487edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|487}}
{{ED intro}}


== Theory ==
== Theory ==
487edo is [[consistency|distinctly consistent]] to the [[13-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 55 -1 -23 }} (counterwürschmidt comma) in the 5-limit, 4375/4374 ([[ragisma]]), 235298/234375 ([[triwellisma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit, [[5632/5625]], [[12005/11979]], [[19712/19683]], [[41503/41472]] in the 11-limit, [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It supports [[semidimfourth]], [[seniority]], and [[vulture]].  
487edo is [[consistency|distinctly consistent]] to the [[13-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 55 -1 -23 }} (counterwürschmidt comma) in the 5-limit, 4375/4374 ([[ragisma]]), 235298/234375 ([[triwellisma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit, [[5632/5625]], [[12005/11979]], [[19712/19683]], [[41503/41472]] in the 11-limit, [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It supports [[semidimfourth]], [[seniority]], and [[vulture]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|487|columns=11}}
{{Harmonics in equal|487}}


=== Subsets and supersets ===
=== Subsets and supersets ===
487edo is the 93rd [[prime edo]].  
487edo is the 93rd [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| {{monzo| 772 -487 }}
| {{monzo| 772 -487 }}
| {{mapping| 487 772 }}
| {{mapping| 487 772 }}
| -0.0958
| −0.0958
| 0.0958
| 0.0958
| 3.89
| 3.89
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| {{monzo| 24 -21 4 }}, {{monzo| 55 -1 -23 }}
| {{monzo| 24 -21 4 }}, {{monzo| 55 -1 -23 }}
| {{mapping| 487 772 1131 }}
| {{mapping| 487 772 1131 }}
| -0.1421
| −0.1421
| 0.1020
| 0.1020
| 4.14
| 4.14
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| 4375/4374, 235298/234375, 33554432/33480783
| 4375/4374, 235298/234375, 33554432/33480783
| {{mapping| 487 772 1131 1367 }}
| {{mapping| 487 772 1131 1367 }}
| -0.0667
| −0.0667
| 0.1577
| 0.1577
| 6.40
| 6.40
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| 4375/4374, 5632/5625, 12005/11979, 41503/41472
| 4375/4374, 5632/5625, 12005/11979, 41503/41472
| {{mapping| 487 772 1131 1367 1685 }}
| {{mapping| 487 772 1131 1367 1685 }}
| -0.0899
| −0.0899
| 0.1485
| 0.1485
| 6.03
| 6.03
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| 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979
| 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979
| {{mapping| 487 772 1131 1367 1685 1802 }}
| {{mapping| 487 772 1131 1367 1685 1802 }}
| -0.0623
| −0.0623
| 0.1490
| 0.1490
| 6.05
| 6.05
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Tritriple]] (5-limit)
| [[Tritriple]] (5-limit)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==

Latest revision as of 23:07, 20 February 2025

← 486edo 487edo 488edo →
Prime factorization 487 (prime)
Step size 2.46407 ¢ 
Fifth 285\487 (702.259 ¢)
Semitones (A1:m2) 47:36 (115.8 ¢ : 88.71 ¢)
Consistency limit 13
Distinct consistency limit 13

487 equal divisions of the octave (abbreviated 487edo or 487ed2), also called 487-tone equal temperament (487tet) or 487 equal temperament (487et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 487 equal parts of about 2.46 ¢ each. Each step represents a frequency ratio of 21/487, or the 487th root of 2.

Theory

487edo is distinctly consistent to the 13-odd-limit. As an equal temperament, it tempers out [24 -21 4⟩ (vulture comma) and [55 -1 -23⟩ (counterwürschmidt comma) in the 5-limit, 4375/4374 (ragisma), 235298/234375 (triwellisma), and 33554432/33480783 (garischisma) in the 7-limit, 5632/5625, 12005/11979, 19712/19683, 41503/41472 in the 11-limit, 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. It supports semidimfourth, seniority, and vulture.

Prime harmonics

Approximation of prime harmonics in 487edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.30 +0.54 -0.45 +0.63 -0.28 +1.00 +0.64 +0.06 +0.40 +0.75
Relative (%) +0.0 +12.3 +22.1 -18.2 +25.7 -11.4 +40.6 +25.9 +2.5 +16.3 +30.6
Steps
(reduced) 		487
(0) 		772
(285) 		1131
(157) 		1367
(393) 		1685
(224) 		1802
(341) 		1991
(43) 		2069
(121) 		2203
(255) 		2366
(418) 		2413
(465)

Subsets and supersets

487edo is the 93rd prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [772 -487⟩ [⟨487 772]] −0.0958 0.0958 3.89
2.3.5 [24 -21 4⟩, [55 -1 -23⟩ [⟨487 772 1131]] −0.1421 0.1020 4.14
2.3.5.7 4375/4374, 235298/234375, 33554432/33480783 [⟨487 772 1131 1367]] −0.0667 0.1577 6.40
2.3.5.7.11 4375/4374, 5632/5625, 12005/11979, 41503/41472 [⟨487 772 1131 1367 1685]] −0.0899 0.1485 6.03
2.3.5.7.11.13 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979 [⟨487 772 1131 1367 1685 1802]] −0.0623 0.1490 6.05

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 131\487 322.79 3087/2560 Seniority
1 157\487 386.86 5/4 Counterwürschmidt
1 182\487 448.46 35/27 Semidimfourth
1 193\487 475.56 320/243 Vulture
1 202\487 497.74 4/3 Gary
1 227\487 559.34 864/625 Tritriple (5-limit)

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

Scales

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