84edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
→Theory: rework
Re-format the interval table
Line 14: Line 14:
84edo is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{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|oquatonic comma]], which maps 5/4 to 9\28.
84edo is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{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|oquatonic comma]], which maps 5/4 to 9\28.


== Table of intervals ==
== Interval table ==
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).  
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
! Degree
! #
! Size (Cents)
! Cents
! Approximate Ratios
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
! Associated ratio
|-
|-
| 0
| 0
| 0.000
| 0.000
| 1/1
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 32: Line 33:
| P1
| P1
| J
| J
| 1/1 exact
|-
|-
| 1
| 1
| 14.286
| 14.286
|
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 42: Line 43:
| ^1
| ^1
| J^
| J^
|
|-
|-
| 2
| 2
| 28.571
| 28.571
|
| Dup 1sn
| Dup 1sn
| ^^1
| ^^1
Line 52: Line 53:
| vA1
| vA1
| Jv&
| Jv&
|
|-
|-
| 3
| 3
| 42.857
| 42.857
|
| Trup 1sn
| Trup 1sn
| ^^^1
| ^^^1
Line 62: Line 63:
| A1
| A1
| J&
| J&
|
|-
|-
| 4
| 4
| 57.143
| 57.143
|
| Trudminor 2nd
| Trudminor 2nd
| vvvm2
| vvvm2
Line 72: Line 73:
| ^A1, vd2
| ^A1, vd2
| J^&, Kvaa
| J^&, Kvaa
|
|-
|-
| 5
| 5
| 71.429
| 71.429
|
| Dudminor 2nd
| Dudminor 2nd
| vvm2
| vvm2
Line 82: Line 83:
| d2
| d2
| Kaa
| Kaa
|
|-
|-
| 6
| 6
| 85.714
| 85.714
|
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 92: Line 93:
| ^d2
| ^d2
| K^aa
| K^aa
|
|-
|-
| 7
| 7
| 100.000
| 100.000
|
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 102: Line 103:
| vm2
| vm2
| Kva
| Kva
|
|-
|-
| 8
| 8
| 114.286
| 114.286
|
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 112: Line 113:
| m2
| m2
| Ka
| Ka
|
|-
|-
| 9
| 9
| 128.571
| 128.571
|
| Dupminor 2nd
| Dupminor 2nd
| ^^m2
| ^^m2
Line 122: Line 123:
| ^m2
| ^m2
| K^a
| K^a
|
|-
|-
| 10
| 10
| 142.857
| 142.857
|
| Trupminor 2nd
| Trupminor 2nd
| ^^^m2
| ^^^m2
Line 132: Line 133:
| vM2
| vM2
| Kv
| Kv
|
|-
|-
| 11
| 11
| 157.143
| 157.143
|
| Trudmajor 2nd
| Trudmajor 2nd
| vvvM2
| vvvM2
Line 142: Line 143:
| M2
| M2
| K
| K
|
|-
|-
| 12
| 12
| 171.429
| 171.429
|
| Dudmajor 2nd
| Dudmajor 2nd
| vvM2
| vvM2
