/11]], [[26/15]]
| [[19/11]], [[26/15]]
|^<sup>3</sup>M6, v<sup>3</sup>m7
| {{UDnote|step=57}}
| {{UDnote|step=57}}
|-
|-
Line 333: Line 392:
| 966.7
| 966.7
| [[7/4]]
| [[7/4]]
|vvm7
| {{UDnote|step=58}}
| {{UDnote|step=58}}
|-
|-
Line 338: Line 398:
| 983.3
| 983.3
| [[30/17]], [[44/25]]
| [[30/17]], [[44/25]]
|vm7
| {{UDnote|step=59}}
| {{UDnote|step=59}}
|-
|-
Line 343: Line 404:
| 1000.0
| 1000.0
| [[16/9]], [[34/19]]
| [[16/9]], [[34/19]]
|m7
| {{UDnote|step=60}}
| {{UDnote|step=60}}
|-
|-
Line 348: Line 410:
| 1016.7
| 1016.7
| [[9/5]]
| [[9/5]]
|^m7
| {{UDnote|step=61}}
| {{UDnote|step=61}}
|-
|-
Line 353: Line 416:
| 1033.3
| 1033.3
| [[20/11]], [[38/21]]
| [[20/11]], [[38/21]]
|^^m7, v~7
| {{UDnote|step=62}}
| {{UDnote|step=62}}
|-
|-
Line 358: Line 422:
| 1050.0
| 1050.0
| [[11/6]]
| [[11/6]]
|~7
| {{UDnote|step=63}}
| {{UDnote|step=63}}
|-
|-
Line 363: Line 428:
| 1066.7
| 1066.7
| [[13/7]], [[24/13]], [[50/27]]
| [[13/7]], [[24/13]], [[50/27]]
|^~7, vvM7
| {{UDnote|step=64}}
| {{UDnote|step=64}}
|-
|-
Line 368: Line 434:
| 1083.3
| 1083.3
| [[15/8]], [[28/15]]
| [[15/8]], [[28/15]]
|vM7
| {{UDnote|step=65}}
| {{UDnote|step=65}}
|-
|-
Line 373: Line 440:
| 1100.0
| 1100.0
| [[17/9]], [[32/17]], [[36/19]]
| [[17/9]], [[32/17]], [[36/19]]
|M7
| {{UDnote|step=66}}
| {{UDnote|step=66}}
|-
|-
Line 378: Line 446:
| 1116.7
| 1116.7
| [[19/10]], [[21/11]], [[40/21]]
| [[19/10]], [[21/11]], [[40/21]]
|^M7
| {{UDnote|step=67}}
| {{UDnote|step=67}}
|-
|-
Line 383: Line 452:
| 1133.3
| 1133.3
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
|^^M7
| {{UDnote|step=68}}
| {{UDnote|step=68}}
|-
|-
Line 388: Line 458:
| 1150.0
| 1150.0
| [[35/18]], [[39/20]], [[64/33]]
| [[35/18]], [[39/20]], [[64/33]]
|^<sup>3</sup>M7, v<sup>3</sup>8
| {{UDnote|step=69}}
| {{UDnote|step=69}}
|-
|-
Line 393: Line 464:
| 1166.7
| 1166.7
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
|vv8
| {{UDnote|step=70}}
| {{UDnote|step=70}}
|-
|-
Line 398: Line 470:
| 1183.3
| 1183.3
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
|v8
| {{UDnote|step=71}}
| {{UDnote|step=71}}
|-
|-
Line 403: Line 476:
| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
|P8
| {{UDnote|step=72}}
| {{UDnote|step=72}}
|}
|}

Latest revision as of 03:11, 11 August 2026

← 71edo 72edo 73edo →
Prime factorization 23 × 32
Step size 16.6667 ¢ 
Fifth 42\72 (700 ¢) (→ 7\12)
Semitones (A1:m2) 6:6 (100 ¢ : 100 ¢)
Consistency limit 17
Distinct consistency limit 11
English Wikipedia has an article on:

72 equal divisions of the octave (abbreviated 72edo or 72ed2), also called 72-tone equal temperament (72tet) or 72 equal temperament (72et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 72 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of 21/72, or the 72nd root of 2.

