72edo

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← 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
Special properties
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 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 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, is consistent in the 17-odd-limit, is the first non-trivial edo to be consistent in the 12- and 13-odd-prime-sum-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 the only regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits 25/24 into two equal 49/48~50/49s, splits 28/27 into two equal 55/54~56/55s, and tunes the octave just. 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.

The 13th harmonic (octave reduced) is so closely mapped on acoustic phi that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.

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)

Subsets and supersets

Since 72 factors into 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.

Intervals

Degrees Cents Approximate Ratios[note 1] Ups and Downs Notation SKULO interval names and notation (K, S, U)
0 0.000 1/1 P1 perfect unison D P1 perfect unison D D
1 16.667 81/80 ^1 up unison ^D K1, L1 comma-wide unison, large unison KD, LD KD
2 33.333 45/44, 64/63 ^^ dup unison ^^D S1, O1 super unison, on unison SD, OD SD
3 50.000 33/32 ^31, v3m2 trup unison, trudminor 2nd ^3D, v3Eb U1, H1, hm2 uber unison, hyper unison, hypominor 2nd UD, HD, uEb UD, uEb
4 66.667 25/24 vvm2 dudminor 2nd vvEb kkA1, sm2 classic aug unison, subminor 2nd kkD#, sEb sD#, (kkD#), sEb
5 83.333 21/20 vm2 downminor 2nd vEb kA1, lm2 comma-narrow aug unison, little minor 2nd kD#, lEb kD#, kEb
6 100.000 35/33, 17/16, 18/17 m2 minor 2nd Eb m2 minor 2nd Eb Eb
7 116.667 15/14, 16/15 ^m2 upminor 2nd ^Eb Km2 classic minor 2nd KEb KEb
8 133.333 27/25, 13/12, 14/13 ^^m2, v~2 dupminor 2nd, downmid 2nd ^^Eb Om2 on minor 2nd OEb SEb
9 150.000 12/11 ~2 mid 2nd v3E N2 neutral 2nd UEb/uE UEb/uE
10 166.667 11/10 ^~2, vvM2 upmid 2nd, dudmajor 2nd vvE oM2 off major 2nd oE sE
11 183.333 10/9 vM2 downmajor 2nd vE kM2 classic/comma-narrow major 2nd kE kE
12 200.000 9/8 M2 major 2nd E M2 major 2nd E E
13 216.667 25/22, 17/15 ^M2 upmajor 2nd ^E LM2 large major 2nd LE KE
14 233.333 8/7 ^^M2 dupmajor 2nd ^^E SM2 supermajor 2nd SE SE
15 250.000 81/70, 15/13 ^3M2,
v3m3
trupmajor 2nd,
trudminor 3rd
^3E,
v3F
HM2, hm3 hypermajor 2nd, hypominor 3rd HE, hF UE, uF
16 266.667 7/6 vvm3 dudminor 3rd vvF sm3 subminor 3rd sF sF
17 283.333 33/28, 13/11, 20/17 vm3 downminor 3rd vF lm3 little minor 3rd lF kF
18 300.000 25/21 m3 minor 3rd F m3 minor 3rd F F
19 316.667 6/5 ^m3 upminor 3rd ^F Km3 classic minor 3rd KF KF
20 333.333 40/33, 17/14 ^^m3, v~3 dupminor 3rd, downmid 3rd ^^F Om3 on minor third OF SF
21 350.000 11/9 ~3 mid 3rd ^3F N3 neutral 3rd UF/uF# UF/uF#
22 366.667 99/80, 16/13, 21/17 ^~3, vvM3 upmid 3rd, dudmajor 3rd vvF# oM3 off major 3rd oF# sF#
23 383.333 5/4 vM3 downmajor 3rd vF# kM3 classic major 3rd kF# kF#
24 400.000 44/35 M3 major 3rd F# M3 major 3rd F# F#
25 416.667 14/11 ^M3 upmajor 3rd ^F# LM3 large major 3rd LF# KF#
26 433.333 9/7 ^^M3 dupmajor 3rd ^^F# SM3 supermajor 3rd SF# SF#
27 450.000 35/27, 13/10 ^3M3, v34 trupmajor 3rd, trud 4th ^3F#, v3G HM3, h4 hypermajor 3rd, hypo 4th HF#, hG UF#, uG
28 466.667 21/16, 17/13 vv4 dud 4th vvG s4 sub 4th sG sG
29 483.333 33/25 v4 down 4th vG l4 little 4th lG kG
30 500.000 4/3 P4 perfect 4th G P4 perfect 4th G G
31 516.667 27/20 ^4 up 4th ^G K4 comma-wide 4th KG KG
32 533.333 15/11 ^^4, v~4 dup 4th, downmid 4th ^^G O4 on 4th OG SG
33 550.000 11/8 ~4 mid 4th ^3G U4/N4 uber 4th / neutral 4th UG UG
34 566.667 25/18, 18/13 ^~4, vvA4 upmid 4th, dudaug 4th vvG# kkA4, sd5 classic aug 4th, sub dim 5th kkG#, sAb SG#, (kkG#), sAb
35 583.333 7/5 vA4, vd5 downaug 4th,

