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=== Higher-limit JI ===
=== 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 ~0.13{{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 ==
== Regular temperament properties ==
Line 769: Line 769:
| 2.3.5
| 2.3.5
| 78732/78125, 531441/524288
| 78732/78125, 531441/524288
| {{mapping| 84 133 195 }}
| {{Mapping| 84 133 195 }}
| +0.498
| +0.498
| 0.531
| 0.531
Line 776: Line 776:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 78732/78125
| 225/224, 1728/1715, 78732/78125
| {{mapping| 84 133 195 236 }}
| {{Mapping| 84 133 195 236 }}
| +0.141
| +0.141
| 0.769
| 0.769
Line 783: Line 783:
| 2.3.5.7.13
| 2.3.5.7.13
| 225/224, 351/350, 640/637, 1701/1690
| 225/224, 351/350, 640/637, 1701/1690
| {{mapping| 84 133 195 236 311 }}
| {{Mapping| 84 133 195 236 311 }}
| −0.013
| −0.013
| 0.754
| 0.754
Line 790: Line 790:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 441/440, 1344/1331, 1728/1715
| 225/224, 441/440, 1344/1331, 1728/1715
| {{mapping| 84 133 195 236 291 }} (84)
| {{Mapping| 84 133 195 236 291 }} (84)
| −0.225
| −0.225
| 1.003
| 1.003
Line 797: Line 797:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 144/143, 225/224, 351/350, 441/440, 975/968
| 144/143, 225/224, 351/350, 441/440, 975/968
| {{mapping| 84 133 195 236 291 311 }} (84)
| {{Mapping| 84 133 195 236 291 311 }} (84)
| −0.292
| −0.292
| 0.928
| 0.928
Line 804: Line 804:
| 2.3.5.7.11
| 2.3.5.7.11
| 99/98, 121/120, 176/175, 78732/78125
| 99/98, 121/120, 176/175, 78732/78125
| {{mapping| 84 133 195 236 290 }} (84e)
| {{Mapping| 84 133 195 236 290 }} (84e)
| +0.601
| +0.601
| 1.151
| 1.151
Line 811: Line 811:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 99/98, 121/120, 176/175, 275/273, 1701/1690
| 99/98, 121/120, 176/175, 275/273, 1701/1690
| {{mapping| 84 133 195 236 290 311 }} (84e)
| {{Mapping| 84 133 195 236 290 311 }} (84e)
| +0.396
| +0.396
| 1.146
| 1.146
Line 829: Line 829:
| 1
| 1
| 19\84
| 19\84
| 271.43
| 271.4
| 7/6
| 7/6
| [[Orwell]] (84e) / [[newspeak]] (84)
| [[Orwell]] (84e) / [[newspeak]] (84)
Line 835: Line 835:
| 1
| 1
| 25\84
| 25\84
| 357.14
| 357.1
| 768/625
| 768/625
| [[Dodifo]]
| [[Dodifo]]
Line 841: Line 841:
| 1
| 1
| 27\84
| 27\84
| 385.71
| 385.7
| 5/4
| 5/4
| [[Mutt]]
| [[Mutt]]
Line 847: Line 847:
| 1
| 1
| 31\84
| 31\84
| 442.86
| 442.9
| 125/81
| 162/125
| [[Sensei]]
| [[Sensei]]
|-
|-
| 1
| 1
| 41\84
| 41\84
| 585.71
| 585.7
| 7/5
| 7/5
| [[Merman]]
| [[Merman]]
Line 859: Line 859:
| 2
| 2
| 5\84
| 5\84
| 71.43
| 71.4
| 25/24
| 25/24
| [[Narayana]]
| [[Narayana]]
Line 865: Line 865:
| 2
| 2
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Bison]]
| [[Bison]]
Line 871: Line 871:
| 2
| 2
| 13\84
| 13\84
| 185.71
| 185.7
| 10/9
| 10/9
| [[Secant]]
| [[Secant]]
Line 877: Line 877:
| 3
| 3
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Nessafof]]
| [[Nessafof]]
|-
| 6
| 5\84
| 71.4
| 25/24
| [[Trivish]]
|-
|-
| 7
| 7
| 5\84
| 5\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(81/80)
| 81/80
| [[Absurdity]]
| [[Absurdity]]
|-
|-
| 12
| 12
| 27\84<br>(1\84)
| 1\84
| 385.71<br>(14.29)
| 14.3
| 5/4<br>(126/125)
| 126/125
| [[Compton]]
| [[Compton]]
|-
| 12
| 2\84
| 28.6
| 64/63
| [[Catler]] (84c)
|-
|-
| 21
| 21
| 41\84<br>(1\84)
| 1\84)
| 585.71<br>(14.29)
| 14.3
| 91875/65536<br>(126/125)
| 126/125
| [[Akjayland]]
| [[Akjayland]]
|-
|-
| 28
| 28
| 49\84<br>(1\84)
| 1\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(105/104)
| 105/104
| [[Oquatonic]]
| [[Oquatonic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== Scales ==
Line 935: Line 947:
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025)
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025)
* [https://www.youtube.com/shorts/Qu6UIA2NmmQ ''84edo groove''] (2026)


; [[Eliora]]
; [[Eliora]]
Retrieved from "https://en.xen.wiki/w/84edo"