84edo: Difference between revisions

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== Intervals ==
== Intervals ==
For this table, the notation of Orwell[9] from the [[4L 5s]] page is taken. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details).
{| class="wikitable"
{| class="wikitable"
|+Table of 84edo intervals
|+Table of 84edo intervals
Line 22: Line 21:
! Approximate Ratios*
! Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
|-
|-
| 0
| 0
Line 30: Line 28:
| P1
| P1
| D
| D
| Perfect 1sn
| P1
| J
|-
|-
| 1
| 1
Line 40: Line 35:
| ^1
| ^1
| ^D
| ^D
| Up 1sn
| ^1
| J^
|-
|-
| 2
| 2
Line 50: Line 42:
| ^^1
| ^^1
| ^^D
| ^^D
| Downaug 1sn
| vA1
| Jv&
|-
|-
| 3
| 3
Line 60: Line 49:
| ^^^1
| ^^^1
| ^^^D
| ^^^D
| Aug 1sn
| A1
| J&
|-
|-
| 4
| 4
Line 70: Line 56:
| vvvm2
| vvvm2
| vvvEb
| vvvEb
| Upaug 1sn, Downdim 2nd
| ^A1, vd2
| J^&, Kvaa
|-
|-
| 5
| 5
Line 80: Line 63:
| vvm2
| vvm2
| vvEb
| vvEb
| Dim 2nd
| d2
| Kaa
|-
|-
| 6
| 6
Line 90: Line 70:
| vm2
| vm2
| vEb
| vEb
| Updim 2nd
| ^d2
| K^aa
|-
|-
| 7
| 7
Line 100: Line 77:
| m2
| m2
| Eb
| Eb
| Downminor 2nd
| vm2
| Kva
|-
|-
| 8
| 8
Line 110: Line 84:
| ^m2
| ^m2
| ^Eb
| ^Eb
| Minor 2nd
| m2
| Ka
|-
|-
| 9
| 9
Line 120: Line 91:
| ^^m2
| ^^m2
| ^^Eb
| ^^Eb
| Upminor 2nd
| ^m2
| K^a
|-
|-
| 10
| 10
Line 130: Line 98:
| ^^^m2
| ^^^m2
| ^^^Eb
| ^^^Eb
| Downmajor 2nd
| vM2
| Kv
|-
|-
| 11
| 11
Line 140: Line 105:
| vvvM2
| vvvM2
| vvvE
| vvvE
| Major 2nd
| M2
| K
|-
|-
| 12
| 12
Line 150: Line 112:
| vvM2
| vvM2
| vvE
| vvE
| Upmajor 2nd
| ^M2
| K^
|-
|-
| 13
| 13
Line 160: Line 119:
| vM2
| vM2
| vE
| vE
| Downaug 2nd
| vA2
| Kv&
|-
|-
| 14
| 14
Line 170: Line 126:
| M2
| M2
| E
| E
| Aug 2nd
| A2
| K&
|-
|-
| 15
| 15
Line 180: Line 133:
| ^M2
| ^M2
| ^E
| ^E
| Upaug 2nd, Downdim 3rd
| ^A2, vd3
| K^&, Lva
|-
|-
| 16
| 16
Line 190: Line 140:
| ^^M2
| ^^M2
| ^^E
| ^^E
| Dim 3rd
| d3
| La
|-
|-
| 17
| 17
Line 200: Line 147:
| ^^^M2
| ^^^M2
| ^^^E
| ^^^E
| Updim 3rd
| ^d3
| L^a
|-
|-
| 18
| 18
Line 210: Line 154:
| vvvm3
| vvvm3
| vvvF
| vvvF
| Down 3rd
| v3
| Lv
|-
|-
| 19
| 19
Line 220: Line 161:
| vvm2
| vvm2
| vvF
| vvF
| Perfect 3rd
| P3
| L
|-
|-
| 20
| 20
Line 230: Line 168:
| vm3
| vm3
| vF
| vF
| Up 3rd
| ^3
| L^
|-
|-
| 21
| 21
Line 240: Line 175:
| m3
| m3
| F
| F
| Downaug 3rd
| vA3
| Lv&
|-
|-
| 22
| 22
Line 250: Line 182:
| ^m3
| ^m3
| ^F
| ^F
| Aug 3rd
| A3
| L&
|-
|-
| 23
| 23
Line 260: Line 189:
| ^^m3
| ^^m3
| ^^F
| ^^F
| Upaug 3rd, Downdim 4th
| ^A3, vd4
| L^&, Mvaa
|-
|-
| 24
| 24
Line 270: Line 196:
| ^^^m3
| ^^^m3
| ^^^F
| ^^^F
| Dim 4th
| d4
| Maa
|-
|-
| 25
| 25
Line 280: Line 203:
| vvvM3
| vvvM3
| vvvF#
| vvvF#
| Updim 4th
| ^d4
| M^aa
|-
|-
| 26
| 26
Line 290: Line 210:
| vvM3
| vvM3
| vvF#
| vvF#
| Downminor 4th
| vm4
| Mva
|-
|-
| 27
| 27
Line 300: Line 217:
| vM3
| vM3
| vF#
| vF#
| Minor 4th
| m4
| Ma
|-
|-
| 28
| 28
Line 310: Line 224:
| M3
| M3
| F#
| F#
| Upminor 4th
| ^m4
| M^a
|-
|-
| 29
| 29
Line 320: Line 231:
| ^M3
| ^M3
| ^F#
| ^F#
| Downmajor 4th
| vM4
| Mv
|-
|-
| 30
| 30
Line 330: Line 238:
| ^^M3
| ^^M3
| ^^F#
| ^^F#
| Major 4th
| M4
| M
|-
|-
| 31
| 31
Line 340: Line 245:
| ^^^M3
| ^^^M3
| ^^^F#
| ^^^F#
| Upmajor 4th
| ^M4
| M^
|-
|-
| 32
| 32
Line 350: Line 252:
| vvv4
| vvv4
| vvvG
| vvvG
| Downaug 4th
| vA4
| Mv&
|-
|-
| 33
| 33
Line 360: Line 259:
| vv4
| vv4
| vvG
| vvG
| Aug 4th
| A4
| M&
|-
|-
| 34
| 34
Line 370: Line 266:
| v4
| v4
| vG
| vG
| Downminor 5th
| vm5
| Nva
|-
|-
| 35
| 35
Line 380: Line 273:
| P4
| P4
| G
| G
| Minor 5th
| m5
| Na
|-
|-
| 36
| 36
Line 390: Line 280:
| ^4
| ^4
| ^G
| ^G
| Upminor 5th
| ^m5
| N^a
|-
|-
| 37
| 37
Line 400: Line 287:
| ^^4
| ^^4
| ^^G
| ^^G
| Downmajor 5th
| vM5
| Nv
|-
|-
| 38
| 38
Line 410: Line 294:
| ^^^4
| ^^^4
| ^^^G
| ^^^G
| Major 5th
| M5
| N
|-
|-
| 39
| 39
Line 420: Line 301:
| vvvA4
| vvvA4
| vvvG#
| vvvG#
| Upmajor 5th
| ^M5
| N^
|-
|-
| 40
| 40
Line 430: Line 308:
| vvA4
| vvA4
| vvG#
| vvG#
| Downaug 5th
| vA5
| Nv&
|-
|-
| 41
| 41
Line 440: Line 315:
| vA4
| vA4
| vG#
| vG#
| Aug 5th
| A5
| N&
|-
|-
| 42
| 42
Line 450: Line 322:
| A4, d5
| A4, d5
| G#, Ab
| G#, Ab
| Upaug 5th, Downdim 6th
| ^A5, vd6
| N^&, Ovaa
|-
|-
| 43
| 43
Line 460: Line 329:
| ^d5
| ^d5
| ^Ab
| ^Ab
| Dim 6th
| d6
| Oaa
|-
|-
| 44
| 44
Line 470: Line 336:
| ^^d5
| ^^d5
| ^^Ab
| ^^Ab
| Updim 6th
| ^d6
| O^aa
|-
|-
| 45
| 45
Line 480: Line 343:
| ^^^d5
| ^^^d5
| ^^^Ab
| ^^^Ab
| Downminor 6th
| vm6
| Ova
|-
|-
| 46
| 46
Line 490: Line 350:
| vvv5
| vvv5
| vvvA
| vvvA
| Minor 6th
| m6
| Oa
|-
|-
| 47
| 47
Line 500: Line 357:
| vv5
| vv5
| vvA
| vvA
| Upminor 6th
| ^m6
| O^a
|-
|-
| 48
| 48
Line 510: Line 364:
| v5
| v5
| vA
| vA
| Downmajor 6th
| vM6
| Ov
|-
|-
| 49
| 49
Line 520: Line 371:
| P5
| P5
| A
| A
| Major 6th
| M6
| O
|-
|-
| 50
| 50
Line 530: Line 378:
| ^5
| ^5
| ^A
| ^A
| Upmajor 6th
| ^M6
| O^
|-
|-
| 51
| 51
Line 540: Line 385:
| ^^5
| ^^5
| ^^A
| ^^A
| Dim 7th
| d7
| Paa
|-
|-
| 52
| 52
Line 550: Line 392:
| ^^^5
| ^^^5
| ^^^A
| ^^^A
| Aug 6th
| A6
| O&
|-
|-
| 53
| 53
Line 560: Line 399:
| vvvm6
| vvvm6
| vvvBb
| vvvBb
| Downminor 7th
| vm7
| Pva
|-
|-
| 54
| 54
Line 570: Line 406:
| vvm6
| vvm6
| vvBb
| vvBb
| Minor 7th
| m7
| Pa
|-
|-
| 55
| 55
Line 580: Line 413:
| vm6
| vm6
| vBb
| vBb
| Upminor 7th
| ^m7
| P^a
|-
|-
| 56
| 56
Line 590: Line 420:
| m6
| m6
| Bb
| Bb
| Downmajor 7th
| vM7
| Pv
|-
|-
| 57
| 57
Line 600: Line 427:
| ^m6
| ^m6
| ^Bb
| ^Bb
| Major 7th
| M7
| P
|-
|-
| 58
| 58
Line 610: Line 434:
| ^^m6
| ^^m6
| ^^Bb
| ^^Bb
| Upmajor 7th
| ^M7
| P^
|-
|-
| 59
| 59
Line 620: Line 441:
| ^^^m6
| ^^^m6
| ^^^Bb
| ^^^Bb
| Downaug 7th
| vA7
| Pv&
|-
|-
| 60
| 60
Line 630: Line 448:
| vvvM6
| vvvM6
| vvvB
| vvvB
| Aug 7th
| A7
| P&
|-
|-
| 61
| 61
Line 640: Line 455:
| vvM6
| vvM6
| vvB
| vvB
| Upaug 7th, Downdim 8th
| ^A7, vd8
| P^&, Qvaa
|-
|-
| 62
| 62
Line 650: Line 462:
| vM6
| vM6
| vB
| vB
| Dim 8th
| d8
| Qaa
|-
|-
| 63
| 63
Line 660: Line 469:
| M6
| M6
| B
| B
| Updim 8th
| ^d8
| Q^aa
|-
|-
| 64
| 64
Line 670: Line 476:
| ^M6
| ^M6
| ^B
| ^B
| Down 8th
| v8
| Qva
|-
|-
| 65
| 65
Line 680: Line 483:
| ^^M6
| ^^M6
| ^^B
| ^^B
| Perfect 8th
| P8
| Qa
|-
|-
| 66
| 66
Line 690: Line 490:
| ^^^M6
| ^^^M6
| ^^^B
| ^^^B
| Up 8th
| ^8
| Q^a
|-
|-
| 67
| 67
Line 700: Line 497:
| vvvm7
| vvvm7
| vvvC
| vvvC
| Downaug 8th
| vA8
| Qv
|-
|-
| 68
| 68
Line 710: Line 504:
| vvm7
| vvm7
| vvC
| vvC
| Aug 8th
| A8
| Q
|-
|-
| 69
| 69
Line 720: Line 511:
| vm7
| vm7
| vC
| vC
| Upaug 8th, Downdim 9th
| ^A8, vd9
| Q^, Rvaa
|-
|-
| 70
| 70
Line 730: Line 518:
| m7
| m7
| C
| C
| Dim 9th
| d9
| Raa
|-
|-
| 71
| 71
Line 740: Line 525:
| ^m7
| ^m7
| ^C
| ^C
| Updim 9th
| ^d9
| R^aa
|-
|-
| 72
| 72
Line 750: Line 532:
| ^^m7
| ^^m7
| ^^C
| ^^C
| Downminor 9th
| vm9
| Rva
|-
|-
| 73
| 73
Line 760: Line 539:
| ^^^m7
| ^^^m7
| ^^^C
| ^^^C
| Minor 9th
| m9
| Ra
|-
|-
| 74
| 74
Line 770: Line 546:
| vvvM7
| vvvM7
| vvvC#
| vvvC#
| Upminor 9th
| ^m9
| R^a
|-
|-
| 75
| 75
Line 780: Line 553:
| vvM7
| vvM7
| vvC#
| vvC#
| Downmajor 9th
| vM9
| Rv
|-
|-
| 76
| 76
Line 790: Line 560:
| vM7
| vM7
| vC#
| vC#
| Major 9th
| M9
| R
|-
|-
| 77
| 77
Line 800: Line 567:
| M7
| M7
| C#
| C#
| Upmajor 9th
| ^M9
| R^
|-
|-
| 78
| 78
Line 810: Line 574:
| ^M7
| ^M7
| ^C#
| ^C#
| Downaug 9th
| vA9
| Rv&
|-
|-
| 79
| 79
Line 820: Line 581:
| ^^M7
| ^^M7
| ^^C#
| ^^C#
| Aug 9th
| A9
| R&
|-
|-
| 80
| 80
Line 830: Line 588:
| ^^^M7
| ^^^M7
| ^^^C#
| ^^^C#
| Upaug 9th, Downdim 10th
| ^A9, vd10
| R^&, Jva
|-
|-
| 81
| 81
Line 840: Line 595:
| vvv8
| vvv8
| vvvD
| vvvD
| Dim 10th
| d10
| Ja
|-
|-
| 82
| 82
Line 850: Line 602:
| vv8
| vv8
| vvD
| vvD
| Updim 10th
| ^d10
| J^a
|-
|-
| 83
| 83
Line 860: Line 609:
| v8
| v8
| vD
| vD
| Down 10th
| v10
| Jv
|-
|-
| 84
| 84
Line 870: Line 616:
| P8
| P8
| D
| D
| Perfect 10th
|}
| P10
<nowiki>*</nowiki> as a 2.3.5.7.13-subgroup temperament
 
== Notation ==
=== 4L 5s (gramitonic) notation ===
The notation of Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".
 
{| class="wikitable center-1 right-2 center-3"
|-
! #
! Cents
! Note
! Name
! Associated Ratio
|-
| 0
| 0.0
| J
| J
| Perfect 0-gramstep
| 1/1
|-
| 8
| 114.3
| K@
| Minor 1-gramstep
| 15/14~16/15
|-
| 11
| 157.1
| K
| Major 1-gramstep
| 11/10~12/11
|-
|-
| 16
| 228.6
| L@
| Diminished 2-gramstep
| 8/7
|-
| 19
| 271.4
| L
| Perfect 2-gramstep
| 7/6
|-
| 27
| 385.7
| M@
| Minor 3-gramstep
| 5/4
|-
| 30
| 428.6
| M
| Major 3-gramstep
| 9/7
|-
| 35
| 500.0
| N@
| Minor 4-gramstep
| 4/3
|-
| 38
| 542.9
| N
| Major 4-gramstep
| 11/8~15/11
|-
| 46
| 657.1
| O@
| Minor 5-gramstep
| 16/11~22/15
|-
| 49
| 700.0
| O
| Major 5-gramstep
| 3/2
|-
| 54
| 771.4
| P@
| Minor 6-gramstep
| 14/9
|-
| 57
| 814.3
| P
| Major 6-gramstep
| 8/5
|-
| 65
| 928.6
| Q@
| Perfect 7-gramstep
| 12/7
|-
| 68
| 971.4
| Q
| Augmented 7-gramstep
| 7/4
|-
| 73
| 1042.9
| R@
| Minor 8-gramstep
| 11/6~20/11
|-
| 76
| 1085.7
| R
| Major 8-gramstep
| 15/8~28/15
|-
| 84
| 1200.0
| J
| Perfect 9-gramstep
| 2/1
|}
|}
<nowiki>*</nowiki> as a 2.3.5.7.13-subgroup temperament


== Regular temperament properties ==
== Regular temperament properties ==

Revision as of 08:50, 24 June 2024

← 83edo 84edo 85edo →
Prime factorization 22 × 3 × 7
Step size 14.2857 ¢ 
Fifth 49\84 (700 ¢) (→ 7\12)
Semitones (A1:m2) 7:7 (100 ¢ : 100 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

84edo is where the orwell temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the edo, referencing the book 1984. Orwell in 84edo comes in two varieties – the 84e val ⟨84 133 195 236 290], supporting the original orwell, and its patent val ⟨84 133 195 236 291] supporting newspeak. 84edo orwell offers mosses of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.

It has fairly good approximation to higher prime harmonics such as 13, 19, 23, 29, and 31. In fact, it is consistent to the no-11 no-17 25-odd-limit. In the 13-limit it is the optimal patent val for the rank-5 temperament tempering out 144/143.

Prime harmonics

Approximation of prime harmonics in 84edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.00 -1.96 -0.60 +2.60 +5.82 +2.33 -4.96 +2.49 +0.30 -1.01 -2.18 +5.80
Relative (%) +0.0 -13.7 -4.2 +18.2 +40.8 +16.3 -34.7 +17.4 +2.1 -7.0 -15.2 +40.6
Steps
(reduced) 		84
(0) 		133
(49) 		195
(27) 		236
(68) 		291
(39) 		311
(59) 		343
(7) 		357
(21) 		380
(44) 		408
(72) 		416
(80) 		438
(18)
Approximation of prime harmonics in 84edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) -0.49 +2.77 +5.92 -2.08 -2.03 -2.60 +6.41 +6.02 +0.78 +6.89 +7.10 +0.55
Relative (%) -3.4 +19.4 +41.5 -14.5 -14.2 -18.2 +44.9 +42.1 +5.5 +48.2 +49.7 +3.8
Steps
(reduced) 		450
(30) 		456
(36) 		467
(47) 		481
(61) 		494
(74) 		498
(78) 		510
(6) 		517
(13) 		520
(16) 		530
(26) 		536
(32) 		544
(40)

Subsets and supersets

84edo is a largely composite number. Since 84 factors as 22 × 3 × 7, 84edo has subset edos 2, 3, 4, 6, 7, 12, 14, 21, 28, 42. Being a small multiple of 12, 84et tempers out the Pythagorean comma, thus supporting the 1/12-octave temperament compton. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.

