981edo: Difference between revisions

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== 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]]. As an equal temperament, it [[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 ===

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=== Subsets and supersets ===

Since 981 ~~= {{factorization|981}}~~, 981edo has subset edos 3, 9, 109, and 327.

Since 981 factors into primes as 32 × 109, 981edo has subset edos 3, 9, 109, and 327.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{monzo| 1555 -981 }}

| {{mapping| 981 1555 }}

| -0.0586

| −0.0586

| 0.0586

| 4.79

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| {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}

| {{mapping| 981 1555 2278 }}

| -0.0722

| −0.0722

| 0.0515

| 4.21

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| 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}

| {{mapping| 981 1555 2278 2754 }}

| -0.0385

| −0.0385

| 0.0562

| 4.59

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| 2401/2400, 4375/4374, 131072/130977, 1771561/1771470

| {{mapping| 981 1555 2278 2754 3394 }}

| -0.0630

| −0.0630

| 0.0545

| 4.46

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| 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470

| {{mapping| 981 1555 2278 2754 3394 3630 }}

| -0.0453

| −0.0453

| 0.0636

| 5.20

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| 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913

| {{mapping| 981 1555 2278 2754 3394 3630 4010 }}

| -0.0473

| −0.0473

| 0.0591

| 4.83

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

! Periods per 8ve

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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|-

| 9

| 258\981 (40\981)

| 258\981 (40\981)

| 315.60 (48.93)

| 315.60 (48.93)

| 6/5 (36/35)

| 6/5 (36/35)

| [[Ennealimmal]] / ennealimmia

|}

<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

[[Category:Ennealimmic]]

[[Category:Ennealimmia]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|981}}
+{{ED intro}}
 == Theory ==
-edo 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.
+edo is a good 13- and 17-limit system, [[consistent]] to the [[17-odd-limit]]. As an equal temperament, it [[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 ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-Since 981 = {{factorization|981}}, 981edo has subset edos 3, 9, 109, and 327.
+Since 981 factors into primes as 3<sup>2</sup> × 109, 981edo has subset edos 3, 9, 109, and 327.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 25: / Line 26: @@
 | {{monzo| 1555 -981 }}
 | {{mapping| 981 1555 }}
-| -0.0586
+| −0.0586
 | 0.0586
 | 4.79
@@ Line 32: / Line 33: @@
 | {{monzo| 1 -27 18 }}, {{monzo| 85 -17 -25 }}
 | {{mapping| 981 1555 2278 }}
-| -0.0722
+| −0.0722
 | 0.0515
 | 4.21
@@ Line 39: / Line 40: @@
 | 2401/2400, 4375/4374, {{monzo| 79 -25 -23 5 }}
 | {{mapping| 981 1555 2278 2754 }}
-| -0.0385
+| −0.0385
 | 0.0562
 | 4.59
@@ Line 46: / Line 47: @@
 | 2401/2400, 4375/4374, 131072/130977, 1771561/1771470
 | {{mapping| 981 1555 2278 2754 3394 }}
-| -0.0630
+| −0.0630
 | 0.0545
 | 4.46
@@ Line 53: / Line 54: @@
 | 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470
 | {{mapping| 981 1555 2278 2754 3394 3630 }}
-| -0.0453
+| −0.0453
 | 0.0636
 | 5.20
@@ Line 60: / Line 61: @@
 | 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913
 | {{mapping| 981 1555 2278 2754 3394 3630 4010 }}
-| -0.0473
+| −0.0473
 | 0.0591
 | 4.83
@@ Line 67: / Line 68: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 80: / Line 83: @@
 |-
 | 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
 |}
-<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
 [[Category:Ennealimmic]]
 [[Category:Ennealimmia]]