Line 152: Line 153:
| ^M2
| ^M2
| K^
| K^
|
|-
|-
| 13
| 13
| 185.714
| 185.714
|
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 162: Line 163:
| vA2
| vA2
| Kv&
| Kv&
|
|-
|-
| 14
| 14
| 200.000
| 200.000
|
| Major 2nd
| Major 2nd
| M2
| M2
Line 172: Line 173:
| A2
| A2
| K&
| K&
|
|-
|-
| 15
| 15
| 214.286
| 214.286
|
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 182: Line 183:
| ^A2, vd3
| ^A2, vd3
| K^&, Lva
| K^&, Lva
|
|-
|-
| 16
| 16
| 228.571
| 228.571
|
| Dupmajor 2nd
| Dupmajor 2nd
| ^^M2
| ^^M2
Line 192: Line 193:
| d3
| d3
| La
| La
|
|-
|-
| 17
| 17
| 242.857
| 242.857
|
| Trupmajor 2nd
| Trupmajor 2nd
| ^^^M2
| ^^^M2
Line 202: Line 203:
| ^d3
| ^d3
| L^a
| L^a
|
|-
|-
| 18
| 18
| 257.143
| 257.143
|
| Trudminor 3rd
| Trudminor 3rd
| vvvm3
| vvvm3
Line 212: Line 213:
| v3
| v3
| Lv
| Lv
|
|-
|-
| 19
| 19
| 271.429
| 271.429
|[[7/6]]
| Dudminor 3rd
| Dudminor 3rd
| vvm2
| vvm2
Line 222: Line 223:
| P3
| P3
| L
| L
| [[7/6]]
|-
|-
| 20
| 20
| 285.714
| 285.714
|
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 232: Line 233:
| ^3
| ^3
| L^
| L^
|
|-
|-
| 21
| 21
| 300.000
| 300.000
|
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 242: Line 243:
| vA3
| vA3
| Lv&
| Lv&
|
|-
|-
| 22
| 22
| 314.286
| 314.286
|
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 252: Line 253:
| A3
| A3
| L&
| L&
|
|-
|-
| 23
| 23
| 328.571
| 328.571
|
| Dupminor 3rd
| Dupminor 3rd
| ^^m3
| ^^m3
Line 262: Line 263:
| ^A3, vd4
| ^A3, vd4
| L^&, Mvaa
| L^&, Mvaa
|
|-
|-
| 24
| 24
| 342.857
| 342.857
|
| Trupminor 3rd
| Trupminor 3rd
| ^^^m3
| ^^^m3
Line 272: Line 273:
| d4
| d4
| Maa
| Maa
|
|-
|-
| 25
| 25
| 357.143
| 357.143
|
| Trudmajor 3rd
| Trudmajor 3rd
| vvvM3
| vvvM3
Line 282: Line 283:
| ^d4
| ^d4
| M^aa
| M^aa
|
|-
|-
| 26
| 26
| 371.429
| 371.429
|
| Dudmajor 3rd
| Dudmajor 3rd
| vvM3
| vvM3
Line 292: Line 293:
| vm4
| vm4
| Mva
| Mva
|
|-
|-
| 27
| 27
| 385.714
| 385.714
|
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 302: Line 303:
| m4
| m4
| Ma
| Ma
|
|-
|-
| 28
| 28
| 400.000
| 400.000
|
| Major 3rd
| Major 3rd
| M3
| M3
Line 312: Line 313:
| ^m4
| ^m4
| M^a
| M^a
|
|-
|-
| 29
| 29
| 414.286
| 414.286
|
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 322: Line 323:
| vM4
| vM4
| Mv
| Mv
|
|-
|-
| 30
| 30
| 428.571
| 428.571
|
| Dupmajor 3rd
| Dupmajor 3rd
