65edo: Difference between revisions

== Theory ==

65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects (which instead elegantly connects primes 7, 13, 37, and 41 to nestoria). Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].

{{Harmonics in equal|65|intervals=prime|columns=15}}

65edo contains [[5edo]] and [[13edo]] as subsets. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].

[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.

== Intervals ==

! [[Cent]]s

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>

! colspan="2" | [[Ups and downs notation]]

| 81/80, 88/87, 93/92, 94/93, 95/94, 96/95, 100/99, 121/120, 115/114, 116/115, 125/124

|-

|-

|-

|-

|-

|-

|-

|-

| 11/10, 32/29

| 10/9, 19/17

| 9/8, 64/57

|-

|-

|-

|-

| 20/17, 27/23, 34/29, 75/64

|-

|-

|-

|-

|-

| 26/21, 47/38, 68/55

|-

| 5/4, 64/51

| 19/15, 24/19, 29/23, 34/27, 81/64

| 23/18, 32/25

| 443.08

| 22/17, 31/24, 40/31, 128/99

| 30/23, 47/36, 72/55

| 29/22, 33/25, 62/47

| 498.46

| 4/3

|-

|-

| 15/11, 34/25, 64/47

| 11/8, 40/29, 62/45

| 25/18, 32/23

| vvG#/vAb

| 24/17, 31/22, 38/27, 45/32

| 17/12, 27/19, 44/31, 64/45

| G#/^Ab

| 627.69

| 36/25, 23/16

| ^A4/v~5

|-

| ~5

|-

| 22/15, 25/17, 47/32

| ^~5

|-

| 683.08

| 34/23, 40/27, 46/31

| 3/2

| 44/29, 50/33, 47/31

| ^5

| 23/15, 55/36, 72/47

| 17/11, 48/31, 31/20, 99/64

| 25/16, 36/23

| 19/12, 27/17, 30/19, 46/29, 128/81

| 812.31

| 8/5, 51/32

| 830.77

| 21/13, 55/34, 76/47

| 849.23

| 18/11, 31/19, 44/27

|-

| 33/20, 38/23

|-

| 886.15

| 5/3, 92/55

| 904.62

| 923.08

| 17/10, 29/17, 46/27, 128/75

| 941.54

| 19/11, 31/18, 50/29, 55/32

| 960.00

| 33/19, 40/23, 47/27, 54/31, 96/55

|-

| 30/17, 44/25, 51/29, 58/33

| 996.92

| 1015.38

| 9/5, 34/19

| 1033.85

| 1052.31

| 11/6, 46/25

| 1070.77

| 1089.23

| 1107.69

| 17/9, 19/10, 36/19, 55/29, 243/128, 256/135

| 1126.15

| 23/12, 44/23, 48/25, 90/47

| 29/15, 31/16, 33/17, 60/31, 64/33

| 45/23, 47/24, 88/45, 92/47, 108/55, 125/64

|-

| 87/55, 93/47, 95/48, 99/50, 115/58, 160/81, 184/93, 188/95, 228/115, 240/121, 248/125

|-

| 2/1

<references group="note" />

== Notation ==

=== Stein–Zimmermann–Gould notation ===

[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:  

{{Sharpness-sharp6-szg}}

 

 

Half-sharps and half-flats can be used to avoid triple arrows:

 

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:

 

 

File:65-EDO_Evo_Sagittal.svg

rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

default [[File:65-EDO_Evo_Sagittal.svg]]

 

==== Revo flavor ====

rect 220 80 340 106 [[33/32]]

default [[File:65-EDO_Revo_Sagittal.svg]]

 

==== Evo-SZ flavor ====

File:65-EDO_Evo-SZ_Sagittal.svg

rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

default [[File:65-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>

 

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

| 243/242, 4000/3993, 5632/5625

 

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

| 5/4

| 99/80<br>(16/15)

| 4/3<br>(81/80)

| 553.85<br>(73.85)

| 11/8<br>(25/24)

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct