84edo: Difference between revisions

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=== Higher-limit JI ===

84edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit (with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics). In fact~~, if we avoid all intervals of 11 and 17 as well as the complex compound prime powers [[27/1|27]] and [[49/1|49]]~~, it is ~~completely~~ [[consistent]] in the no-37's no-47's 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~1.3{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too (when supported by context). ~~Except 17, the~~ only missing primes are thus ~~[[37/32|~~37]], [[47~~/32|47]]~~, ~~[[67/64|~~67]], [[71~~/64|71]]~~, ~~[[79/64|~~79]] and [[83~~/64|83]]~~, which ~~coincidentally~~ are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).

84edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is [[consistent]] in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~1.3{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).

== Regular temperament properties ==

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 === Higher-limit JI ===
-edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit (with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics). In fact, if we avoid all intervals of 11 and 17 as well as the complex compound prime powers [[27/1|27]] and [[49/1|49]], it is completely [[consistent]] in the no-37's no-47's 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~1.3{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too (when supported by context). Except 17, the only missing primes are thus [[37/32|37]], [[47/32|47]], [[67/64|67]], [[71/64|71]], [[79/64|79]] and [[83/64|83]], which coincidentally are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).
+edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is [[consistent]] in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~1.3{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).
 == Regular temperament properties ==