| ^^M3
| ^^M3
Line 332: Line 333:
| M4
| M4
| M
| M
|
|-
|-
| 31
| 31
| 442.857
| 442.857
|
| Trupmajor 3rd
| Trupmajor 3rd
| ^^^M3
| ^^^M3
Line 342: Line 343:
| ^M4
| ^M4
| M^
| M^
|
|-
|-
| 32
| 32
| 457.143
| 457.143
|
| Trud 4th
| Trud 4th
| vvv4
| vvv4
Line 352: Line 353:
| vA4
| vA4
| Mv&
| Mv&
|
|-
|-
| 33
| 33
| 471.429
| 471.429
|
| Dud 4th
| Dud 4th
| vv4
| vv4
Line 362: Line 363:
| A4
| A4
| M&
| M&
|
|-
|-
| 34
| 34
| 485.714
| 485.714
|
| Down 4th
| Down 4th
| v4
| v4
Line 372: Line 373:
| vm5
| vm5
| Nva
| Nva
|
|-
|-
| 35
| 35
| 500.000
| 500.000
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 382: Line 383:
| m5
| m5
| Na
| Na
|
|-
|-
| 36
| 36
| 514.286
| 514.286
|
| Up 4th
| Up 4th
| ^4
| ^4
Line 392: Line 393:
| ^m5
| ^m5
| N^a
| N^a
|
|-
|-
| 37
| 37
| 528.571
| 528.571
|
| Dup 4th
| Dup 4th
| ^^4
| ^^4
Line 402: Line 403:
| vM5
| vM5
| Nv
| Nv
|
|-
|-
| 38
| 38
| 542.857
| 542.857
|
| Trup 4th
| Trup 4th
| ^^^4
| ^^^4
Line 412: Line 413:
| M5
| M5
| N
| N
| [[11/8]] in the 84b val
|-
|-
| 39
| 39
| 557.143
| 557.143
|
| Trudaug 4th
| Trudaug 4th
| vvvA4
| vvvA4
Line 422: Line 423:
| ^M5
| ^M5
| N^
| N^
|
|-
|-
| 40
| 40
| 571.429
| 571.429
|
| Dudaug 4th
| Dudaug 4th
| vvA4
| vvA4
Line 432: Line 433:
| vA5
| vA5
| Nv&
| Nv&
|
|-
|-
| 41
| 41
| 585.714
| 585.714
|
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 442: Line 443:
| A5
| A5
| N&
| N&
|
|-
|-
| 42
| 42
| 600.000
| 600.000
|
| Aug 4th, Dim 5th
| Aug 4th, Dim 5th
| A4, d5
| A4, d5
Line 452: Line 453:
| ^A5, vd6
| ^A5, vd6
| N^&, Ovaa
| N^&, Ovaa
|
|-
|-
| 43
| 43
| 614.286
| 614.286
|
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 462: Line 463:
| d6
| d6
| Oaa
| Oaa
|
|-
|-
| 44
| 44
| 628.571
| 628.571
|
| Dupdim 5th
| Dupdim 5th
| ^^d5
| ^^d5
Line 472: Line 473:
| ^d6
| ^d6
| O^aa
| O^aa
|
|-
|-
| 45
| 45
| 642.857
| 642.857
|
| Trupdim 5th
| Trupdim 5th
| ^^^d5
| ^^^d5
Line 482: Line 483:
| vm6
| vm6
| Ova
| Ova
|
|-
|-
| 46
| 46
| 657.143
| 657.143
|
| Trud 5th
| Trud 5th
| vvv5
| vvv5
Line 492: Line 493:
| m6
| m6
| Oa
| Oa
|
|-
|-
| 47
| 47
| 671.429
| 671.429
|
| Dud 5th
| Dud 5th
| vv5
| vv5
Line 502: Line 503:
| ^m6
| ^m6
| O^a
| O^a
|
|-
|-
| 48
| 48
| 685.714
| 685.714
|
| Down 5th
| Down 5th
| v5
| v5
Line 512: Line 513:
| vM6
| vM6
| Ov
| Ov
|
|-
|-
| 49
| 49
| 700.000
| 700.000
