981edo: Difference between revisions

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== Theory ==
== Theory ==
981edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. The equal temperament [[Tempering out|tempers out]] [[2080/2079]], [[2401/2400]], 2431/2430, [[4096/4095]], [[4225/4224]], [[4375/4374]], 4459/4455 and 4914/4913 in the 17-limit. It provides the [[optimal patent val]] for 13-limit [[ennealimmic]], the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit [[ennealimmia]], which also tempers out 4096/4095.  
981edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. The equal temperament [[Tempering out|tempers out]] [[2080/2079]], [[2401/2400]], [[2431/2430]], [[4096/4095]], [[4225/4224]], [[4375/4374]], 4459/4455 and [[4914/4913]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit [[ennealimmic]], the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit [[ennealimmia]], which also tempers out 4096/4095.  


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 981 = 3<sup>2</sup> × 109, 981edo has subset edos 3, 9, 109, and 327.
Since 981 = {{factorization|981}}, 981edo has subset edos 3, 9, 109, and 327.


== Regular temperament properties ==
== Regular temperament properties ==
Line 24: Line 24:
| 2.3
| 2.3
| {{monzo| 1555 -981 }}
| {{monzo| 1555 -981 }}
| [{{val| 981 1555 }}]
| {{mapping| 981 1555 }}
| -0.0586
| -0.0586
| 0.0586
| 0.0586
Line 31: Line 31:
| 2.3.5
| 2.3.5
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| [{{val| 981 1555 2278 }}]
| {{mapping| 981 1555 2278 }}
| -0.0722
| -0.0722
| 0.0515
| 0.0515
Line 38: Line 38:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
| [{{val| 981 1555 2278 2754 }}]
| {{mapping| 981 1555 2278 2754 }}
| -0.0385
| -0.0385
| 0.0562
| 0.0562
Line 45: Line 45:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
| [{{val| 981 1555 2278 2754 3394 }}]
| {{mapping| 981 1555 2278 2754 3394 }}
| -0.0630
| -0.0630
| 0.0545
| 0.0545
Line 52: Line 52:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
| [{{val| 981 1555 2278 2754 3394 3630 }}]
| {{mapping| 981 1555 2278 2754 3394 3630 }}
| -0.0453
| -0.0453
| 0.0636
| 0.0636
Line 59: Line 59:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
| [{{val| 981 1555 2278 2754 3394 3630 4010 }}]
| {{mapping| 981 1555 2278 2754 3394 3630 4010 }}
| -0.0473
| -0.0473
| 0.0591
| 0.0591
Line 70: Line 70:
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 77: Line 77:
| 500.306
| 500.306
| 8192/6137
| 8192/6137
| [[Langwidge]] (2.17.19 subgroup)
| [[Protolangwidge]]
|-
|-
| 9
| 9

Revision as of 11:18, 2 November 2023

← 980edo 981edo 982edo →
Prime factorization 32 × 109
Step size 1.22324 ¢ 
Fifth 574\981 (702.141 ¢)
Semitones (A1:m2) 94:73 (115 ¢ : 89.3 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

Theory

981edo is a good 13- and 17-limit system, consistent to the 17-odd-limit. The equal temperament tempers out 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4225/4224, 4375/4374, 4459/4455 and 4914/4913 in the 17-limit. It provides the optimal patent val for 13-limit ennealimmic, the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit ennealimmia, which also tempers out 4096/4095.

Prime harmonics

Approximation of prime harmonics in 981edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.186 +0.231 -0.019 +0.364 -0.161 +0.243 -0.265 +0.472 +0.392 -0.081
Relative (%) +0.0 +15.2 +18.9 -1.5 +29.8 -13.1 +19.9 -21.7 +38.6 +32.1 -6.7
Steps
(reduced) 		981
(0) 		1555
(574) 		2278
(316) 		2754
(792) 		3394
(451) 		3630
(687) 		4010
(86) 		4167
(243) 		4438
(514) 		4766
(842) 		4860
(936)

Subsets and supersets

Since 981 = 32 × 109, 981edo has subset edos 3, 9, 109, and 327.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [1555 -981⟩ [⟨981 1555]] -0.0586 0.0586 4.79
2.3.5 [1 -27 18⟩, [85 -17 -25⟩ [⟨981 1555 2278]] -0.0722 0.0515 4.21
2.3.5.7 2401/2400, 4375/4374, [79 -25 -23 5⟩ [⟨981 1555 2278 2754]] -0.0385 0.0562 4.59
2.3.5.7.11 2401/2400, 4375/4374, 131072/130977, 1771561/1771470 [⟨981 1555 2278 2754 3394]] -0.0630 0.0545 4.46
2.3.5.7.11.13 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470 [⟨981 1555 2278 2754 3394 3630]] -0.0453 0.0636 5.20
2.3.5.7.11.13.17 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913 [⟨981 1555 2278 2754 3394 3630 4010]] -0.0473 0.0591 4.83

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 409\981 500.306 8192/6137 Protolangwidge
9 258\981
(40\981) 		315.60
(48.93) 		6/5
(36/35) 		Ennealimmal / ennealimmia

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

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