487edo: Difference between revisions

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+counterwuerschmidt
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories
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== Theory ==
== Theory ==
487et 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]]. 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]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 24: Line 24:
| 2.3
| 2.3
| {{monzo| 772 -487 }}
| {{monzo| 772 -487 }}
| [{{val| 487 772 }}]
| {{mapping| 487 772 }}
| -0.0958
| -0.0958
| 0.0958
| 0.0958
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| 2.3.5
| 2.3.5
| {{monzo| 24 -21 4 }}, {{monzo| 55 -1 -23 }}
| {{monzo| 24 -21 4 }}, {{monzo| 55 -1 -23 }}
| [{{val| 487 772 1131 }}]
| {{mapping| 487 772 1131 }}
| -0.1421
| -0.1421
| 0.1020
| 0.1020
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, 235298/234375, 33554432/33480783
| 4375/4374, 235298/234375, 33554432/33480783
| [{{val| 487 772 1131 1367 }}]
| {{mapping| 487 772 1131 1367 }}
| -0.0667
| -0.0667
| 0.1577
| 0.1577
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| 2.3.5.7.11
| 2.3.5.7.11
| 4375/4374, 5632/5625, 12005/11979, 41503/41472
| 4375/4374, 5632/5625, 12005/11979, 41503/41472
| [{{val| 487 772 1131 1367 1685 }}]
| {{mapping| 487 772 1131 1367 1685 }}
| -0.0899
| -0.0899
| 0.1485
| 0.1485
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979
| 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979
| [{{val| 487 772 1131 1367 1685 1802 }}]
| {{mapping| 487 772 1131 1367 1685 1802 }}
| -0.0623
| -0.0623
| 0.1490
| 0.1490
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 103: Line 103:
| [[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


== Scales ==
== Scales ==
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* [[Silver17]]
* [[Silver17]]


[[Category:487edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Silver]]
[[Category:Silver]]

Revision as of 06:14, 3 November 2023

← 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

Template:EDO intro

Theory

487edo is distinctly consistent to the 13-odd-limit. The equal temperament 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 it is distinct

Scales

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