70edo: Difference between revisions

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This tuning was singled out by [[William Stoney]] in his article "Theoretical Possibilities for Equally Tempered Systems" (in the book [https://monoskop.org/images/c/c3/Lincoln_Harry_B_ed_The_Computer_and_Music_1970.pdf The Computer and Music]) as one of the six best systems of size 72 or smaller, along with [[72edo|72]], [[65edo|65]], [[58edo|58]], [[53edo|53]], and [[41edo|41]]. These other systems have had notice paid to them, but the same does not seem to be true of 70, which seems to have been ignored ever since, despite its excellent perfect fifth, which is the 4th number in the convergent sequence to the [[Logarithmic approximants #Argent temperament|silver ratio]], following [[29edo]], [[12edo]], and [[5edo]] and preceding [[169edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth, and this perfect fifth, 41\70, is the true center of the diatonic tuning spectrum, as it is the [[geometric mean]] of [[5edo|3\5edo]] and [[7edo|4\7edo]].

The [[patent val]] for 70edo [[tempering out|tempers out]] [[2048/2025]], making it a [[diaschismic]] system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out [[32805/32768]]. In the [[7-limit]], the patent val tempers out [[126/125]], [[2430/2401]] and [[5120/5103]], and provides the optimum patent val for the [[kumonga]] temperament. The 70c val tempers out [[50/49]], making it a tuning for [[doublewide]] even better than the optimal patent val. The 70cd val tempers out [[225/224]] and [[3125/3087]] instead. The alternative mapping begins to make more sense in the [[11-limit]] and higher, where the patent val tempers out [[99/98]] and [[121/120]] in the 11-limit, [[169/168]] and [[352/351]] in the [[13-limit]], and [[221/220]] in the [[17-limit]]. 70cd on the other hand, with flat 5 and 7, tempers out [[100/99]] and [[245/242]] in the 11-limit, [[105/104]] and [[196/195]] in the 13-limit, and [[154/153]] and [[170/169]] in the 17-limit. 70 also makes sense as a no-5 or -7 system, tempering out [[131769/131072]] in the 11-limit, [[352/351]] and [[2197/2187]] in the 13-limit, and [[289/288]] and [[1089/1088]] in the 17-limit.

The [[patent val]] for 70edo [[tempering out|tempers out]] [[2048/2025]], making it a [[diaschismic]] system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out [[32805/32768]]. In the [[7-limit]], the patent val tempers out [[126/125]], [[2430/2401]] and [[5120/5103]], and provides the optimum patent val for the [[kumonga]] temperament. The 70c val tempers out [[50/49]], making it a tuning for [[doublewide]] even better than the optimal patent val. The 70cd val tempers out [[225/224]] and [[3125/3087]] instead. The alternative mapping begins to make more sense in the [[11-limit]] and higher, where the patent val tempers out [[99/98]] and [[121/120]] in the 11-limit, [[169/168]] and [[352/351]] in the [[13-limit]], and [[221/220]] in the [[17-limit]]. 70cd on the other hand, with flat 5 and 7, tempers out [[100/99]] and [[245/242]] in the 11-limit, [[105/104]] and [[196/195]] in the 13-limit, and [[154/153]] and [[170/169]] in the 17-limit. 70 also makes sense as a no-5 or -7 system, tempering out [[131769/131072]] in the 11-limit, [[352/351]] and [[2197/2187]] in the 13-limit, and [[289/288]] and [[1089/1088]] in the 17-limit. (''See [[regular temperament]] for more about what all this means and how to use it.'')

The 17-limit [[k*N subgroups|2*70]] subgroup, on which 70 is tuned like [[140edo]], is 2.3.25.35.11.13.17.

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 This tuning was singled out by [[William Stoney]] in his article "Theoretical Possibilities for Equally Tempered Systems" (in the book [https://monoskop.org/images/c/c3/Lincoln_Harry_B_ed_The_Computer_and_Music_1970.pdf The Computer and Music]) as one of the six best systems of size 72 or smaller, along with [[72edo|72]], [[65edo|65]], [[58edo|58]], [[53edo|53]], and [[41edo|41]]. These other systems have had notice paid to them, but the same does not seem to be true of 70, which seems to have been ignored ever since, despite its excellent perfect fifth, which is the 4th number in the convergent sequence to the [[Logarithmic approximants #Argent temperament|silver ratio]], following [[29edo]], [[12edo]], and [[5edo]] and preceding [[169edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth, and this perfect fifth, 41\70, is the true center of the diatonic tuning spectrum, as it is the [[geometric mean]] of [[5edo|3\5edo]] and [[7edo|4\7edo]].
-The [[patent val]] for 70edo [[tempering out|tempers out]] [[2048/2025]], making it a [[diaschismic]] system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out [[32805/32768]]. In the [[7-limit]], the patent val tempers out [[126/125]], [[2430/2401]] and [[5120/5103]], and provides the optimum patent val for the [[kumonga]] temperament. The 70c val tempers out [[50/49]], making it a tuning for [[doublewide]] even better than the optimal patent val. The 70cd val tempers out [[225/224]] and [[3125/3087]] instead. The alternative mapping begins to make more sense in the [[11-limit]] and higher, where the patent val tempers out [[99/98]] and [[121/120]] in the 11-limit, [[169/168]] and [[352/351]] in the [[13-limit]], and [[221/220]] in the [[17-limit]]. 70cd on the other hand, with flat 5 and 7, tempers out [[100/99]] and [[245/242]] in the 11-limit, [[105/104]] and [[196/195]] in the 13-limit, and [[154/153]] and [[170/169]] in the 17-limit. 70 also makes sense as a no-5 or -7 system, tempering out [[131769/131072]] in the 11-limit, [[352/351]] and [[2197/2187]] in the 13-limit, and [[289/288]] and [[1089/1088]] in the 17-limit.
+The [[patent val]] for 70edo [[tempering out|tempers out]] [[2048/2025]], making it a [[diaschismic]] system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out [[32805/32768]]. In the [[7-limit]], the patent val tempers out [[126/125]], [[2430/2401]] and [[5120/5103]], and provides the optimum patent val for the [[kumonga]] temperament. The 70c val tempers out [[50/49]], making it a tuning for [[doublewide]] even better than the optimal patent val. The 70cd val tempers out [[225/224]] and [[3125/3087]] instead. The alternative mapping begins to make more sense in the [[11-limit]] and higher, where the patent val tempers out [[99/98]] and [[121/120]] in the 11-limit, [[169/168]] and [[352/351]] in the [[13-limit]], and [[221/220]] in the [[17-limit]]. 70cd on the other hand, with flat 5 and 7, tempers out [[100/99]] and [[245/242]] in the 11-limit, [[105/104]] and [[196/195]] in the 13-limit, and [[154/153]] and [[170/169]] in the 17-limit. 70 also makes sense as a no-5 or -7 system, tempering out [[131769/131072]] in the 11-limit, [[352/351]] and [[2197/2187]] in the 13-limit, and [[289/288]] and [[1089/1088]] in the 17-limit. (''See [[regular temperament]] for more about what all this means and how to use it.'')
 The 17-limit [[k*N subgroups|2*70]] subgroup, on which 70 is tuned like [[140edo]], is 2.3.25.35.11.13.17.