981edo: 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| 1555 -981 }}
| {{monzo| 1555 -981 }}
| {{mapping| 981 1555 }}
| {{mapping| 981 1555 }}
| -0.0586
| &minus;0.0586
| 0.0586
| 0.0586
| 4.79
| 4.79
Line 32: Line 24:
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
| {{mapping| 981 1555 2278 }}
| {{mapping| 981 1555 2278 }}
| -0.0722
| &minus;0.0722
| 0.0515
| 0.0515
| 4.21
| 4.21
Line 39: Line 31:
| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
| {{mapping| 981 1555 2278 2754 }}
| {{mapping| 981 1555 2278 2754 }}
| -0.0385
| &minus;0.0385
| 0.0562
| 0.0562
| 4.59
| 4.59
Line 46: Line 38:
| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
| {{mapping| 981 1555 2278 2754 3394 }}
| {{mapping| 981 1555 2278 2754 3394 }}
| -0.0630
| &minus;0.0630
| 0.0545
| 0.0545
| 4.46
| 4.46
Line 53: Line 45:
| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
| {{mapping| 981 1555 2278 2754 3394 3630 }}
| {{mapping| 981 1555 2278 2754 3394 3630 }}
| -0.0453
| &minus;0.0453
| 0.0636
| 0.0636
| 5.20
| 5.20
Line 60: Line 52:
| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
| {{mapping| 981 1555 2278 2754 3394 3630 4010 }}
| {{mapping| 981 1555 2278 2754 3394 3630 4010 }}
| -0.0473
| &minus;0.0473
| 0.0591
| 0.0591
| 4.83
| 4.83
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 80: Line 67:
|-
|-
| 9
| 9
| 258\981<br>(40\981)
| 258\981<br />(40\981)
| 315.60<br>(48.93)
| 315.60<br />(48.93)
| 6/5<br>(36/35)
| 6/5<br />(36/35)
| [[Ennealimmal]] / ennealimmia
| [[Ennealimmal]] / ennealimmia
|}
{{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}}


[[Category:Ennealimmic]]
[[Category:Ennealimmic]]
[[Category:Ennealimmia]]
[[Category:Ennealimmia]]

Revision as of 03:03, 16 November 2024

← 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 409\981 | 500.306 | 8192/6137 | Protolangwidge |- | 9 | 258\981
(40\981) | 315.60
(48.93) | 6/5
(36/35) | Ennealimmal / ennealimmia Template:Rank-2 end Template:Orf

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