|[[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 522: Line 523:
| M6
| M6
| O
| O
| [[3/2]]
|-
|-
| 50
| 50
| 714.286
| 714.286
|
| Up 5th
| Up 5th
| ^5
| ^5
Line 532: Line 533:
| ^M6
| ^M6
| O^
| O^
|
|-
|-
| 51
| 51
| 728.571
| 728.571
|
| Dup 5th
| Dup 5th
| ^^5
| ^^5
Line 542: Line 543:
| d7
| d7
| Paa
| Paa
|
|-
|-
| 52
| 52
| 742.857
| 742.857
|
| Trup 5th
| Trup 5th
| ^^^5
| ^^^5
Line 552: Line 553:
| A6
| A6
| O&
| O&
|
|-
|-
| 53
| 53
| 757.143
| 757.143
|
| Trudminor 6th
| Trudminor 6th
| vvvm6
| vvvm6
Line 562: Line 563:
| vm7
| vm7
| Pva
| Pva
|
|-
|-
| 54
| 54
| 771.429
| 771.429
|
| Dudminor 6th
| Dudminor 6th
| vvm6
| vvm6
Line 572: Line 573:
| m7
| m7
| Pa
| Pa
|
|-
|-
| 55
| 55
| 785.714
| 785.714
|
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 582: Line 583:
| ^m7
| ^m7
| P^a
| P^a
|
|-
|-
| 56
| 56
| 800.000
| 800.000
|
| Minor 6th
| Minor 6th
| m6
| m6
Line 592: Line 593:
| vM7
| vM7
| Pv
| Pv
|
|-
|-
| 57
| 57
| 814.286
| 814.286
|[[5/3]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 602: Line 603:
| M7
| M7
| P
| P
| [[5/3]]
|-
|-
| 58
| 58
| 828.571
| 828.571
|
| Dupminor 6th
| Dupminor 6th
| ^^m6
| ^^m6
Line 612: Line 613:
| ^M7
| ^M7
| P^
| P^
|
|-
|-
| 59
| 59
| 842.857
| 842.857
|
| Trupminor 6th
| Trupminor 6th
| ^^^m6
| ^^^m6
Line 622: Line 623:
| vA7
| vA7
| Pv&
| Pv&
|
|-
|-
| 60
| 60
| 857.143
| 857.143
|[[105/64]]
| Trudmajor 6th
| Trudmajor 6th
| vvvM6
| vvvM6
Line 632: Line 633:
| A7
| A7
| P&
| P&
| [[105/64]]
|-
|-
| 61
| 61
| 871.429
| 871.429
|
| Dudmajor 6th
| Dudmajor 6th
| vvM6
| vvM6
Line 642: Line 643:
| ^A7, vd8
| ^A7, vd8
| P^&, Qvaa
| P^&, Qvaa
|
|-
|-
| 62
| 62
| 885.714
| 885.714
|
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 652: Line 653:
| d8
| d8
| Qaa
| Qaa
|
|-
|-
| 63
| 63
| 900.000
| 900.000
|
| Major 6th
| Major 6th
| M6
| M6
Line 662: Line 663:
| ^d8
| ^d8
| Q^aa
| Q^aa
|
|-
|-
| 64
| 64
| 914.286
| 914.286
|
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 672: Line 673:
| v8
| v8
| Qva
| Qva
|
|-
|-
| 65
| 65
| 928.571
| 928.571
|
| Dupmajor 6th
| Dupmajor 6th
| ^^M6
| ^^M6
Line 682: Line 683:
| P8
| P8
| Qa
| Qa
|
|-
|-
| 66
| 66
| 942.857
| 942.857
|
| Trupmajor 6th
| Trupmajor 6th
| ^^^M6
| ^^^M6
Line 692: Line 693:
| ^8
| ^8
| Q^a
| Q^a
|
|-
|-
| 67
| 67
| 957.143
| 957.143
|
| Trudminor 7th
| Trudminor 7th
| vvvm7
| vvvm7
Line 702: Line 703:
| vA8
| vA8
| Qv
| Qv
|
|-
|-
| 68
| 68
| 971.429
| 971.429
|
| Dudminor 7th
| Dudminor 7th
| vvm7
| vvm7
Line 712: Line 713:
| A8
| A8
| Q
| Q
|
|-
|-
| 69
| 69
| 985.714
| 985.714
|
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 722: Line 723:
| ^A8, vd9
| ^A8, vd9
| Q^, Rvaa
| Q^, Rvaa
|
|-
|-
| 70
| 70
| 1000.000
| 1000.000
|
| Minor 7th
| Minor 7th
| m7
| m7
Line 732: Line 733:
| d9
| d9
| Raa
| Raa
|
|-
|-
| 71
| 71
| 1014.286
| 1014.286
|
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 742: Line 743:
| ^d9
| ^d9
| R^aa
| R^aa
|
|-
|-
| 72
| 72
| 1028.571
| 1028.571
|
| Dupminor 7th
| Dupminor 7th
| ^^m7
| ^^m7
Line 752: Line 753:
| vm9
| vm9
| Rva
| Rva
|
|-
|-
| 73
| 73
| 1042.857
| 1042.857
|
| Trupminor 7th
| Trupminor 7th
| ^^^m7
| ^^^m7
Line 762: Line 763:
| m9
| m9
| Ra
| Ra
|
|-
|-
| 74
| 74
| 1057.143
| 1057.143
|
| Trudmajor 7th
| Trudmajor 7th
| vvvM7
| vvvM7
Line 772: Line 773:
| ^m9
| ^m9
| R^a
| R^a
|
|-
|-
| 75
| 75
| 1071.429
| 1071.429
|
| Dudmajor 7th
| Dudmajor 7th
| vvM7
| vvM7
Line 782: Line 783:
| vM9
| vM9
| Rv
| Rv
|
|-
|-
| 76
| 76
| 1085.714
| 1085.714
|
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 792: Line 793:
| M9
| M9
| R
| R
|
|-
|-
| 77
| 77
| 1100.000
| 1100.000
|
| Major 7th
| Major 7th
| M7
| M7
Line 802: Line 803:
| ^M9
| ^M9
| R^
| R^
|
|-
|-
|-
|-
| 78
| 78
| 1114.286
| 1114.286
|
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 813: Line 814:
| vA9
| vA9
| Rv&
| Rv&
|
|-
|-
| 79
| 79
| 1128.571
| 1128.571
|
| Dupmajor 7th
| Dupmajor 7th
| ^^M7
| ^^M7
Line 823: Line 824:
| A9
| A9
| R&
| R&
|
|-
|-
| 80
| 80
| 1142.857
| 1142.857
|
| Trupmajor 7th
| Trupmajor 7th
| ^^^M7
| ^^^M7
Line 833: Line 834:
| ^A9, vd10
| ^A9, vd10
| R^&, Jva
| R^&, Jva
|
|-
|-
| 81
| 81
| 1157.143
| 1157.143
|
| Trud 8ve
| Trud 8ve
| vvv8
| vvv8
Line 843: Line 844:
| d10
| d10
| Ja
| Ja
|
|-
|-
| 82
| 82
| 1171.429
| 1171.429
|
| Dud 8ve
| Dud 8ve
| vv8
| vv8
Line 853: Line 854:
| ^d10
| ^d10
| J^a
| J^a
|
|-
|-
| 83
| 83
| 1185.714
| 1185.714
| Down 8ve
|
| Down 8ve
| v8
| v8
| vD
| vD
Line 863: Line 864:
| v10
| v10
| Jv
| Jv
|
|-
|-
| 84
| 84
| 1200.000
| 1200.000
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
| P8
| P8
Line 873: Line 874:
| P10
| P10
| J
| J
| [[2/1]] exact
|-
|-
|}
|}