downdim 5th

vG#, vAb kA4, ld5 comma-narrow aug 4th, little dim 5th kG#, lAb kG#, kAb
36 600.000 99/70, 17/12 A4, d5 aug 4th, dim 5th G#, Ab A4, d5 aug 4th, dim 5th G#, Ab G#, Ab
37 616.667 10/7 ^A4, ^d5 upaug 4th, updim 5th ^G#, ^Ab LA4, Kd5 large aug 4th, comma-wide dim 5th LG#, KAb KG#, KAb
38 633.333 36/25, 13/9 v~5, ^^d5 downmid 5th,

dupdim 5th

^^Ab SA4, KKd5 super aug 4th, classic dim 5th SG#, KKAb SG#, SAb, (KKAb)
39 650.000 16/11 ~5 mid 5th v3A u5/N5 unter 5th / neutral 5th uA uA
40 666.667 22/15 vv5, ^~5 dud 5th, upmid 5th vvA o5 off 5th oA sA
41 683.333 40/27 v5 down 5th vA k5 comma-narrow 5th kA kA
42 700.000 3/2 P5 perfect 5th A P5 perfect 5th A A
43 716.667 50/33 ^5 up 5th ^A L5 large fifth LA KA
44 733.333 32/21 ^^5 dup 5th ^^A S5 super fifth SA SA
45 750.000 54/35, 17/11 ^35, v3m6 trup 5th, trudminor 6th ^3A, v3Bb H5, hm6 hyper fifth, hypominor 6th HA, hBb UA, uBb
46 766.667 14/9 vvm6 dudminor 6th vvBb sm6 superminor 6th sBb sBb
47 783.333 11/7 vm6 downminor 6th vBb lm6 little minor 6th lBb kBb
48 800.000 35/22 m6 minor 6th Bb m6 minor 6th Bb Bb
49 816.667 8/5 ^m6 upminor 6th ^Bb Km6 classic minor 6th kBb kBb
50 833.333 81/50, 13/8 ^^m6, v~6 dupminor 6th, downmid 6th ^^Bb Om6 on minor 6th oBb sBb
51 850.000 18/11 ~6 mid 6th v3B N6 neutral 6th UBb, uB UBb, uB
52 866.667 33/20, 28/17 ^~6, vvM6 upmid 6th, dudmajor 6th vvB oM6 off major 6th oB sB
53 883.333 5/3 vM6 downmajor 6th vB kM6 classic major 6th kB kB
54 900.000 27/16 M6 major 6th B M6 major 6th B B
55 916.667 56/33, 17/10 ^M6 upmajor 6th ^B LM6 large major 6th LB KB
56 933.333 12/7 ^^M6 dupmajor 6th ^^B SM6 supermajor 6th SB SB
57 950.000 121/70 ^3M6,
v3m7
trupmajor 6th,
trudminor 7th
^3B,
v3C
HM6, hm7 hypermajor 6th, hypominor 7th HB, hC UB, uC
58 966.667 7/4 vvm7 dudminor 7th vvC sm7 subminor 7th sC sC
59 983.333 44/25 vm7 downminor 7th vC lm7 little minor 7th lC kC
60 1000.000 16/9 m7 minor 7th C m7 minor 7th C C
61 1016.667 9/5 ^m7 upminor 7th ^C Km7 classic/comma-wide minor 7th KC KC
62 1033.333 20/11 ^^m7, v~7 dupminor 7th, downmid 7th ^^C Om7 on minor 7th OC SC
63 1050.000 11/6 ~7 mid 7th ^3C N7, hd8 neutral 7th, hypo dim 8ve UC/uC#, hDb UC/uC#, uDb
64 1066.667 50/27 ^~7, vvM7 upmid 7th, dudmajor 7th vvC# oM7, sd8 off major 7th, sub dim 8ve oC#, sDb sC#, sDb
65 1083.333 15/8 vM7 downmajor 7th vC# kM7, ld8 classic major 7th, little dim 8ve kC#, lDb kC#, kDb
66 1100.000 66/35, 17/9 M7 major 7th C# M7, d8 major 7th, dim 8ve C#, Db C#, Db
67 1116.667 21/11 ^M7 upmajor 7th ^C# LM7, Kd8 large major 7th, comma-wide dim 8ve LC#, KDb KC#, KDb
68 1133.333 27/14, 48/25 ^^M7 dupmajor 7th ^^C# SM7, KKd8 supermajor 7th, classic dim 8ve SC#, KKDb SC#, SDb, (KKDb)
69 1150.000 35/18 ^3M7, v38 trupmajor 7th, trud octave ^3C#, v3D HM7, u8, h8 hypermajor 7th, unter 8ve, hypo 8ve HC#, uD, hD UC#, uDb, uD
70 1166.667 49/25 vv8 dud octave vvD s8, o8 sub 8ve, off 8ve sD, oD sD
71 1183.333 99/50 v8 down octave vD k8, l8 comma-narrow 8ve, little 8ve kD, lD kD
72 1200.000 2/1 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:

  • −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
  • −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
  • +3 degrees (the trup ring) corrects 4/3 to 11/8 via 33/32
  • +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
  • 0 degrees (the plain ring) corrects 256/243 to 17/16 via 4131/4096
  • 0 degrees (the plain ring) corrects 32/27 to 19/16 via 513/512

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.

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:

72edo Sagittal.png

Ups and downs

Using Helmholtz–Ellis accidentals, 72edo can also be notated using ups and downs notation:

Semitones 0 16 13 12 23 56 1 1+16 1+13 1+12 1+23 1+56 2 2+16 2+13
Sharp Symbol
Heji18.svg
Heji19.svg
Heji20.svg
HeQu1.svg
Heji23.svg
Heji24.svg
Heji25.svg
Heji26.svg
Heji27.svg
HeQu3.svg
Heji30.svg
Heji31.svg
Heji32.svg
Heji33.svg
Heji34.svg
Flat Symbol
Heji17.svg
Heji16.svg
HeQd1.svg
Heji13.svg
Heji12.svg
Heji11.svg
Heji10.svg
Heji9.svg
HeQd3.svg
Heji6.svg
Heji5.svg
Heji4.svg
Heji3.svg
Heji2.svg

In some cases, certain notes may be best notated using semi- and sesquisharps and flats with arrows:

Semitones 0 16 13 12 23 56 1 1+16 1+13 1+12 1+23 1+56 2 2+16
Sharp Symbol
Heji18.svg
Heji19.svg
HeQu1-sd1.svg
HeQu1.svg
HeQu1-su1.svg
Heji24.svg
Heji25.svg
Heji26.svg
HeQu3-sd1.svg
HeQu3.svg
HeQu3-su1.svg
Heji31.svg
Heji32.svg
Heji33.svg
Flat Symbol
Heji17.svg
HeQd1-su1.svg
HeQd1.svg
HeQd1-sd1.svg
Heji12.svg
Heji11.svg
Heji10.svg
HeQd3-su1.svg
HeQd3.svg
HeQd3-sd1.svg
Heji5.svg
Heji4.svg
Heji3.svg

Ivan Wyschnegradsky's notation

Step Offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp Symbol
Heji18.svg
Wyschnegradsky's 1/6 sharp.svg
Wyschnegradsky's 1/3 sharp.svg
HeQu1.svg
Wyschnegradsky's 2/3 sharp.svg
Wyschnegradsky's 5/6 sharp.svg
Heji25.svg
Wyschnegradsky's 7/6 sharp.svg
Wyschnegradsky's 4/3 sharp.svg
HeQu3.svg
Wyschnegradsky's 5/3 sharp.svg
Wyschnegradsky's 11/6 sharp.svg
Heji32.svg
Wyschnegradsky's 1/6 sharp.svgHeji32.svg
Flat Symbol
Wyschnegradsky's 1/6 flat.svg
Wyschnegradsky's 1/3 flat.svg
Wyschnegradsky's Half flat.svg
Wyschnegradsky's 2/3 flat.svg
Wyschnegradsky's 5/6 flat.svg
Heji11.svg
Wyschnegradsky's 7/6 flat.svg
Wyschnegradsky's 4/3 flat.svg
Wyschnegradsky's 3/2 flat.svg
Wyschnegradsky's 5/3 flat.svg
Wyschnegradsky's 11/6 flat.svg
Heji4.svg
Wyschnegradsky's 1/6 flat.svgHeji4.svg

JI approximation

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

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.

plot72.png

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

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

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 2] 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 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
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 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
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

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 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
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)
Decades

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

Scales

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).

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

Music

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

External links

Notes

  1. Based on treating 72edo as a 17-limit temperament; other approaches are also possible. For lower limits see Table of 72edo intervals.
  2. Ratios longer than 10 digits are presented by placeholders with informative hints.