Intervals

Table of 84edo intervals
# Cents Approximate Ratios* Ups and Downs Notation
0 0.000 1/1 Perfect 1sn P1 D
1 14.286 81/80, 126/125 Up 1sn ^1 ^D
2 28.571 50/49, 64/63 Dup 1sn ^^1 ^^D
3 42.857 36/35, 40/39, 49/48 Trup 1sn ^^^1 ^^^D
4 57.143 27/26 Trudminor 2nd vvvm2 vvvEb
5 71.429 25/24, 26/25, 28/27 Dudminor 2nd vvm2 vvEb
6 85.714 21/20 Downminor 2nd vm2 vEb
7 100.000 Minor 2nd m2 Eb
8 114.286 15/14, 16/15 Upminor 2nd ^m2 ^Eb
9 128.571 14/13 Dupminor 2nd ^^m2 ^^Eb
10 142.857 13/12 Trupminor 2nd ^^^m2 ^^^Eb
11 157.143 Trudmajor 2nd vvvM2 vvvE
12 171.429 Dudmajor 2nd vvM2 vvE
13 185.714 10/9 Downmajor 2nd vM2 vE
14 200.000 9/8 Major 2nd M2 E
15 214.286 Upmajor 2nd ^M2 ^E
16 228.571 8/7 Dupmajor 2nd ^^M2 ^^E
17 242.857 15/13 Trupmajor 2nd ^^^M2 ^^^E
18 257.143 Trudminor 3rd vvvm3 vvvF
19 271.429 7/6 Dudminor 3rd vvm2 vvF
20 285.714 Downminor 3rd vm3 vF
21 300.000 32/27 Minor 3rd m3 F
22 314.286 6/5 Upminor 3rd ^m3 ^F
23 328.571 Dupminor 3rd ^^m3 ^^F
24 342.857 39/32 Trupminor 3rd ^^^m3 ^^^F
25 357.143 16/13 Trudmajor 3rd vvvM3 vvvF#
26 371.429 26/21 Dudmajor 3rd vvM3 vvF#
27 385.714 5/4 Downmajor 3rd vM3 vF#
28 400.000 Major 3rd M3 F#
29 414.286 Upmajor 3rd ^M3 ^F#
30 428.571 9/7 Dupmajor 3rd ^^M3 ^^F#
31 442.857 Trupmajor 3rd ^^^M3 ^^^F#
32 457.143 13/10 Trud 4th vvv4 vvvG
33 471.429 21/16 Dud 4th vv4 vvG
34 485.714 Down 4th v4 vG
35 500.000 4/3 Perfect 4th P4 G
36 514.286 27/20 Up 4th ^4 ^G
37 528.571 Dup 4th ^^4 ^^G
38 542.857 Trup 4th ^^^4 ^^^G
39 557.143 18/13 Trudaug 4th vvvA4 vvvG#
40 571.429 Dudaug 4th vvA4 vvG#
41 585.714 7/5 Downaug 4th vA4 vG#
42 600.000 Aug 4th, Dim 5th A4, d5 G#, Ab
43 614.286 10/7 Updim 5th ^d5 ^Ab
44 628.571 Dupdim 5th ^^d5 ^^Ab
45 642.857 13/9 Trupdim 5th ^^^d5 ^^^Ab
46 657.143 Trud 5th vvv5 vvvA
47 671.429 Dud 5th vv5 vvA
48 685.714 40/27 Down 5th v5 vA
49 700.000 3/2 Perfect 5th P5 A
50 714.286 Up 5th ^5 ^A
51 728.571 32/21 Dup 5th ^^5 ^^A
52 742.857 20/13 Trup 5th ^^^5 ^^^A
53 757.143 Trudminor 6th vvvm6 vvvBb
54 771.429 14/9 Dudminor 6th vvm6 vvBb
55 785.714 Downminor 6th vm6 vBb
56 800.000 Minor 6th m6 Bb
57 814.286 8/5 Upminor 6th ^m6 ^Bb
58 828.571 21/13 Dupminor 6th ^^m6 ^^Bb
59 842.857 13/8 Trupminor 6th ^^^m6 ^^^Bb
60 857.143 64/39 Trudmajor 6th vvvM6 vvvB
61 871.429 Dudmajor 6th vvM6 vvB
62 885.714 5/3 Downmajor 6th vM6 vB
63 900.000 27/16 Major 6th M6 B
64 914.286 Upmajor 6th ^M6 ^B
65 928.571 12/7 Dupmajor 6th ^^M6 ^^B
66 942.857 Trupmajor 6th ^^^M6 ^^^B
67 957.143 26/15 Trudminor 7th vvvm7 vvvC
68 971.429 7/4 Dudminor 7th vvm7 vvC
69 985.714 Downminor 7th vm7 vC
70 1000.000 16/9 Minor 7th m7 C
71 1014.286 9/5 Upminor 7th ^m7 ^C
72 1028.571 Dupminor 7th ^^m7 ^^C
73 1042.857 Trupminor 7th ^^^m7 ^^^C
74 1057.143 24/13 Trudmajor 7th vvvM7 vvvC#
75 1071.429 13/7 Dudmajor 7th vvM7 vvC#
76 1085.714 15/8, 28/15 Downmajor 7th vM7 vC#
77 1100.000 Major 7th M7 C#
78 1114.286 40/21 Upmajor 7th ^M7 ^C#
79 1128.571 25/13, 27/14, 48/25 Dupmajor 7th ^^M7 ^^C#
80 1142.857 52/27 Trupmajor 7th ^^^M7 ^^^C#
81 1157.143 35/18, 39/20, 96/49 Trud 8ve vvv8 vvvD
82 1171.429 49/25, 63/32 Dud 8ve vv8 vvD
83 1185.714 125/63, 160/81 Down 8ve v8 vD
84 1200.000 2/1 Perfect 8ve P8 D

