63edo: Difference between revisions

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== Theory ==

63edo [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap| 29 & 34d }} temperament in the 7-, 11- and 13-limit.

63edo is almost [[consistent]] to the [[15-odd-limit]]; the only inconsistency is that [[10/9]] is mapped to 9\63 (1\7, the same as what [[11/10]] is mapped to consistently) so that it is almost 11{{cent}} out of tune. This corresponds to 63edo exaggerating the syntonic comma, [[81/80]], to two steps, so that it finds a somewhat flat mean-tone between ~10/9 and ~9/8.

As an equal temperament, it [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap| 29 & 34d }} temperament in the 7-, 11- and 13-limit.

63 is also a fascinating division to look at in the [[47-limit]]. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 [[subgroup]], and is a great candidate for a [[gentle]] tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of [[23/16]], therefore tempering out [[736/729]]. Its diesis (+12 fifths) can represent [[33/32]], [[32/31]], [[30/29]], [[29/28]], [[28/27]], as well as [[91/88]], and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits [[11/9]] into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.

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| 10

| 190.5

| [[29/26]], [[39/35]], [[49/44]]

| [[19/17]], [[29/26]], [[39/35]], [[49/44]]

|-

| 11

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| 53

| 1009.5

| [[52/29]], [[70/39]], [[88/49]]

| [[34/19]], [[52/29]], [[70/39]], [[88/49]]

|-

| 54

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== Notation ==

=== Ups and downs notation ===

63edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[56edo #Sagittal notation|56edo]].

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</imagemap>

=== ~~Ups and downs notation~~ ===

== Approximation to JI ==

~~Using [[Helmholtz–Ellis]] accidentals, 63edo can be notated using [[ups and downs notation]]:~~

=== Interval mappings ===

~~== Approximation to JI ==~~

=== Zeta peak index ===

{~~| class="wikitable center-all"~~

{{ZPI

|-

| zpi = 321

~~! colspan~~=~~"3" | Tuning~~

| steps = 63.0192885705350

~~! colspan="3"~~ | ~~Strength~~

| step size = 19.0417890652143

~~! colspan~~=~~"2" | Closest edo~~

| tempered height = 6.768662

~~! colspan="2" | Integer limit~~

| pure height = 6.534208

|-

| integral = 1.049023

~~! ZPI~~

| gap = 15.412920

~~! Steps per octave~~

| octave = 1199.63271110850

~~! Step size (cents)~~

| consistent = 8

~~! Height~~

| distinct = 8

~~! Integral~~

}}

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[321zpi]]~~

| 63.0192885705350

| 19.0417890652143

| 6.768662

| 1.049023

| 15.412920

| ~~63edo~~

| 1199.63271110850

| 8

|}

== Regular temperament properties ==

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| 26\63<br>(1\63)

| 495.24<br>(19.05)

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

| 4/3<br>(64/63)

| [[Sevond]]

|-

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* Timeywimey (original/default tuning): 16 10 7 4 11 5 10

* Sandcastle (original/default tuning): 8 10 8 11 8 8 10

== Instruments ==

* [[Lumatone mapping for 63edo]]

* [[Skip fretting system 63 3 17]]

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/IYLzF4ogl_w ''microtonal improvisation in 63edo''] (2025)

; [[Cam Taylor]]

* [https://soundcloud.com/cam-taylor-2-1/12tone63edo1 ''Improvisation in 12-tone fifths chain''] (2015)

@@ Line 3: / Line 3: @@
 == Theory ==
-edo [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap| 29 & 34d }} temperament in the 7-, 11- and 13-limit.
+edo is almost [[consistent]] to the [[15-odd-limit]]; the only inconsistency is that [[10/9]] is mapped to 9\63 (1\7, the same as what [[11/10]] is mapped to consistently) so that it is almost 11{{cent}} out of tune. This corresponds to 63edo exaggerating the syntonic comma, [[81/80]], to two steps, so that it finds a somewhat flat mean-tone between ~10/9 and ~9/8.
+As an equal temperament, it [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap| 29 & 34d }} temperament in the 7-, 11- and 13-limit.
 is also a fascinating division to look at in the [[47-limit]]. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 [[subgroup]], and is a great candidate for a [[gentle]] tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of [[23/16]], therefore tempering out [[736/729]]. Its diesis (+12 fifths) can represent [[33/32]], [[32/31]], [[30/29]], [[29/28]], [[28/27]], as well as [[91/88]], and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits [[11/9]] into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.
@@ Line 70: / Line 72: @@
 | 10
 | 190.5
-| [[29/26]], [[39/35]], [[49/44]]
+| [[19/17]], [[29/26]], [[39/35]], [[49/44]]
 |-
 | 11
@@ Line 242: / Line 244: @@
 | 53
 | 1009.5
-| [[52/29]], [[70/39]], [[88/49]]
+| [[34/19]], [[52/29]], [[70/39]], [[88/49]]
 |-
 | 54
@@ Line 563: / Line 565: @@
 == Notation ==
+=== Ups and downs notation ===
+edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
+{{Sharpness-sharp7a}}
+Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
+{{Sharpness-sharp7}}
 === Sagittal notation ===
 This notation uses the same sagittal sequence as [[56edo #Sagittal notation|56edo]].
@@ Line 590: / Line 601: @@
 </imagemap>
-=== Ups and downs notation ===
+== Approximation to JI ==
-Using [[Helmholtz–Ellis]] accidentals, 63edo can be notated using [[ups and downs notation]]:
+=== Interval mappings ===
-{{Sharpness-sharp7}}
+{{Q-odd-limit intervals}}
-== Approximation to JI ==
 === Zeta peak index ===
-{| class="wikitable center-all"
+{{ZPI
-|-
+| zpi = 321
-! colspan="3" | Tuning
+| steps = 63.0192885705350
-! colspan="3" | Strength
+| step size = 19.0417890652143
-! colspan="2" | Closest edo
+| tempered height = 6.768662
-! colspan="2" | Integer limit
+| pure height = 6.534208
-|-
+| integral = 1.049023
-! ZPI
+| gap = 15.412920
-! Steps per octave
+| octave = 1199.63271110850
-! Step size (cents)
+| consistent = 8
-! Height
+| distinct = 8
-! Integral
+}}
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[321zpi]]
-| 63.0192885705350
-| 19.0417890652143
-| 6.768662
-| 1.049023
-| 15.412920
-| 63edo
-| 1199.63271110850
-| 8
-| 8
-|}
 == Regular temperament properties ==
@@ Line 723: / Line 716: @@
 | 26\63<br>(1\63)
 | 495.24<br>(19.05)
-| 4/3<br>(81/80)
+| 4/3<br>(64/63)
 | [[Sevond]]
 |-
@@ Line 738: / Line 731: @@
 * Timeywimey (original/default tuning): 16 10 7 4 11 5 10
 * Sandcastle (original/default tuning): 8 10 8 11 8 8 10
+== Instruments ==
+* [[Lumatone mapping for 63edo]]
+* [[Skip fretting system 63 3 17]]
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/IYLzF4ogl_w ''microtonal improvisation in 63edo''] (2025)
 ; [[Cam Taylor]]
 * [https://soundcloud.com/cam-taylor-2-1/12tone63edo1 ''Improvisation in 12-tone fifths chain''] (2015)