285edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 452 -285 }}
| {{monzo| 452 -285 }}
| {{mapping| 285 452 }}
| {{mapping| 285 452 }}
| -0.3795
| &minus;0.3795
| 0.3794
| 0.3794
| 9.01
| 9.01
Line 32: Line 24:
| 67108864/66430125, {{monzo| -14 -19 19 }}
| 67108864/66430125, {{monzo| -14 -19 19 }}
| {{mapping| 285 452 662 }}
| {{mapping| 285 452 662 }}
| -0.4043
| &minus;0.4043
| 0.3117
| 0.3117
| 7.41
| 7.41
Line 39: Line 31:
| 3136/3125, 5120/5103, 40353607/39858075
| 3136/3125, 5120/5103, 40353607/39858075
| {{mapping| 285 452 662 800 }}
| {{mapping| 285 452 662 800 }}
| -0.2673
| &minus;0.2673
| 0.3596
| 0.3596
| 8.54
| 8.54
Line 46: Line 38:
| 3025/3024, 3136/3125, 5120/5103, 12005/11979
| 3025/3024, 3136/3125, 5120/5103, 12005/11979
| {{mapping| 285 452 662 800 986 }}
| {{mapping| 285 452 662 800 986 }}
| -0.2289
| &minus;0.2289
| 0.3307
| 0.3307
| 7.85
| 7.85
Line 53: Line 45:
| 352/351, 676/675, 847/845, 3025/3024, 12005/11979
| 352/351, 676/675, 847/845, 3025/3024, 12005/11979
| {{mapping| 285 452 662 800 986 1055 }}
| {{mapping| 285 452 662 800 986 1055 }}
| -0.2618
| &minus;0.2618
| 0.3107
| 0.3107
| 7.38
| 7.38
Line 59: Line 51:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 74: Line 60:
|-
|-
| 3
| 3
| 59\285<br>(36\285)
| 59\285<br />(36\285)
| 248.42<br>(151.58)
| 248.42<br />(151.58)
| 15/13<br>(12/11)
| 15/13<br />(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
| 3
| 3
| 59\285<br>(23\285)
| 59\285<br />(23\285)
| 496.84<br>(96.84)
| 496.84<br />(96.84)
| 4/3<br>(256/243)
| 4/3<br />(256/243)
| [[Misty]]
| [[Misty]]
|-
|-
| 19
| 19
| 118\285<br>(2\285)
| 118\285<br />(2\285)
| 496.84<br>(8.42)
| 496.84<br />(8.42)
| 4/3<br>(15625/15552)
| 4/3<br />(15625/15552)
| [[Enneadecal]] (5-limit)
| [[Enneadecal]] (5-limit)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 02:48, 16 November 2024

← 284edo 285edo 286edo →
Prime factorization 3 × 5 × 19
Step size 4.21053 ¢ 
Fifth 167\285 (703.158 ¢)
Semitones (A1:m2) 29:20 (122.1 ¢ : 84.21 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

285edo has a sharp tendency. The equal temperament tempers out the misty comma and the enneadeca in the 5-limit; 3136/3125 and 5120/5103 in the 7-limit; 3025/3024 and 3388/3375 in the 11-limit; 352/351, 676/675, 847/845, 1001/1000, and 2080/2079 in the 13-limit. It supports the 13-limit hemimist temperament.

Prime harmonics

Approximation of prime harmonics in 285edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.20 +1.05 -0.40 +0.26 +1.58 +0.31 +1.43 -0.91 +2.00 +0.23
Relative (%) +0.0 +28.6 +25.0 -9.6 +6.2 +37.5 +7.3 +34.1 -21.5 +47.5 +5.4
Steps
(reduced) 		285
(0) 		452
(167) 		662
(92) 		800
(230) 		986
(131) 		1055
(200) 		1165
(25) 		1211
(71) 		1289
(149) 		1385
(245) 		1412
(272)

Subsets and supersets

Since 285 factors into 3 × 5 × 19, 285edo has subset edos 3, 5, 15, 19, 57, and 95.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [452 -285⟩ | [⟨285 452]] | −0.3795 | 0.3794 | 9.01 |- | 2.3.5 | 67108864/66430125, [-14 -19 19⟩ | [⟨285 452 662]] | −0.4043 | 0.3117 | 7.41 |- | 2.3.5.7 | 3136/3125, 5120/5103, 40353607/39858075 | [⟨285 452 662 800]] | −0.2673 | 0.3596 | 8.54 |- | 2.3.5.7.11 | 3025/3024, 3136/3125, 5120/5103, 12005/11979 | [⟨285 452 662 800 986]] | −0.2289 | 0.3307 | 7.85 |- | 2.3.5.7.11.13 | 352/351, 676/675, 847/845, 3025/3024, 12005/11979 | [⟨285 452 662 800 986 1055]] | −0.2618 | 0.3107 | 7.38 |}

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 109\285 | 458.95 | 125/96 | Majvam |- | 3 | 59\285
(36\285) | 248.42
(151.58) | 15/13
(12/11) | Hemimist |- | 3 | 59\285
(23\285) | 496.84
(96.84) | 4/3
(256/243) | Misty |- | 19 | 118\285
(2\285) | 496.84
(8.42) | 4/3
(15625/15552) | Enneadecal (5-limit) Template:Rank-2 end Template:Orf

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