46edo: Difference between revisions

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== Theory ==

In the opinion of some, 46edo is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]] or [[53edo]]. 46edo's fifth is slightly sharp of just, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], but it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap edo]]. It is distinctly [[consistent]] in the [[9-odd-limit]], and it is consistent to the [[13-odd-limit]] or the no-15 no-19 [[23-odd-limit]].

[[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)1/10, is only 0.01 cents flat of 3\46 octaves.

46edo is also notable for being the smallest equal division to approximate harmonics 3, 5, 7, 11, and 13 with less than 25% [[relative interval error]].

[[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]], and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)1/10, is only 0.01 cents flat of 3\46 octaves.

[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.

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! Approximate ratios<ref name="interval ratios" group="note">{{sg|limit=2.3.5.7.11.13.17.23 subgroup}} However, ratios of 15 are not included here, as except for 15/8 and 16/15 themselves 46edo has intervals involving the 15th harmonic poorly approximated in general. This is because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being [[consistent]] in the [[15-odd-limit]], as there is a discrepancy approximating [[15/13]] and [[26/15]]—9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.</ref>

! colspan="3" | [[Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: v5A1 and ^^d2)

! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)

! colspan="2" | [[Solfege]]

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| downminor

| zo

| {a, b, 0, 1}

| [a, b, 0, 1>

| 7/6, 7/4

|-

| minor

| fourthward wa

| {a, b}, b ~~< -1~~

| [a, b>, b < −1

| 32/27, 16/9

|-

| upminor

| gu

| {a, b, ~~-1}~~

| [a, b, −1>

| 6/5, 9/5

|-

| dupminor

| ilo

| {a, b, 0, 0, 1}

| [a, b, 0, 0, 1>

| 11/9, 11/6

|-

| dudmajor

| lu

| {a, b, 0, 0, ~~-1}~~

| [a, b, 0, 0, −1>

| 12/11, 18/11

|-

| downmajor

| yo

| {a, b, 1}

| [a, b, 1>

| 5/4, 5/3

|-

| major

| fifthward wa

| {a, b}, b > 1

| [a, b>, b > 1

| 9/8, 27/16

|-

| upmajor

| ru

| {a, b, 0, ~~-1}~~

| [a, b, 0, −1>

| 9/7, 12/7

|}

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| zo

| 6:7:9

| ~~0-10-27~~

| 0–10–27

| C vEb G

| Cvm

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| gu

| 10:12:15

| ~~0-12-27~~

| 0–12–27

| C ^Eb G

| C^m

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| ilo

| 18:22:27

| ~~0-13-27~~

| 0–13–27

| C ^^Eb G

| C^^m

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| lu

| 22:27:33

| ~~0-14-27~~

| 0–14–27

| C vvE G

| Cvv

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| yo

| 4:5:6

| ~~0-15-27~~

| 0–15–27

| C vE G

| Cv

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| ru

| 14:18:21

| ~~0-17-27~~

| 0–17–27

| C ^E G

| C^

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For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see [[icositritonic]].

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 214~~

~~| steps = 46.0089748051542~~

~~| step size = 26.0818678330031~~

~~| tempered height = 7.495674~~

~~| pure height = 7.451949~~

~~| integral = 1.356067~~

~~| gap = 17.747832~~

~~| octave = 1199.76592031814~~

~~| consistent = 14~~

~~| distinct = 11~~

}}

== Regular temperament properties ==

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| [[Rodan]]

| [[1L 4s]] (5-tone) [[1L 5s]] (6-tone) [[5L 6s]] (11-tone) 5L 11s (16-tone) 5L 16s (21-tone) 5L 21s (26-tone) 5L 26s (31-tone) 5L 31s (36-tone) 5L 36s (41-tone)

| 10:9 ~QE 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE~~, Pathological~~

| 10:9 ~QE 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE

|-

| 11\46

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| [[Amity]] / [[hitchcock]]

| [[4L 3s]] (7-tone) [[7L 4s]] (11-tone) 7L 11s (18-tone) 7L 18s (25-tone) 7L 25s (32-tone) 7L 32s (39-tone)

| 7:6 ~ QE 6:1 5:1 4:1 3:1 2:1 ~ QE~~! Pathological~~

| 7:6 ~ QE 6:1 5:1 4:1 3:1 2:1 ~ QE

|-

| 15\46

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| [[Magus]] / [[amigo]]

| [[1L 2s]] (3-tone) [[3L 1s]] (4-tone) [[3L 4s]] (7-tone) [[3L 7s]] (10-tone) 3L 10s (13-tone) 3L 13s (16-tone) 3L 16s (19-tone) 3L 19s (21-tone) 3L 21s (24-tone) 3L 24s (27-tone) 3L 27s (30-tone) 3L 30s (33-tone) 3L 33s (36-tone) 3L 36s (39-tone) 3L 39s (42-tone)

| 16:15 ~ QE 15:1 14:1 13:1 12:1 11:1 10:1 9:1 8:1 7:1 6:1 5:1 4:1 3:1 ~~~ Pathological~~ 2:1 ~ QE~~, Pathological~~

| 16:15 ~ QE 15:1 14:1 13:1 12:1 11:1 10:1 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE

|-

| 17\46

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| [[Bison]]

