77edo: Difference between revisions

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== Theory ==

With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the {{nowrap|31 &~~amp;~~ 46}} temperament, and [[starling]], the [[~~126/125~~]] [[~~planar temperament~~]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] ~~extensions dwynwen and~~ valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[~~Unicorn~~ family #Alicorn|alicorn]] and 11- and 13-limit [[~~Unicorn~~ family #Camahueto|camahueto]].

With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]] less flat than that, 77edo represents an excellent tuning choice for both [[valentine]] (hence also [[Carlos Alpha]]), the {{nowrap|31 & 46}} temperament, and [[starling]], the [[rank-3 temperament]] [[tempering out]] [[126/125]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extension [[valentino]], as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[unicorn family #Camahueto|camahueto]].

77et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.

77et tempers out the [[schisma]] (32805/32768) in the [[5-limit]]; [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit; [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit; and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.

The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.

~~77edo is an excellent edo for [[Carlos Alpha]], since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only −0.042912 cents.~~

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 77 factors into primes as {{nowrap| 7 × 11 }}, 77edo contains [[7edo]] and [[11edo]] as subset edos.

Since 77 factors into primes as {{nowrap|7 × 11}}, 77edo contains [[7edo]] and [[11edo]] as subset edos.

== Intervals ==

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 == Theory ==
-With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the {{nowrap|31 &amp; 46}} temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].
+With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]] less flat than that, 77edo represents an excellent tuning choice for both [[valentine]] (hence also [[Carlos Alpha]]), the {{nowrap|31 & 46}} temperament, and [[starling]], the [[rank-3 temperament]] [[tempering out]] [[126/125]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extension [[valentino]], as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[unicorn family #Camahueto|camahueto]].
-et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
+et tempers out the [[schisma]] (32805/32768) in the [[5-limit]]; [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit; [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit; and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
 The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.
-edo is an excellent edo for [[Carlos Alpha]], since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only −0.042912 cents.
 === Prime harmonics ===
@@ Line 16: / Line 14: @@
 === Subsets and supersets ===
-Since 77 factors into primes as {{nowrap| 7 × 11 }}, 77edo contains [[7edo]] and [[11edo]] as subset edos.
+Since 77 factors into primes as {{nowrap|7 × 11}}, 77edo contains [[7edo]] and [[11edo]] as subset edos.
 == Intervals ==