80edo: Difference between revisions

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== Theory ==

80edo is the first edo 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-limit|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. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just, with inconsistencies usually arising through not cancelling the over-sharpness of compound harmonics [[21/1|21]], [[27/1|27]], [[35/1|35]], [[45/1|45]] and [[49/1|49]] (and their octave-equivalents), which may be seen as an interesting limitation. 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. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's [[41-limit]]. If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].

80edo is the first edo 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-limit|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. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just, with inconsistencies usually arising through not cancelling the over-sharpness of compound harmonics [[21/1|21]], [[27/1|27]], [[35/1|35]], [[45/1|45]] and [[49/1|49]] (and their octave-equivalents), which may be seen as an interesting limitation. 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. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies. If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].

80et [[Tempering out|tempers out]] [[2048/2025]] in the 5-limit; [[1728/1715]], [[3136/3125]], [[4000/3969]], and [[4375/4374]] 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 [[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]]. The last comma is notable as it equates 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. It 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.

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 == Theory ==
-edo is the first edo 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-limit|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. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just, with inconsistencies usually arising through not cancelling the over-sharpness of compound harmonics [[21/1|21]], [[27/1|27]], [[35/1|35]], [[45/1|45]] and [[49/1|49]] (and their octave-equivalents), which may be seen as an interesting limitation. 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. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's [[41-limit]]. If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].
+edo is the first edo 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-limit|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. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just, with inconsistencies usually arising through not cancelling the over-sharpness of compound harmonics [[21/1|21]], [[27/1|27]], [[35/1|35]], [[45/1|45]] and [[49/1|49]] (and their octave-equivalents), which may be seen as an interesting limitation. 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. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's [[41-limit]], or even the [[43-limit]] if you are fine with [[43/32]] being slightly flat causing more inconsistencies. If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent [[Ringer scale]] described in the [[#Ringer 80|Ringer 80 section of this article]].
 et [[Tempering out|tempers out]] [[2048/2025]] in the 5-limit; [[1728/1715]], [[3136/3125]], [[4000/3969]], and [[4375/4374]] 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 [[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]]. The last comma is notable as it equates 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. It 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.