82edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
82edo's [[patent val]] is [[contorted]] in the [[11-limit]], | 82edo's [[patent val]] is [[contorted]] in the [[11-limit]], since {{nowrap|82 {{=}} 2 × 41}}. In the [[13-limit]] the patent val tempers out [[169/168]] and [[676/675]], and in the [[17-limit]] tempers out [[273/272]]. It provides the optimal patent val for [[soothsaying]] temperament and [[support]]s [[baladic]] temperament. The 82d val tempers out [[50/49]] and is an excellent tuning for [[astrology]] and [[byhearted]], surpassing their optimal patent vals. The alternative 82e val tempers out [[121/120]] instead. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|82}} | {{Harmonics in equal|82}} | ||
=== Subsets and supersets === | |||
82edo contains [[2edo]] and [[41edo]] as subsets. [[164edo]], which doubles it, is a notable tuning. | |||
A step of 82edo is exactly 30 [[mina]]s. | |||
== Intervals == | == Intervals == | ||
{| class="wikitable right-1 right-2 left-3 left-4 left-5" | {| class="wikitable right-1 right-2 left-3 left-4 left-5" | ||
| | |- | ||
! # | ! rowspan="2" | # | ||
! Cents | ! rowspan="2" | Cents | ||
! | ! rowspan="2" | Approximate ratios* | ||
! Additional | ! colspan="2" | Additional ratios | ||
! | |- | ||
! Using the 82e val | |||
! Using the patent val | |||
|- | |- | ||
| 0 | | 0 | ||
| Line 25: | Line 32: | ||
| 1 | | 1 | ||
| 14.634 | | 14.634 | ||
| | | ''65/64'', 91/90 | ||
| | | ''55/54'' | ||
| | | | ||
|- | |- | ||
| 2 | | 2 | ||
| 29.268 | | 29.268 | ||
| 49/48, 50/49, ''81/80'', ''126/125'' | |||
| | | | ||
| | | 45/44, 55/54 | ||
|- | |- | ||
| 3 | | 3 | ||
| 43.902 | | 43.902 | ||
| | | 40/39 | ||
| | | ''33/32'', ''45/44'' | ||
| | | | ||
|- | |- | ||
| 4 | | 4 | ||
| 58.537 | | 58.537 | ||
| ''25/24'', 28/27, ''36/35'' | |||
| | | | ||
| | | 33/32 | ||
|- | |- | ||
| 5 | | 5 | ||
| 73.171 | | 73.171 | ||
| | | 26/25, ''27/26'' | ||
| 22/21 | | 22/21 | ||
| | | | ||
| Line 55: | Line 62: | ||
| 6 | | 6 | ||
| 87.805 | | 87.805 | ||
| | | 19/18, 20/19, 21/20 | ||
| | | | ||
| 22/21 | | ''22/21'' | ||
|- | |- | ||
| 7 | | 7 | ||
| 102.439 | | 102.439 | ||
| 18/17 | | 17/16, 18/17 | ||
| | | | ||
| | | | ||
| Line 67: | Line 74: | ||
| 8 | | 8 | ||
| 117.073 | | 117.073 | ||
| 16/15 | | 15/14, 16/15 | ||
| | | | ||
| | | | ||
| Line 86: | Line 93: | ||
| 160.976 | | 160.976 | ||
| | | | ||
| 12/11 | | 11/10, ''12/11'' | ||
| | | | ||
|- | |- | ||
| Line 93: | Line 100: | ||
| 10/9, 21/19 | | 10/9, 21/19 | ||
| | | | ||
| 11/10 | | ''11/10'' | ||
|- | |- | ||
| 13 | | 13 | ||
| Line 129: | Line 136: | ||
| 7/6 | | 7/6 | ||
| | | | ||
| 22/19 | | ''22/19'' | ||
|- | |- | ||
| 19 | | 19 | ||
| Line 135: | Line 142: | ||
| 20/17 | | 20/17 | ||
| | | | ||
| 13/11 | | ''13/11'' | ||
|- | |- | ||
| 20 | | 20 | ||
| Line 158: | Line 165: | ||
| 336.585 | | 336.585 | ||
| 17/14 | | 17/14 | ||
| 11/9 | | ''11/9'' | ||
| | | | ||
|- | |- | ||
| Line 187: | Line 194: | ||
| 28 | | 28 | ||
| 409.756 | | 409.756 | ||
| 24/19 | | 19/15, 24/19 | ||
| | | | ||
| 14/11 | | ''14/11'' | ||
|- | |- | ||
| 29 | | 29 | ||
| Line 200: | Line 207: | ||
| 439.024 | | 439.024 | ||
| 9/7 | | 9/7 | ||
| 22/17 | | ''22/17'' | ||
| | | | ||
|- | |- | ||
| Line 237: | Line 244: | ||
| 19/14 | | 19/14 | ||
| | | | ||
| 15/11 | | ''15/11'' | ||
|- | |- | ||
| 37 | | 37 | ||
| 541.463 | | 541.463 | ||
| 26/19 | | 26/19 | ||
| 15/11 | | ''11/8'', 15/11 | ||
| | | | ||
|- | |- | ||
| Line 253: | Line 260: | ||
| 39 | | 39 | ||
| 570.732 | | 570.732 | ||
| 18/13 | | ''18/13'' | ||
| | | | ||
| | | | ||
| Line 265: | Line 272: | ||
| 41 | | 41 | ||
| 600.000 | | 600.000 | ||
| 24/17 | | 17/12, 24/17 | ||
| | | | ||
| | | | ||
| Line 275: | Line 282: | ||
| | | | ||
|} | |} | ||
<nowiki />* As a no-11 19-limit temperament | |||
== Notation == | |||
=== Ups and downs notation === | |||
60edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals: | |||
{{Sharpness-sharp8}} | |||
== Approximation to JI == | |||
=== Zeta peak index === | |||
{{ZPI | |||
| zpi = 448 | |||
| steps = 81.9541455954050 | |||
| step size = 14.6423343356444 | |||
| tempered height = 6.653983 | |||
| pure height = 5.154524 | |||
| integral = 0.941321 | |||
| gap = 14.718732 | |||
| octave = 1200.67141552284 | |||
| consistent = 8 | |||
| distinct = 8 | |||
}} | |||
== Instruments == | |||
* [[Lumatone mapping for 82edo]] | |||
== Music == | |||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/p9xUY8EU7Zg ''microtonal improvisation in 82edo''] (2025) | |||