Each step of 72edo is called a morion (plural moria). This produces a twelfth-tone tuning, with the whole tone measuring 200 ¢, 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 Ivan Wyschnegradsky, Julián Carrillo (who is better associated with 96edo), Georg Friedrich Haas, Ezra Sims, Rick Tagawa, James Tenney, and the jazz musician Joe Maneri.

Theory

72edo approximates 11-limit just intonation exceptionally well. It is the second edo (after 58) to be consistent in the 17-odd-limit, and the second edo (also after 58) to be distinctly consistent in the 11-odd-limit, but it is the first edo to be consistent to distance 2 in the 11-odd-limit, meaning every interval in the 11-odd-limit is approximated with less than 25% relative error (about 4 cents). It also has pretty good accuracy for the 19-limit, being almost consistent to the entire 21-odd-limit with the only inconsistency occurring at 19/13 and its octave complement. It is the ninth zeta integral edo.

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.

The octave reduced 13th harmonic is mapped on 50\72, an interval inherited from 36edo (25\36) that is a very close approximation to acoustic phi, and the 17th and 19th harmonics come from 12edo.

72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo circulating temperament.

Prime harmonics

Approximation of prime harmonics in 72edo
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)
Approximation of prime harmonics in 72edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -1.34 +4.27 +5.15 +1.16 -6.84 +7.50 -0.22 +4.03 +3.64 +5.54 +2.13
Relative (%) -8.1 +25.6 +30.9 +7.0 -41.0 +45.0 -1.3 +24.2 +21.8 +33.3 +12.8
Steps
(reduced) 		375
(15) 		386
(26) 		391
(31) 		400
(40) 		412
(52) 		424
(64) 		427
(67) 		437
(5) 		443
(11) 		446
(14) 		454
(22)

As a tuning of other temperaments

72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits 25/24 into two equal 49/48~50/49's, and splits 28/27 into two equal 55/54~56/55's (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for 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.

Subsets and supersets

Since 72 factors into primes as 23 × 32, 72edo has subset edos 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36. 144edo, which doubles it, provides a possible correction to its approximate harmonic 13, though unlike 72 it is not consistent to the 13-odd-limit.

Intervals

# Cents Approximate ratios[note 1] Ups and downs notation

(EUs: v6A1 and d2)

0 0.0 1/1 P1 D
1 16.7 81/80, 91/90, 99/98, 100/99, 105/104 ^1 ^D, ^E♭♭
2 33.3 45/44, 49/48, 50/49, 55/54, 64/63 ^^ ^^D, ^^E♭♭
3 50.0 33/32, 36/35, 40/39 ^31, v3m2 ^3D, v3E♭
4 66.7 25/24, 26/25, 27/26, 28/27 vvm2 vvD♯, vvE♭
5 83.3 20/19, 21/20, 22/21 vm2 vD♯, vE♭
6 100.0 17/16, 18/17, 19/18 m2 D♯, E♭
7 116.7 15/14, 16/15 ^m2 ^D♯, ^E♭
8 133.3 13/12, 14/13, 27/25 ^^m2, v~2 ^^D♯, ^^E♭
9 150.0 12/11 ~2 ^3D♯, v3E
10 166.7 11/10, 21/19 ^~2, vvM2 vvD𝄪, vvE
11 183.3 10/9 vM2 vD𝄪, vE
12 200.0 9/8, 19/17 M2 E
13 216.7 17/15, 25/22 ^M2 ^E, ^F♭
14 233.3 8/7 ^^M2 ^^E, ^^F♭
15 250.0 15/13, 22/19 ^3M2, v3m3 ^3E, v3F
16 266.7 7/6 vvm3 vvE♯, vvF
17 283.3 13/11, 20/17 vm3 vE♯, vF
18 300.0 19/16, 25/21, 32/27 m3 F
19 316.7 6/5 ^m3 ^F, ^G♭♭
20 333.3 17/14, 39/32, 40/33 ^^m3, v~3 ^^F, ^^G♭♭
21 350.0 11/9, 27/22 ~3 ^3F, v3G♭
22 366.7 16/13, 21/17, 26/21 ^~3, vvM3 vvF♯, vvG♭
23 383.3 5/4 vM3 vF♯, vG♭
24 400.0 24/19 M3 F♯, G♭
25 416.7 14/11, 19/15 ^M3 ^F♯, ^G♭
26 433.3 9/7 ^^M3 ^^F♯, ^^G♭
27 450.0 13/10, 22/17 ^3M3, v34 ^3F♯, v3G
28 466.7 17/13, 21/16 vv4 vvF𝄪, vvG
29 483.3 33/25 v4 vF𝄪, vG
30 500.0 4/3 P4 G
31 516.7 27/20 ^4 ^G, ^A♭♭
32 533.3 15/11, 19/14, 26/19 ^^4, v~4 ^^G, ^^A♭♭
33 550.0 11/8 ~4 ^3G, v3A♭
34 566.7 18/13, 25/18 ^~4, vvA4 vvG♯, vvA♭
35 583.3 7/5 vA4, vd5 vG♯, vA♭
36 600.0 17/12, 24/17 A4, d5 G♯, A♭
37 616.7 10/7 ^A4, ^d5 ^G♯, ^A♭
38 633.3 13/9, 36/25 v~5, ^^d5 ^^G♯, ^^A♭
39 650.0 16/11 ~5 ^3G♯, v3A
40 666.7 19/13, 22/15, 28/19 vv5, ^~5 vvG𝄪, vvA
41 683.3 40/27 v5 vG𝄪, vA
42 700.0 3/2 P5 A
43 716.7 50/33 ^5 ^A, ^B♭♭
44 733.3 26/17, 32/21 ^^5 ^^A, ^^B♭♭
45 750.0 17/11, 20/13 ^35, v3m6 ^3A, v3B♭
46 766.7 14/9 vvm6 vvA♯, vvB♭
47 783.3 11/7, 30/19 vm6 vA♯, vB♭
48 800.0 19/12 m6 A♯, B♭
49 816.7 8/5 ^m6 ^A♯, ^B♭
50 833.3 13/8, 21/13, 34/21 ^^m6, v~6 ^^A♯, ^^B♭
51 850.0 18/11, 44/27 ~6 ^3A♯, v3B
52 866.7 28/17, 33/20, 64/39 ^~6, vvM6 vvA𝄪, vvB
53 883.3 5/3 vM6 vA𝄪, vB
54 900.0 27/16, 32/19, 42/25 M6 B
55 916.7 17/10, 22/13 ^M6 ^B, ^C♭
56 933.3 12/7 ^^M6 ^^B, ^^C♭
57 950.0 19/11, 26/15 ^3M6, v3m7 ^3B, v3C
58 966.7 7/4 vvm7 vvB♯, vvC
59 983.3 30/17, 44/25 vm7 vB♯, vC
60 1000.0 16/9, 34/19 m7 C
61 1016.7 9/5 ^m7 ^C, ^D♭♭
62 1033.3 20/11, 38/21 ^^m7, v~7 ^^C, ^^D♭♭
63 1050.0 11/6 ~7 ^3C, v3D♭
64 1066.7 13/7, 24/13, 50/27 ^~7, vvM7 vvC♯, vvD♭
65 1083.3 15/8, 28/15 vM7 vC♯, vD♭
66 1100.0 17/9, 32/17, 36/19 M7 C♯, D♭
67 1116.7 19/10, 21/11, 40/21 ^M7 ^C♯, ^D♭
68 1133.3 25/13, 27/14, 48/25, 52/27 ^^M7 ^^C♯, ^^D♭
69 1150.0 35/18, 39/20, 64/33 ^3M7, v38 ^3C♯, v3D
70 1166.7 49/25, 55/28, 63/32, 88/45, 96/49 vv8 vvC𝄪, vvD
71 1183.3 99/50, 160/81, 180/91, 196/99, 208/105 v8 vC𝄪, vD
72 1200.0 2/1 P8 D
  1. ↑ As a 19-limit temperament, inconsistent intervals in italic. For a table of intervals by prime limit, see Table of 72edo intervals.

Proposed interval names and solfèges

Table of proposed interval names and solfèges
# Cents Ups and downs notation SKULO interval names and notation (K, S, U)
0 0.0 P1 perfect unison D P1 perfect unison D D
1 16.7 ^1 up unison ^D K1, L1 comma-wide unison, large unison KD, LD KD
2 33.3 ^^ dup unison ^^D S1, O1 super unison, on unison SD, OD SD
3 50.0 ^31, v3m2 trup unison, trudminor 2nd ^3D, v3Eb U1, H1, hm2 uber unison, hyper unison, hypominor 2nd UD, HD, uEb UD, uEb
4 66.7 vvm2 dudminor 2nd vvEb kkA1, sm2 classic aug unison, subminor 2nd kkD#, sEb sD#, (kkD#), sEb
5 83.3 vm2 downminor 2nd vEb kA1, lm2 comma-narrow aug unison, little minor 2nd kD#, lEb kD#, kEb
6 100.0 m2 minor 2nd Eb m2 minor 2nd Eb Eb
7 116.7 ^m2 upminor 2nd ^Eb Km2 classic minor 2nd KEb KEb
8 133.3 ^^m2, v~2 dupminor 2nd, downmid 2nd ^^Eb Om2 on minor 2nd OEb SEb
9 150.0 ~2 mid 2nd v3E N2 neutral 2nd UEb/uE UEb/uE
10 166.7 ^~2, vvM2 upmid 2nd, dudmajor 2nd vvE oM2 off major 2nd oE sE
11 183.3 vM2 downmajor 2nd vE kM2 classic/comma-narrow major 2nd kE kE
12 200.0 M2 major 2nd E M2 major 2nd E E
13 216.7 ^M2 upmajor 2nd ^E LM2 large major 2nd LE KE
14 233.3 ^^M2 dupmajor 2nd ^^E SM2 supermajor 2nd SE SE
15 250.0 ^3M2,
v3m3 		trupmajor 2nd,
trudminor 3rd 		^3E,
v3F 		HM2, hm3 hypermajor 2nd, hypominor 3rd HE, hF UE, uF
16 266.7 vvm3 dudminor 3rd vvF sm3 subminor 3rd sF sF
17 283.3 vm3 downminor 3rd vF lm3 little minor 3rd lF kF
18 300.0 m3 minor 3rd F m3 minor 3rd F F
19 316.7 ^m3 upminor 3rd ^F Km3 classic minor 3rd KF KF
20 333.3 ^^m3, v~3 dupminor 3rd, downmid 3rd ^^F Om3 on minor third OF SF
21 350.0 ~3 mid 3rd ^3F N3 neutral 3rd UF/uF# UF/uF#
22 366.7 ^~3, vvM3 upmid 3rd, dudmajor 3rd vvF# oM3 off major 3rd oF# sF#
23 383.3 vM3 downmajor 3rd vF# kM3 classic major 3rd kF# kF#
24 400.0 M3 major 3rd F# M3 major 3rd F# F#
25 416.7 ^M3 upmajor 3rd ^F# LM3 large major 3rd LF# KF#
26 433.3 ^^M3 dupmajor 3rd ^^F# SM3 supermajor 3rd SF# SF#
27 450.0 ^3M3, v34 trupmajor 3rd, trud 4th ^3F#, v3G HM3, h4 hypermajor 3rd, hypo 4th HF#, hG UF#, uG
28 466.7 vv4 dud 4th vvG s4 sub 4th sG sG
29 483.3 v4 down 4th vG l4 little 4th lG kG
30 500.0 P4 perfect 4th G P4 perfect 4th G G
31 516.7 ^4 up 4th ^G K4 comma-wide 4th KG KG
32 533.3 ^^4, v~4 dup 4th, downmid 4th ^^G O4 on 4th OG SG
33 550.0 ~4 mid 4th ^3G U4/N4 uber 4th / neutral 4th UG UG
34 566.7 ^~4, vvA4 upmid 4th, dudaug 4th vvG# kkA4, sd5 classic aug 4th, sub dim 5th kkG#, sAb SG#, (kkG#), sAb
35 583.3 vA4, vd5 downaug 4th,
downdim 5th 		vG#, vAb kA4, ld5 comma-narrow aug 4th, little dim 5th kG#, lAb kG#, kAb
36 600.0 A4, d5 aug 4th, dim 5th G#, Ab A4, d5 aug 4th, dim 5th G#, Ab G#, Ab
37 616.7 ^A4, ^d5 upaug 4th, updim 5th ^G#, ^Ab LA4, Kd5 large aug 4th, comma-wide dim 5th LG#, KAb KG#, KAb
38 633.3 v~5, ^^d5 downmid 5th,
dupdim 5th 		^^Ab SA4, KKd5 super aug 4th, classic dim 5th SG#, KKAb SG#, SAb, (KKAb)
39 650.0 ~5 mid 5th v3A u5/N5 unter 5th / neutral 5th uA uA
40 666.7 vv5, ^~5 dud 5th, upmid 5th vvA o5 off 5th oA sA
41 683.3 v5 down 5th vA k5 comma-narrow 5th kA kA
42 700.0 P5 perfect 5th A P5 perfect 5th A A
43 716.7 ^5 up 5th ^A L5 large fifth LA KA
44 733.3 ^^5 dup 5th ^^A S5 super fifth SA SA
45 750.0 ^35, v3m6 trup 5th, trudminor 6th ^3A, v3Bb H5, hm6 hyper fifth, hypominor 6th HA, hBb UA, uBb
46 766.7 vvm6 dudminor 6th vvBb sm6 superminor 6th sBb sBb
47 783.3 vm6 downminor 6th vBb lm6 little minor 6th lBb kBb
48 800.0 m6 minor 6th Bb m6 minor 6th Bb Bb
49 816.7 ^m6 upminor 6th ^Bb Km6 classic minor 6th kBb kBb
50 833.3 ^^m6, v~6 dupminor 6th, downmid 6th ^^Bb Om6 on minor 6th oBb sBb
51 850.0 ~6 mid 6th v3B N6 neutral 6th UBb, uB UBb, uB
52 866.7 ^~6, vvM6 upmid 6th, dudmajor 6th vvB oM6 off major 6th oB sB
53 883.3 vM6 downmajor 6th vB kM6 classic major 6th kB kB
54 900.0 M6 major 6th B M6 major 6th B B
55 916.7 ^M6 upmajor 6th ^B LM6 large major 6th LB KB
56 933.3 ^^M6 dupmajor 6th ^^B SM6 supermajor 6th SB SB
57 950.0 ^3M6,
v3m7 		trupmajor 6th,
trudminor 7th 		^3B,
v3C 		HM6, hm7 hypermajor 6th, hypominor 7th HB, hC UB, uC
58 966.7 vvm7 dudminor 7th vvC sm7 subminor 7th sC sC
59 983.3 vm7 downminor 7th vC lm7 little minor 7th lC kC
60 1000.0 m7 minor 7th C m7 minor 7th C C
61 1016.7 ^m7 upminor 7th ^C Km7 classic/comma-wide minor 7th KC KC
62 1033.3 ^^m7, v~7 dupminor 7th, downmid 7th ^^C Om7 on minor 7th OC SC
63 1050.0 ~7 mid 7th ^3C N7, hd8 neutral 7th, hypo dim 8ve UC/uC#, hDb UC/uC#, uDb
64 1066.7 ^~7, vvM7 upmid 7th, dudmajor 7th vvC# oM7, sd8 off major 7th, sub dim 8ve oC#, sDb sC#, sDb
65 1083.3 vM7 downmajor 7th vC# kM7, ld8 classic major 7th, little dim 8ve kC#, lDb kC#, kDb
66 1100.0 M7 major 7th C# M7, d8 major 7th, dim 8ve C#, Db C#, Db
67 1116.7 ^M7 upmajor 7th ^C# LM7, Kd8 large major 7th, comma-wide dim 8ve LC#, KDb KC#, KDb
68 1133.3 ^^M7 dupmajor 7th ^^C# SM7, KKd8 supermajor 7th, classic dim 8ve SC#, KKDb SC#, SDb, (KKDb)
69 1150.0 ^3M7, v38 trupmajor 7th, trud octave ^3C#, v3D HM7, u8, h8 hypermajor 7th, unter 8ve, hypo 8ve HC#, uD, hD UC#, uDb, uD
70 1166.7 vv8 dud octave vvD s8, o8 sub 8ve, off 8ve sD, oD sD
71 1183.3 v8 down octave vD k8, l8 comma-narrow 8ve, little 8ve kD, lD kD
72 1200.0 P8 perfect octave D P8 perfect octave D D

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality Color Monzo format Examples
dudminor 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
dupminor,
downmid 		luyo (a b 1 0 -1) 15/11
tho (a b 0 0 0 1) 13/8, 13/9
mid ilo (a b 0 0 1) 11/9, 11/6
lu (a b 0 0 -1) 12/11, 18/11
upmid,
dudmajor 		logu (a b -1 0 1) 11/10
thu (a b 0 0 0 -1) 16/13, 18/13
downmajor yo (a b 1) 5/4, 5/3
major fifthward wa (a b), b > 1 9/8, 27/16
dupmajor ru (a b 0 -1) 9/7, 12/7
trupmajor,
trudminor 		thogu (a b -1 0 0 1) 13/10
thuyo (a b 1 0 0 -1) 15/13

All 72edo 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 dudminor
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.

Relationship between primes and rings

In 72tet, there are 6 rings. 12edo is the plain ring; thus every 6 degrees is the 3-limit.

Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:

Thus the product of a ratio's monzo with ⟨0 0 -1 -2 3 2 0 0], modulo 6, specifies which ring the ratio lies on.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

72edo can also be notated with Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Semitones 0 1⁄6 1⁄3 1⁄2 2⁄3 5⁄6 1 1 1⁄6 1 1⁄3 1 1⁄2 1 2⁄3 1 5⁄6 2 2 1⁄6
Sharp symbol   
 
  
   
    
   
  
   
    
     
    
   
  
   
Flat symbol
 
  
   
    
   
  
   
    
     
    
   
  
   

Half-sharps and half-flats can be used to avoid triple arrows:

Semitones 0 1⁄6 1⁄3 1⁄2 2⁄3 5⁄6 1 1 1⁄6 1 1⁄3 1 1⁄2 1 2⁄3 1 5⁄6 2 2 1⁄6
Sharp symbol   
 
   
  
   
   
  
   
   
  
   
   
  
   
Flat symbol
 
   
  
   
   
  
   
   
  
   
   
  
   

Sagittal notation

This notation uses the same sagittal sequence as edos 65- and 79edo, and is a superset of the notations for edos 36, 24, 18, 12, 8, and 6.

Evo flavor

72-EDO_Evo_Sagittal.svg

Evo-SZ flavor

72-EDO_Evo-SZ_Sagittal.svg

Revo flavor

72-EDO_Revo_Sagittal.svg

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 72edo in the Revo flavor of Sagittal:

Ivan Wyschnegradsky's notation

Semitones 0 1⁄6 1⁄3 1⁄2 2⁄3 5⁄6 1 1 1⁄6 1 1⁄3 1 1⁄2 1 2⁄3 1 5⁄6 2 2 1⁄6
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Approximation to JI

alt : Your browser has no SVG support.
Selected intervals approximated in 72edo

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 mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 72edo
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

Zeta properties

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.

plot72.png

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 [⟨72 114 167 202 249 266]] +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
2.3.5.7.11.13.17.19 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272 [⟨72 114 167 202 249 266 294 306]] +0.780 0.762 4.57
  • 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are 99, 270, 224, 494, and 217, respectively.

Commas

Commas tempered out by 72edo include…

Prime
limit 		Ratio[note 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 comma
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 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 comma
7 19683/19600 [-4 9 -2 -2⟩ 7.32 Cataharry comma
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 comma
11 3025/3024 [-4 -3 2 -1 2⟩ 0.57 Lehmerisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Wizardharry comma
11 6250/6237 [1 -4 5 -1 -1⟩ 3.60 Liganellus comma
11 9801/9800 [-3 4 -2 -2 2⟩ 0.18 Kalisma
11 (14 digits) [16 -3 0 0 6⟩ 2.04 Nexus comma
13 169/168 [-3 -1 0 -1 0 2⟩ 10.27 Buzurgisma
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 Minor minthma
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
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
  1. ↑ 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.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 1\72 16.7 105/104 Quincy
1 5\72 83.3 21/20 Marvolo
1 7\72 116.7 15/14 Miracle / benediction / manna
1 17\72 283.3 13/11 Neominor
1 19\72 316.7 6/5 Catakleismic
1 25\72 416.7 14/11 Sqrtphi
1 29\72 483.3 45/34 Hemiseven
1 31\72 516.7 27/20 Gravity / marvo / zarvo
1 35\72 583.3 7/5 Cotritone
2 5\72 83.3 21/20 Harry
2 7\72 116.7 15/14 Semimiracle
2 11\72 183.3 10/9 Unidec / hendec
2 21\72
(19\72) 		316.7
(283.3) 		6/5
(13/11) 		Bikleismic
2 23\72
(13\72) 		383.3
(216.7) 		5/4
(17/15) 		Wizard / lizard / gizzard
3 11\72 183.3 10/9 Mirkat
3 19\72
(5\72) 		316.7
(83.3) 		6/5
(21/20) 		Tritikleismic
4 19\72
(1\72) 		316.7
(16.7) 		6/5
(105/104) 		Quadritikleismic
8 34\72
(2\72) 		566.7
(33.3) 		168/121
(55/54) 		Octowerck / octowerckis
8 35\72
(1\72) 		583.3
(16.7) 		7/5
(100/99) 		Octoid / octopus
9 19\72
(3\72) 		316.7
(50.0) 		6/5
(36/35) 		Ennealimmal / ennealimnic / ennealiminal
9 23\72
(1\72) 		383.3
(16.7) 		5/4
(105/104) 		Enneaportent
12 23\72
(1\72) 		383.3
(16.7) 		5/4
(100/99) 		Compton / comptone
18 19\72
(1\72) 		316.7
(16.7) 		6/5
(105/104) 		Hemiennealimmal
24 23\72
(1\72) 		383.3
(16.7) 		5/4
(105/104) 		Hours
36 23\72
(1\72) 		383.3
(16.7) 		5/4
(81/80) 		Gamelstearn

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly stretching the octave, using tunings such as 114edt, 380zpi or 186ed6. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 380zpi and 186ed6 are milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

Scales

Miracle-tempered scales
Maeve Gutierrez's scales
Budjarn Lambeth's scales
Gene Ward Smith's scales
Iannis Xenakis' scales
Others

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

Instruments

If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music).

One can also use a skip fretting system:

Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:

Music

Tolgahan Çoğulu
Bryan Deister
Ambient Esoterica
Jake Freivald
English Wikipedia has an article on:
Georg Friedrich Haas
Budjarn Lambeth
Claudi Meneghin
Mintsoda_15
Prent Rodgers
Gene Ward Smith
Ivan Wyschnegradsky
James Tenney
Xeno Ov Eleas

External links

Retrieved from "https://en.xen.wiki/index.php?title=72edo&oldid=235570"