* as a 2.3.5.7.13-subgroup temperament

Notation

4L 5s (gramitonic) notation

The notation of Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".

# Cents Note Name Associated Ratio
0 0.0 J Perfect 0-gramstep 1/1
8 114.3 K@ Minor 1-gramstep 15/14~16/15
11 157.1 K Major 1-gramstep 11/10~12/11
16 228.6 L@ Diminished 2-gramstep 8/7
19 271.4 L Perfect 2-gramstep 7/6
27 385.7 M@ Minor 3-gramstep 5/4
30 428.6 M Major 3-gramstep 9/7
35 500.0 N@ Minor 4-gramstep 4/3
38 542.9 N Major 4-gramstep 11/8~15/11
46 657.1 O@ Minor 5-gramstep 16/11~22/15
49 700.0 O Major 5-gramstep 3/2
54 771.4 P@ Minor 6-gramstep 14/9
57 814.3 P Major 6-gramstep 8/5
65 928.6 Q@ Perfect 7-gramstep 12/7
68 971.4 Q Augmented 7-gramstep 7/4
73 1042.9 R@ Minor 8-gramstep 11/6~20/11
76 1085.7 R Major 8-gramstep 15/8~28/15
84 1200.0 J Perfect 9-gramstep 2/1

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 78732/78125, 531441/524288 [⟨84 133 195]] +0.498 0.531 3.72
2.3.5.7 225/224, 1728/1715, 78732/78125 [⟨84 133 195 236]] +0.141 0.769 5.39
2.3.5.7.11 225/224, 441/440, 1344/1331, 1728/1715 [⟨84 133 195 236 291]] (84) -0.225 1.003 7.02
2.3.5.7.11 99/98, 121/120, 176/175, 78732/78125 [⟨84 133 195 236 290]] (84e) +0.601 1.151 8.05

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 19\84 271.43 7/6 Orwell (84e)
Newspeak (84)
1 25\84 357.14 768/625 Dodifo
1 27\84 385.71 5/4 Mutt
1 31\84 442.86 125/81 Sensei
1 41\84 585.71 7/5 Merman
2 5\84 71.43 25/24 Narayana
2 11\84 157.14 35/32 Bison
2 13\84 185.71 10/9 Secant
3 11\84 157.14 35/32 Nessafof
7 5\84 500.00
(14.29) 		4/3
(81/80) 		Absurdity
12 27\84
(1\84) 		385.71
(14.29) 		5/4
(126/125) 		Compton
21 41\84
(1\84) 		585.71
(14.29) 		91875/65536
(126/125) 		Akjayland
28 49\84
(1\84) 		500.00
(14.29) 		4/3
(105/104) 		Oquatonic

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

Scales

MOS

Brightest mode is listed.

Other

Music

John Cage
Eliora
JUMBLE
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