Revision as of 09:00, 21 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.

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.

Interval table

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).

Table of 84edo intervals
# Cents Approximate Ratios Ups and Downs Notation 4L 5s Notation
0 0.000 1/1 Perfect 1sn P1 D Perfect 1sn P1 J
1 14.286 Up 1sn ^1 ^D Up 1sn ^1 J^
2 28.571 Dup 1sn ^^1 ^^D Downaug 1sn vA1 Jv&
3 42.857 Trup 1sn ^^^1 ^^^D Aug 1sn A1 J&
4 57.143 Trudminor 2nd vvvm2 vvvEb Upaug 1sn, Downdim 2nd ^A1, vd2 J^&, Kvaa
5 71.429 Dudminor 2nd vvm2 vvEb Dim 2nd d2 Kaa
6 85.714 Downminor 2nd vm2 vEb Updim 2nd ^d2 K^aa
7 100.000 Minor 2nd m2 Eb Downminor 2nd vm2 Kva
8 114.286 Upminor 2nd ^m2 ^Eb Minor 2nd m2 Ka
9 128.571 Dupminor 2nd ^^m2 ^^Eb Upminor 2nd ^m2 K^a
10 142.857 Trupminor 2nd ^^^m2 ^^^Eb Downmajor 2nd vM2 Kv
11 157.143 Trudmajor 2nd vvvM2 vvvE Major 2nd M2 K
12 171.429 Dudmajor 2nd vvM2 vvE Upmajor 2nd ^M2 K^
13 185.714 Downmajor 2nd vM2 vE Downaug 2nd vA2 Kv&
14 200.000 Major 2nd M2 E Aug 2nd A2 K&
15 214.286 Upmajor 2nd ^M2 ^E Upaug 2nd, Downdim 3rd ^A2, vd3 K^&, Lva
16 228.571 Dupmajor 2nd ^^M2 ^^E Dim 3rd d3 La
17 242.857 Trupmajor 2nd ^^^M2 ^^^E Updim 3rd ^d3 L^a
18 257.143 Trudminor 3rd vvvm3 vvvF Down 3rd v3 Lv
19 271.429 7/6 Dudminor 3rd vvm2 vvF Perfect 3rd P3 L
20 285.714 Downminor 3rd vm3 vF Up 3rd ^3 L^
21 300.000 Minor 3rd m3 F Downaug 3rd vA3 Lv&
22 314.286 Upminor 3rd ^m3 ^F Aug 3rd A3 L&
23 328.571 Dupminor 3rd ^^m3 ^^F Upaug 3rd, Downdim 4th ^A3, vd4 L^&, Mvaa
24 342.857 Trupminor 3rd ^^^m3 ^^^F Dim 4th d4 Maa
25 357.143 Trudmajor 3rd vvvM3 vvvF# Updim 4th ^d4 M^aa
26 371.429 Dudmajor 3rd vvM3 vvF# Downminor 4th vm4 Mva
27 385.714 Downmajor 3rd vM3 vF# Minor 4th m4 Ma
28 400.000 Major 3rd M3 F# Upminor 4th ^m4 M^a
29 414.286 Upmajor 3rd ^M3 ^F# Downmajor 4th vM4 Mv
30 428.571 Dupmajor 3rd ^^M3 ^^F# Major 4th M4 M
31 442.857 Trupmajor 3rd ^^^M3 ^^^F# Upmajor 4th ^M4 M^
32 457.143 Trud 4th vvv4 vvvG Downaug 4th vA4 Mv&
33 471.429 Dud 4th vv4 vvG Aug 4th A4 M&
34 485.714 Down 4th v4 vG Downminor 5th vm5 Nva
35 500.000 Perfect 4th P4 G Minor 5th m5 Na
36 514.286 Up 4th ^4 ^G Upminor 5th ^m5 N^a
37 528.571 Dup 4th ^^4 ^^G Downmajor 5th vM5 Nv
38 542.857 Trup 4th ^^^4 ^^^G Major 5th M5 N
39 557.143 Trudaug 4th vvvA4 vvvG# Upmajor 5th ^M5 N^
40 571.429 Dudaug 4th vvA4 vvG# Downaug 5th vA5 Nv&
41 585.714 Downaug 4th vA4 vG# Aug 5th A5 N&
42 600.000 Aug 4th, Dim 5th A4, d5 G#, Ab Upaug 5th, Downdim 6th ^A5, vd6 N^&, Ovaa