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[6L 2s]] (8-tone) 8L 6s (14-tone) 8L 14s (22-tone) 8L 22s (30-tone) 8L 30s (38-tone

| 17:6 11:6 6:5 ~ QE 5:1 4:1 3:1 2:1 ~ QE~~, Pathological~~

| 17:6 11:6 6:5 ~ QE 5:1 4:1 3:1 2:1 ~ QE

|-

| 7\46

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| [[Abigail]]

| 2L 2s (4-tone) [[4L 2s]] (6-tone) [[6L 2s]] (8-tone) 6L 8s (14-tone) 6L 14s (20-tone) 6L 20s (26-tone) 6L 26s (32-tone) 6L 32s (38-tone) 6L 38s (44-tone)

| 15:8 8:7 ~ QE 8:1 7:1 6:1 5:1 4:1 3:1 ~~~ Pathological~~ 2:1 ~ QE~~, Pathological~~

| 15:8 8:7 ~ QE 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE

|-

| 9\46

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| [[Vines]]

| 2L 2s (4-tone) [[4L 2s]] (6-tone) [[4L 6s]] (10-tone) 4L 10s (14-tone) 4L 14s (18-tone) 4L 18s (22-tone) 4L 22s (26-tone) 4L 26s (30-tone) 4L 30s (34-tone) 4L 34s (38-tone) 4L 38s (42-tone)

| 12:11 ~ QE 11:1 10:1 9:1 8:1 7:1 6:1 5:1 4:1 3:1 ~~~ Pathological~~ 2:1 ~ QE~~, Pathological~~

| 12:11 ~ QE 11:1 10:1 9:1 8:1 7:1 6:1 5:1 4:1 3:1 2:1 ~ QE

|-

| 23

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== Instruments ==

* [[Skip fretting system 46 2 11]]: A skip-fretting system for playing 46-edo on a 23-edo stringed instrument.

=== Lumatone ===

* [[Skip fretting system 46 7 11]]

* [[Lumatone mapping for 46edo]]

=== Skip fretting ===

'''[[Skip fretting system 46 2 11]]''' is a [[skip fretting]] system for playing 46-edo on a 23-edo stringed instrument.

'''Skip fretting system 46 7 11''' is another skip fretting system for 46edo. The examples on this page are for 7-string [[guitar]].

; Harmonics

1/1: string 2 open

2/1: string 3 fret 5

3/2: not easily accessible

5/4: string 5 fret 4

== Music ==

=== Modern renditions ===

; {{W|Johann Sebastian Bach}}

* [https://www.youtube.com/watch?v=wMEdFl2puL0 "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)

* [https://www.youtube.com/watch?v=4yetEubmIk0 "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

