Xenharmonic Wiki

72edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Jkarimak (talk | contribs)
191 edits
m →Intervals: style
TallKite (talk | contribs)
autopatrolled, Sysops
3,990 edits
updated ups/down notation, general cleanup
Line 1: Line 1:
__FORCETOC__
__FORCETOC__


= Theory =
<b>72-tone equal temperament</b>, or <b>72-edo</b>, 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 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic,_Turkish,_Persian|Arabic]] music, and has itself been used to tune Turkish music.
<b>72-tone equal temperament</b>, or <b>72-edo</b>, 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 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic,_Turkish,_Persian|Arabic]] music, and has itself been used to tune Turkish music.


Line 8: Line 9:


72 is an excellent tuning for [[Gamelismic_clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family#Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
72 is an excellent tuning for [[Gamelismic_clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family#Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
=Commas=
Commas tempered out by 72edo include...
{| class="wikitable"
|-
! | 3-limit
|-
| |<nowiki> Pythagorean comma = 531441/524288 = |-19 12</nowiki>&gt;
|}
{| class="wikitable"
|-
! | 5-limit
|-
| |<nowiki> kleisma = 15625/15552 = |-6 -5 6</nowiki>&gt;
ampersand = 34171875/33554432 = |-25 7 6&gt;
graviton = 129140163/128000000 = |-13 17 -6&gt;
ennealimma = 7629394531250/7625597484987 = |1 -27 18&gt;
|}
{| class="wikitable"
|-
! | 7-limit
! | 11-limit
! | 13-limit
|-
| | ...............................
225/224
1029/1024
2401/2400
4375/4374
16875/16807
19683/19600
420175/419904
250047/250000
| | .......................
243/242
385/384
441/440
540/539
1375/1372
3025/3024
4000/3993
6250/6237
9801/9800
| | .......................
169/168
325/324
351/350
364/363
625/624
676/675
729/728
1001/1000
1575/1573
1716/1715
2080/2079
6656/6655
|}
=Temperaments=
It 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.
See also [[List_of_edo-distinct_72et_rank_two_temperaments|List of edo-distinct 72et rank two temperaments]].
=Harmonic Scale=
Mode 8 of the harmonic series -- [[overtone_scales|overtones 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).
{| class="wikitable"
|-
| | Overtones 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
| |
|}


=Intervals=
=Intervals=
Line 306: Line 31:
| style="text-align:center;" | ^1
| style="text-align:center;" | ^1
| style="text-align:center;" | up unison
| style="text-align:center;" | up unison
| style="text-align:center;" | D^
| style="text-align:center;" | ^D
|-
|-
| | 2
| | 2
Line 313: Line 38:
| style="text-align:center;" | ^^
| style="text-align:center;" | ^^
| style="text-align:center;" | double-up unison
| style="text-align:center;" | double-up unison
| style="text-align:center;" | D^^
| style="text-align:center;" | ^^D
|-
|-
| | 3
| | 3
Line 322: Line 47:


triple-down minor 2nd
triple-down minor 2nd
| style="text-align:center;" | D^<span style="font-size: 90%; vertical-align: super;">3</span>, Ebv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>D, v<span style="font-size: 90%; vertical-align: super;">3</span>Eb
|-
|-
| | 4
| | 4
Line 329: Line 54:
| style="text-align:center;" | vvm2
| style="text-align:center;" | vvm2
| style="text-align:center;" | double-downminor 2nd
| style="text-align:center;" | double-downminor 2nd
| style="text-align:center;" | Ebvv
| style="text-align:center;" | vvEb
|-
|-
| | 5
| | 5
Line 336: Line 61:
| style="text-align:center;" | vm2
| style="text-align:center;" | vm2
| style="text-align:center;" | downminor 2nd
| style="text-align:center;" | downminor 2nd
| style="text-align:center;" | Ebv
| style="text-align:center;" | vEb
|-
|-
| | 6
| | 6
Line 350: Line 75:
| style="text-align:center;" | ^m2
| style="text-align:center;" | ^m2
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | Eb^
| style="text-align:center;" | ^Eb
|-
|-
| | 8
| | 8
Line 357: Line 82:
| style="text-align:center;" | v~2
| style="text-align:center;" | v~2
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" | Eb^^
| style="text-align:center;" | ^^Eb
|-
|-
| | 9
| | 9
Line 364: Line 89:
| style="text-align:center;" | ~2
| style="text-align:center;" | ~2
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | Ev<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>E
|-
|-
| | 10
| | 10
Line 371: Line 96:
| style="text-align:center;" | ^~2
| style="text-align:center;" | ^~2
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | Evv
| style="text-align:center;" | vvE
|-
|-
| | 11
| | 11
Line 378: Line 103:
| style="text-align:center;" | vM2
| style="text-align:center;" | vM2
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | Ev
| style="text-align:center;" | vE
|-
|-
| | 12
| | 12
Line 392: Line 117:
| style="text-align:center;" | ^M2
| style="text-align:center;" | ^M2
| style="text-align:center;" | upmajor 2nd
| style="text-align:center;" | upmajor 2nd
| style="text-align:center;" | E^
| style="text-align:center;" | ^E
|-
|-
| | 14
| | 14
Line 399: Line 124:
| style="text-align:center;" | ^^M2
| style="text-align:center;" | ^^M2
| style="text-align:center;" | double-upmajor 2nd
| style="text-align:center;" | double-upmajor 2nd
| style="text-align:center;" | E^^
| style="text-align:center;" | ^^E
|-
|-
| | 15
| | 15
Line 408: Line 133:


triple-down minor 3rd
triple-down minor 3rd
| style="text-align:center;" | E^<span style="font-size: 90%; vertical-align: super;">3</span>, Fv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>E, v<span style="font-size: 90%; vertical-align: super;">3</span>F
|-
|-
| | 16
| | 16
Line 415: Line 140:
| style="text-align:center;" | vvm3
| style="text-align:center;" | vvm3
| style="text-align:center;" | double-downminor 3rd
| style="text-align:center;" | double-downminor 3rd
| style="text-align:center;" | Fvv
| style="text-align:center;" | vvF
|-
|-
| | 17
| | 17
Line 422: Line 147:
| style="text-align:center;" | vm3
| style="text-align:center;" | vm3
| style="text-align:center;" | downminor 3rd
| style="text-align:center;" | downminor 3rd
| style="text-align:center;" | Fv
| style="text-align:center;" | vF
|-
|-
| | 18
| | 18
Line 436: Line 161:
| style="text-align:center;" | ^m3
| style="text-align:center;" | ^m3
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | F^
| style="text-align:center;" | ^F
|-
|-
| | 20
| | 20
Line 443: Line 168:
| style="text-align:center;" | v~3
| style="text-align:center;" | v~3
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | F^^
| style="text-align:center;" | ^^F
|-
|-
| | 21
| | 21
Line 450: Line 175:
| style="text-align:center;" | ~3
| style="text-align:center;" | ~3
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | F^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F
|-
|-
| | 22
| | 22
Line 457: Line 182:
| style="text-align:center;" | ^~3
| style="text-align:center;" | ^~3
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | F#vv
| style="text-align:center;" | vvF#
|-
|-
| | 23
| | 23
Line 464: Line 189:
| style="text-align:center;" | vM3
| style="text-align:center;" | vM3
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | F#v
| style="text-align:center;" | vF#
|-
|-
| | 24
| | 24
Line 478: Line 203:
| style="text-align:center;" | ^M3
| style="text-align:center;" | ^M3
| style="text-align:center;" | upmajor 3rd
| style="text-align:center;" | upmajor 3rd
| style="text-align:center;" | F#^
| style="text-align:center;" | ^F#
|-
|-
| | 26
| | 26
Line 485: Line 210:
| style="text-align:center;" | ^^M3
| style="text-align:center;" | ^^M3
| style="text-align:center;" | double-upmajor 3rd
| style="text-align:center;" | double-upmajor 3rd
| style="text-align:center;" | F#^^
| style="text-align:center;" | ^^F#
|-
|-
| | 27
| | 27
Line 494: Line 219:


triple-down 4th
triple-down 4th
| style="text-align:center;" | F#^<span style="font-size: 90%; vertical-align: super;">3</span>, Gv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F#, v<span style="font-size: 90%; vertical-align: super;">3</span>G
|-
|-
| | 28
| | 28
Line 501: Line 226:
| style="text-align:center;" | vv4
| style="text-align:center;" | vv4
| style="text-align:center;" | double-down 4th
| style="text-align:center;" | double-down 4th
| style="text-align:center;" | Gvv
| style="text-align:center;" | vvG
|-
|-
| | 29
| | 29
Line 508: Line 233:
| style="text-align:center;" | v4
| style="text-align:center;" | v4
| style="text-align:center;" | down 4th
| style="text-align:center;" | down 4th
| style="text-align:center;" | Gv
| style="text-align:center;" | vG
|-
|-
| | 30
| | 30
Line 522: Line 247:
| style="text-align:center;" | ^4
| style="text-align:center;" | ^4
| style="text-align:center;" | up 4th
| style="text-align:center;" | up 4th
| style="text-align:center;" | G^
| style="text-align:center;" | ^G
|-
|-
| | 32
| | 32
| | 533.333
| | 533.333
| | 15/11
| | 15/11
| style="text-align:center;" | ^^4
| style="text-align:center;" | v~4
| style="text-align:center;" | double-up 4th
| style="text-align:center;" | downmid 4th
| style="text-align:center;" | G^^
| style="text-align:center;" | ^^G
|-
|-
| | 33
| | 33
| | 550
| | 550
| | 11/8
| | 11/8
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4
| style="text-align:center;" | ~4
| style="text-align:center;" | triple-up 4th
| style="text-align:center;" | mid 4th
| style="text-align:center;" | G^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>G
|-
|-
| | 34
| | 34
| | 566.667
| | 566.667
| | 25/18
| | 25/18
| style="text-align:center;" | vvA4
| style="text-align:center;" | ^~4
| style="text-align:center;" | double-down aug 4th
| style="text-align:center;" | upmid 4th
| style="text-align:center;" | G#vv
| style="text-align:center;" | vvG#
|-
|-
| | 35
| | 35
Line 550: Line 275:
| style="text-align:center;" | vA4, vd5
| style="text-align:center;" | vA4, vd5
| style="text-align:center;" | downaug 4th, updim 5th
| style="text-align:center;" | downaug 4th, updim 5th
| style="text-align:center;" | G#v, Abv
| style="text-align:center;" | vG#, vAb
|-
|-
| | 36
| | 36
Line 564: Line 289:
| style="text-align:center;" | ^A4, ^d5
| style="text-align:center;" | ^A4, ^d5
| style="text-align:center;" | upaug 4th, downdim 5th
| style="text-align:center;" | upaug 4th, downdim 5th
| style="text-align:center;" | G#^, Ab^
| style="text-align:center;" | ^G#, ^Ab
|-
|-
| | 38
| | 38
| | 633.333
| | 633.333
| | 36/25
| | 36/25
| style="text-align:center;" | ^^d5
| style="text-align:center;" | v~5
| style="text-align:center;" | double-updim 5th
| style="text-align:center;" | downmid 5th
| style="text-align:center;" | Ab^^
| style="text-align:center;" | ^^Ab
|-
|-
| | 39
| | 39
| | 650
| | 650
| | 16/11
| | 16/11
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>5
| style="text-align:center;" | ~5
| style="text-align:center;" | triple-down 5th
| style="text-align:center;" | mid 5th
| style="text-align:center;" | Av<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>A
|-
|-
| | 40
| | 40
| | 666.667
| | 666.667
| | 22/15
| | 22/15
| style="text-align:center;" | vv5
| style="text-align:center;" | ^~5
| style="text-align:center;" | double-down 5th
| style="text-align:center;" | upmid 5th
| style="text-align:center;" | Avv
| style="text-align:center;" | vvA
|-
|-
| | 41
| | 41
Line 592: Line 317:
| style="text-align:center;" | v5
| style="text-align:center;" | v5
| style="text-align:center;" | down 5th
| style="text-align:center;" | down 5th
| style="text-align:center;" | Av
| style="text-align:center;" | vA
|-
|-
| | 42
| | 42
Line 606: Line 331:
| style="text-align:center;" | ^5
| style="text-align:center;" | ^5
| style="text-align:center;" | up 5th
| style="text-align:center;" | up 5th
| style="text-align:center;" | A^
| style="text-align:center;" | ^A
|-
|-
| | 44
| | 44
Line 613: Line 338:
| style="text-align:center;" | ^^5
| style="text-align:center;" | ^^5
| style="text-align:center;" | double-up 5th
| style="text-align:center;" | double-up 5th
| style="text-align:center;" | A^^
| style="text-align:center;" | ^^A
|-
|-
| | 45
| | 45
Line 622: Line 347:


triple-down minor 6th
triple-down minor 6th
| style="text-align:center;" | A^<span style="font-size: 90%; vertical-align: super;">3</span>, Bbv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>A, v<span style="font-size: 90%; vertical-align: super;">3</span>Bb
|-
|-
| | 46
| | 46
Line 629: Line 354:
| style="text-align:center;" | vvm6
| style="text-align:center;" | vvm6
| style="text-align:center;" | double-downminor 6th
| style="text-align:center;" | double-downminor 6th
| style="text-align:center;" | Bbvv
| style="text-align:center;" | vvBb
|-
|-
| | 47
| | 47
Line 636: Line 361:
| style="text-align:center;" | vm6
| style="text-align:center;" | vm6
| style="text-align:center;" | downminor 6th
| style="text-align:center;" | downminor 6th
| style="text-align:center;" | Bbv
| style="text-align:center;" | vBb
|-
|-
| | 48
| | 48
Line 650: Line 375:
| style="text-align:center;" | ^m6
| style="text-align:center;" | ^m6
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | Bb^
| style="text-align:center;" | ^Bb
|-
|-
| | 50
| | 50
Line 657: Line 382:
| style="text-align:center;" | v~6
| style="text-align:center;" | v~6
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | Bb^^
| style="text-align:center;" | ^^Bb
|-
|-
| | 51
| | 51
Line 664: Line 389:
| style="text-align:center;" | ~6
| style="text-align:center;" | ~6
| style="text-align:center;" | mid 6th
| style="text-align:center;" | mid 6th
| style="text-align:center;" | Bv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>B
|-
|-
| | 52
| | 52
Line 671: Line 396:
| style="text-align:center;" | ^~6
| style="text-align:center;" | ^~6
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | Bvv
| style="text-align:center;" | vvB
|-
|-
| | 53
| | 53
Line 678: Line 403:
| style="text-align:center;" | vM6
| style="text-align:center;" | vM6
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | Bv
| style="text-align:center;" | vB
|-
|-
| | 54
| | 54
Line 692: Line 417:
| style="text-align:center;" | ^M6
| style="text-align:center;" | ^M6
| style="text-align:center;" | upmajor 6th
| style="text-align:center;" | upmajor 6th
| style="text-align:center;" | B^
| style="text-align:center;" | ^B
|-
|-
| | 56
| | 56
Line 699: Line 424:
| style="text-align:center;" | ^^M6
| style="text-align:center;" | ^^M6
| style="text-align:center;" | double-upmajor 6th
| style="text-align:center;" | double-upmajor 6th
| style="text-align:center;" | B^^
| style="text-align:center;" | ^^B
|-
|-
| | 57
| | 57
Line 708: Line 433:


triple-down minor 7th
triple-down minor 7th
| style="text-align:center;" | B^<span style="font-size: 90%; vertical-align: super;">3</span>, Cv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>B, v<span style="font-size: 90%; vertical-align: super;">3</span>C
|-
|-
| | 58
| | 58
Line 715: Line 440:
| style="text-align:center;" | vvm7
| style="text-align:center;" | vvm7
| style="text-align:center;" | double-downminor 7th
| style="text-align:center;" | double-downminor 7th
| style="text-align:center;" | Cvv
| style="text-align:center;" | vvC
|-
|-
| | 59
| | 59
Line 722: Line 447:
| style="text-align:center;" | vm7
| style="text-align:center;" | vm7
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | Cv
| style="text-align:center;" | vC
|-
|-
| | 60
| | 60
Line 736: Line 461:
| style="text-align:center;" | ^m7
| style="text-align:center;" | ^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | C^
| style="text-align:center;" | ^C
|-
|-
| | 62
| | 62
Line 743: Line 468:
| style="text-align:center;" | v~7
| style="text-align:center;" | v~7
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | C^^
| style="text-align:center;" | ^^C
|-
|-
| | 63
| | 63
Line 750: Line 475:
| style="text-align:center;" | ~7
| style="text-align:center;" | ~7
| style="text-align:center;" | mid 7th
| style="text-align:center;" | mid 7th
| style="text-align:center;" | C^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C
|-
|-
| | 64
| | 64
Line 757: Line 482:
| style="text-align:center;" | ^~7
| style="text-align:center;" | ^~7
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | C#vv
| style="text-align:center;" | vvC#
|-
|-
| | 65
| | 65
Line 764: Line 489:
| style="text-align:center;" | vM7
| style="text-align:center;" | vM7
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | C#v
| style="text-align:center;" | vC#
|-
|-
| | 66
| | 66
Line 778: Line 503:
| style="text-align:center;" | ^M7
| style="text-align:center;" | ^M7
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | C#^
| style="text-align:center;" | ^C#
|-
|-
| | 68
| | 68
Line 785: Line 510:
| style="text-align:center;" | ^^M7
| style="text-align:center;" | ^^M7
| style="text-align:center;" | double-upmajor 7th
| style="text-align:center;" | double-upmajor 7th
| style="text-align:center;" | C#^^
| style="text-align:center;" | ^^C#
|-
|-
| | 69
| | 69
Line 794: Line 519:


triple-down octave
triple-down octave
| style="text-align:center;" | C#^<span style="font-size: 90%; vertical-align: super;">3</span>, Dv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C#, v<span style="font-size: 90%; vertical-align: super;">3</span>D
|-
|-
| | 70
| | 70
Line 801: Line 526:
| style="text-align:center;" | vv8
| style="text-align:center;" | vv8
| style="text-align:center;" | double-down octave
| style="text-align:center;" | double-down octave
| style="text-align:center;" | Dvv
| style="text-align:center;" | vvD
|-
|-
| | 71
| | 71
Line 808: Line 533:
| style="text-align:center;" | v8
| style="text-align:center;" | v8
| style="text-align:center;" | down octave
| style="text-align:center;" | down octave
| style="text-align:center;" | Dv
| style="text-align:center;" | vD
|-
|-
| | 72
| | 72
Line 866: Line 591:
| style="text-align:center;" | 9/7, 12/7
| style="text-align:center;" | 9/7, 12/7
|}
|}
All 72edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:
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:


{| class="wikitable"
{| class="wikitable"
Line 880: Line 605:
| style="text-align:center;" | 6:7:9
| style="text-align:center;" | 6:7:9
| style="text-align:center;" | 0-16-42
| style="text-align:center;" | 0-16-42
| style="text-align:center;" | C Ebvv G
| style="text-align:center;" | C vvEb G
| style="text-align:center;" | C.vvm
| style="text-align:center;" | Cvvm
| style="text-align:center;" | C double-down minor
| style="text-align:center;" | C double-down minor
|-
|-
Line 887: Line 612:
| style="text-align:center;" | 10:12:15
| style="text-align:center;" | 10:12:15
| style="text-align:center;" | 0-19-42
| style="text-align:center;" | 0-19-42
| style="text-align:center;" | C Eb^ G
| style="text-align:center;" | C ^Eb G
| style="text-align:center;" | C.^m
| style="text-align:center;" | C^m
| style="text-align:center;" | C upminor
| style="text-align:center;" | C upminor
|-
|-
Line 894: Line 619:
| style="text-align:center;" | 18:22:27
| style="text-align:center;" | 18:22:27
| style="text-align:center;" | 0-21-42
| style="text-align:center;" | 0-21-42
| style="text-align:center;" | C Ev<span style="font-size: 90%; vertical-align: super;">3</span> G
| style="text-align:center;" | C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
| style="text-align:center;" | C~
| style="text-align:center;" | C~
| style="text-align:center;" | C mid
| style="text-align:center;" | C mid
Line 901: Line 626:
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 0-23-42
| style="text-align:center;" | 0-23-42
| style="text-align:center;" | C Ev G
| style="text-align:center;" | C vE G
| style="text-align:center;" | C.v
| style="text-align:center;" | Cv
| style="text-align:center;" | C downmajor or C dot down
| style="text-align:center;" | C downmajor or C down
|-
|-
| style="text-align:center;" | ru
| style="text-align:center;" | ru
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 0-26-42
| style="text-align:center;" | 0-26-42
| style="text-align:center;" | C E^^ G
| style="text-align:center;" | C ^^E G
| style="text-align:center;" | C.^^
| style="text-align:center;" | C^^
| style="text-align:center;" | C double-upmajor or C dot double-up
| style="text-align:center;" | C double-upmajor or C double-up
|}
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
=Commas=
Commas tempered out by 72edo include...
{| class="wikitable"
|-
! | 3-limit
|-
| |<nowiki> Pythagorean comma = 531441/524288 = |-19 12</nowiki>&gt;
|}
{| class="wikitable"
|-
! | 5-limit
|-
| |<nowiki> kleisma = 15625/15552 = |-6 -5 6</nowiki>&gt;
ampersand = 34171875/33554432 = |-25 7 6&gt;
graviton = 129140163/128000000 = |-13 17 -6&gt;
ennealimma = 7629394531250/7625597484987 = |1 -27 18&gt;
|}
{| class="wikitable"
|-
! | 7-limit
! | 11-limit
! | 13-limit
|-
| | ...............................
225/224
1029/1024
2401/2400
4375/4374
16875/16807
19683/19600
420175/419904
250047/250000
| | .......................
243/242
385/384
441/440
540/539
1375/1372
3025/3024
4000/3993
6250/6237
9801/9800
| | .......................
169/168
325/324
351/350
364/363
625/624
676/675
729/728
1001/1000
1575/1573
1716/1715
2080/2079
6656/6655
|}
=Temperaments=
It 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.
See also [[List_of_edo-distinct_72et_rank_two_temperaments|List of edo-distinct 72et rank two temperaments]].
=Scales=
[[smithgw72a|smithgw72a]], [[smithgw72b|smithgw72b]], [[smithgw72c|smithgw72c]], [[smithgw72d|smithgw72d]], [[smithgw72e|smithgw72e]], [[smithgw72f|smithgw72f]], [[smithgw72g|smithgw72g]], [[smithgw72h|smithgw72h]], [[smithgw72i|smithgw72i]], [[smithgw72j|smithgw72j]]
[[blackjack|blackjack]], [[miracle_8|miracle_8]], [[miracle_10|miracle_10]], [[miracle_12|miracle_12]], [[miracle_12a|miracle_12a]], [[miracle_24hi|miracle_24hi]], [[miracle_24lo|miracle_24lo]]
[[keenanmarvel|keenanmarvel]], [[xenakis_chrome|xenakis_chrome]], [[xenakis_diat|xenakis_diat]], [[xenakis_schrome|xenakis_schrome]]
[[genus24255et72|Euler(24255) genus in 72 equal]]
[[JuneGloom|JuneGloom]]
==Harmonic Scale==
Mode 8 of the harmonic series -- [[overtone_scales|overtones 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).
{| class="wikitable"
|-
| | Overtones 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
| |
|}


=Linear temperaments=
=Linear temperaments=
Line 1,080: Line 1,092:


''[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 June Gloom #9]'' by Prent Rodgers
''[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 June Gloom #9]'' by Prent Rodgers
=Scales=
[[smithgw72a|smithgw72a]], [[smithgw72b|smithgw72b]], [[smithgw72c|smithgw72c]], [[smithgw72d|smithgw72d]], [[smithgw72e|smithgw72e]], [[smithgw72f|smithgw72f]], [[smithgw72g|smithgw72g]], [[smithgw72h|smithgw72h]], [[smithgw72i|smithgw72i]], [[smithgw72j|smithgw72j]]
[[blackjack|blackjack]], [[miracle_8|miracle_8]], [[miracle_10|miracle_10]], [[miracle_12|miracle_12]], [[miracle_12a|miracle_12a]], [[miracle_24hi|miracle_24hi]], [[miracle_24lo|miracle_24lo]]
[[keenanmarvel|keenanmarvel]], [[xenakis_chrome|xenakis_chrome]], [[xenakis_diat|xenakis_diat]], [[xenakis_schrome|xenakis_schrome]]
[[genus24255et72|Euler(24255) genus in 72 equal]]
[[JuneGloom|JuneGloom]]


=External links=
=External links=
Retrieved from "https://en.xen.wiki/w/72edo"