80edo: Difference between revisions
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The '''80 equal | The '''80 equal divisions of the octave''' ('''80edo'''), or the '''80(-tone) equal temperament''' ('''80tet''', '''80et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the octave into 80 [[equal]]ly-sized steps. Each step is exactly 15 [[cent]]s. | ||
== Theory == | == Theory == | ||
80et is the first equal temperament that represents the [[19-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits in the 29-prime-limit are consistent, and its patent val generally does well at approximating (29-prime-limited) harmonic series segments, such as modes 16 through 30 but especially modes 8 through 15, with inconsistencies usually caused by not cancelling the over-sharpness of compound harmonics 21, 27, 35, 45 and 49 (and their octave-equivalents), which may be seen as an interesting limitation. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just. This means it can be used as a general-purpose approximate 29-prime-limit system with a relatively manageable number of tones, with some care taken around inconsistency. | 80et is the first equal temperament that represents the [[19-odd-limit]] [[tonality diamond]] [[consistent]]ly, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits in the 29-prime-limit are consistent, and its patent val generally does well at approximating (29-prime-limited) harmonic series segments, such as modes 16 through 30 but especially modes 8 through 15, with inconsistencies usually caused by not cancelling the over-sharpness of compound harmonics 21, 27, 35, 45 and 49 (and their octave-equivalents), which may be seen as an interesting limitation. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just. This means it can be used as a general-purpose approximate 29-prime-limit system with a relatively manageable number of tones, with some care taken around inconsistency. | ||
80et [[ | 80et [[Tempering out|tempers out]] [[2048/2025]], [[3136/3125]], [[1728/1715]], [[4375/4374]] and [[Octagar family|4000/3969]] in the [[7-limit]], [[176/175]], [[540/539]] and [[4000/3993]] in the [[11-limit]], [[169/168]], [[325/324]], [[351/350]], [[352/351]], [[364/363]] and [[Sinbadmic tetrad|1001/1000]] in the [[13-limit]], [[136/135]], [[221/220]], [[256/255]], [[289/288]], 561/560, 595/594, 715/714, [[936/935]] and 1275/1274 in the [[17-limit]], 190/189, 286/285, 361/360, 400/399, 456/455, 476/475, 969/968, 1331/1330, [[1445/1444]], 1521/1520, 1540/1539 and 1729/1728 in the [[19-limit]], 208/207, 253/252, 323/322 and 460/459 in the [[23-limit]] and 320/319 in the [[29-limit]], equating a sharp [[29/16]] with a near-perfect [[20/11]], although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals. | ||
80et provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[canou family]], especially in higher limits. | 80et provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[canou family]], especially in higher limits. | ||
=== Prime harmonics === | |||
{{primes in edo|80|columns=10}} | {{primes in edo|80|columns=10}} | ||