43 614.286 Updim 5th ^d5 ^Ab Dim 6th d6 Oaa
44 628.571 Dupdim 5th ^^d5 ^^Ab Updim 6th ^d6 O^aa
45 642.857 Trupdim 5th ^^^d5 ^^^Ab Downminor 6th vm6 Ova
46 657.143 Trud 5th vvv5 vvvA Minor 6th m6 Oa
47 671.429 Dud 5th vv5 vvA Upminor 6th ^m6 O^a
48 685.714 Down 5th v5 vA Downmajor 6th vM6 Ov
49 700.000 3/2 Perfect 5th P5 A Major 6th M6 O
50 714.286 Up 5th ^5 ^A Upmajor 6th ^M6 O^
51 728.571 Dup 5th ^^5 ^^A Dim 7th d7 Paa
52 742.857 Trup 5th ^^^5 ^^^A Aug 6th A6 O&
53 757.143 Trudminor 6th vvvm6 vvvBb Downminor 7th vm7 Pva
54 771.429 Dudminor 6th vvm6 vvBb Minor 7th m7 Pa
55 785.714 Downminor 6th vm6 vBb Upminor 7th ^m7 P^a
56 800.000 Minor 6th m6 Bb Downmajor 7th vM7 Pv
57 814.286 5/3 Upminor 6th ^m6 ^Bb Major 7th M7 P
58 828.571 Dupminor 6th ^^m6 ^^Bb Upmajor 7th ^M7 P^
59 842.857 Trupminor 6th ^^^m6 ^^^Bb Downaug 7th vA7 Pv&
60 857.143 105/64 Trudmajor 6th vvvM6 vvvB Aug 7th A7 P&
61 871.429 Dudmajor 6th vvM6 vvB Upaug 7th, Downdim 8th ^A7, vd8 P^&, Qvaa
62 885.714 Downmajor 6th vM6 vB Dim 8th d8 Qaa
63 900.000 Major 6th M6 B Updim 8th ^d8 Q^aa
64 914.286 Upmajor 6th ^M6 ^B Down 8th v8 Qva
65 928.571 Dupmajor 6th ^^M6 ^^B Perfect 8th P8 Qa
66 942.857 Trupmajor 6th ^^^M6 ^^^B Up 8th ^8 Q^a
67 957.143 Trudminor 7th vvvm7 vvvC Downaug 8th vA8 Qv
68 971.429 Dudminor 7th vvm7 vvC Aug 8th A8 Q
69 985.714 Downminor 7th vm7 vC Upaug 8th, Downdim 9th ^A8, vd9 Q^, Rvaa
70 1000.000 Minor 7th m7 C Dim 9th d9 Raa
71 1014.286 Upminor 7th ^m7 ^C Updim 9th ^d9 R^aa
72 1028.571 Dupminor 7th ^^m7 ^^C Downminor 9th vm9 Rva
73 1042.857 Trupminor 7th ^^^m7 ^^^C Minor 9th m9 Ra
74 1057.143 Trudmajor 7th vvvM7 vvvC# Upminor 9th ^m9 R^a
75 1071.429 Dudmajor 7th vvM7 vvC# Downmajor 9th vM9 Rv
76 1085.714 Downmajor 7th vM7 vC# Major 9th M9 R
77 1100.000 Major 7th M7 C# Upmajor 9th ^M9 R^
78 1114.286 Upmajor 7th ^M7 ^C# Downaug 9th vA9 Rv&
79 1128.571 Dupmajor 7th ^^M7 ^^C# Aug 9th A9 R&
80 1142.857 Trupmajor 7th ^^^M7 ^^^C# Upaug 9th, Downdim 10th ^A9, vd10 R^&, Jva
81 1157.143 Trud 8ve vvv8 vvvD Dim 10th d10 Ja
82 1171.429 Dud 8ve vv8 vvD Updim 10th ^d10 J^a
83 1185.714 Down 8ve v8 vD Down 10th v10 Jv
84 1200.000 2/1 Perfect 8ve P8 D Perfect 10th P10 J

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
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 19\84 271.43 7/6 Orwell (84e)
Newspeak (84p)
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

Scales

MOS

Brightest mode is listed.

Other

Music

John Cage
Eliora
JUMBLE
Retrieved from "https://en.xen.wiki/index.php?title=84edo&oldid=146486"