@@ Line 9: / Line 9: @@
 == Theory ==
 In the opinion of some, 46edo is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]] or [[53edo]]. 46edo's fifth is slightly sharp of just, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], but it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap edo]]. It is distinctly [[consistent]] in the [[9-odd-limit]], and it is consistent to the [[13-odd-limit]] or the no-15 no-19 [[23-odd-limit]].
-[[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves.
+edo is also notable for being the smallest equal division to approximate harmonics 3, 5, 7, 11, and 13 with less than 25% [[relative interval error]].
+[[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]], and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves.
 [[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.
@@ Line 29: / Line 31: @@
 ! Approximate ratios<ref name="interval ratios" group="note">{{sg|limit=2.3.5.7.11.13.17.23&nbsp;subgroup}} However, ratios of 15 are not included here, as except for 15/8 and 16/15 themselves 46edo has intervals involving the 15th harmonic poorly approximated in general. This is because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being [[consistent]] in the [[15-odd-limit]], as there is a discrepancy approximating [[15/13]] and [[26/15]]&mdash;9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.</ref>
 ! colspan="3" | [[Ups and downs notation]]
+([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>5</sup>A1 and ^^d2)
 ! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
 ! colspan="2" | [[Solfege]]
@@ Line 611: / Line 614: @@
 | downminor
 | zo
-| {a, b, 0, 1}
+| [a, b, 0, 1>
 | 7/6, 7/4
 |-
 | minor
 | fourthward wa
-| {a, b}, b < -1
+| [a, b>, b &lt; −1
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| {a, b, -1}
+| [a, b, −1>
 | 6/5, 9/5
 |-
 | dupminor
 | ilo
-| {a, b, 0, 0, 1}
+| [a, b, 0, 0, 1>
 | 11/9, 11/6
 |-
 | dudmajor
 | lu
-| {a, b, 0, 0, -1}
+| [a, b, 0, 0, −1>
 | 12/11, 18/11
 |-
 | downmajor
 | yo
-| {a, b, 1}
+| [a, b, 1>
 | 5/4, 5/3
 |-
 | major
 | fifthward wa
-| {a, b}, b > 1
+| [a, b>, b &gt; 1
 | 9/8, 27/16
 |-
 | upmajor
 | ru
-| {a, b, 0, -1}
+| [a, b, 0, −1>
 | 9/7, 12/7
 |}
@@ Line 662: / Line 665: @@
 | zo
 | 6:7:9
-| 0-10-27
+| 0–10–27
 | C vEb G
 | Cvm
@@ Line 669: / Line 672: @@
 | gu
 | 10:12:15
-| 0-12-27
+| 0–12–27
 | C ^Eb G
 | C^m
@@ Line 676: / Line 679: @@
 | ilo
 | 18:22:27
-| 0-13-27
+| 0–13–27
 | C ^^Eb G
 | C^^m
@@ Line 683: / Line 686: @@
 | lu
 | 22:27:33
-| 0-14-27
+| 0–14–27
 | C vvE G
 | Cvv
@@ Line 690: / Line 693: @@
 | yo
 | 4:5:6
-| 0-15-27
+| 0–15–27
 | C vE G
 | Cv
@@ Line 697: / Line 700: @@
 | ru
 | 14:18:21
-| 0-17-27
+| 0–17–27
 | C ^E G
 | C^
@@ Line 775: / Line 778: @@
 For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see [[icositritonic]].
-=== Zeta peak index ===
-{{ZPI
-| zpi = 214
-| steps = 46.0089748051542
-| step size = 26.0818678330031
-| tempered height = 7.495674
-| pure height = 7.451949
-| integral = 1.356067
-| gap = 17.747832
-| octave = 1199.76592031814
-| consistent = 14
-| distinct = 11
-}}
 == Regular temperament properties ==
@@ Line 1,080: / Line 1,069: @@
 | [[Rodan]]
 | [[1L&nbsp;4s]] (5-tone)<br>[[1L&nbsp;5s]] (6-tone)<br>[[5L&nbsp;6s]] (11-tone)<br>5L&nbsp;11s (16-tone)<br>5L&nbsp;16s (21-tone)<br>5L&nbsp;21s (26-tone)<br>5L&nbsp;26s (31-tone)<br>5L&nbsp;31s (36-tone)<br>5L&nbsp;36s (41-tone)
-| 10:9 ~QE<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE, Pathological
+| 10:9 ~QE<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
 | 11\46
@@ Line 1,092: / Line 1,081: @@
 | [[Amity]] / [[hitchcock]]
 | [[4L&nbsp;3s]] (7-tone)<br>[[7L&nbsp;4s]] (11-tone)<br>7L&nbsp;11s (18-tone)<br>7L&nbsp;18s (25-tone)<br>7L&nbsp;25s (32-tone)<br>7L&nbsp;32s (39-tone)
-| 7:6 ~ QE<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE! Pathological
+| 7:6 ~ QE<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
 | 15\46
@@ Line 1,098: / Line 1,087: @@
 | [[Magus]] / [[amigo]]
 | [[1L&nbsp;2s]] (3-tone)<br>[[3L&nbsp;1s]] (4-tone)<br>[[3L&nbsp;4s]] (7-tone)<br>[[3L&nbsp;7s]] (10-tone)<br>3L&nbsp;10s (13-tone)<br>3L&nbsp;13s (16-tone)<br>3L&nbsp;16s (19-tone)<br>3L&nbsp;19s (21-tone)<br>3L&nbsp;21s (24-tone)<br>3L&nbsp;24s (27-tone)<br>3L&nbsp;27s (30-tone)<br>3L&nbsp;30s (33-tone)<br>3L&nbsp;33s (36-tone)<br>3L&nbsp;36s (39-tone)<br>3L&nbsp;39s (42-tone)
-| 16:15 ~ QE<br>15:1<br>14:1<br>13:1<br>12:1<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
+| 16:15 ~ QE<br>15:1<br>14:1<br>13:1<br>12:1<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
 | 17\46
@@ Line 1,153: / Line 1,142: @@
 | [[Bison]]
 | 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[6L&nbsp;2s]] (8-tone)<br>8L&nbsp;6s (14-tone)<br>8L&nbsp;14s (22-tone)<br>8L&nbsp;22s (30-tone)<br>8L&nbsp;30s (38-tone
-| 17:6<br>11:6<br>6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE, Pathological
+| 17:6<br>11:6<br>6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
 | 7\46
@@ Line 1,165: / Line 1,154: @@
 | [[Abigail]]
 | 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[6L&nbsp;2s]] (8-tone)<br>6L&nbsp;8s (14-tone)<br>6L&nbsp;14s (20-tone)<br>6L&nbsp;20s (26-tone)<br>6L&nbsp;26s (32-tone)<br>6L&nbsp;32s (38-tone)<br>6L&nbsp;38s (44-tone)
-| 15:8<br>8:7 ~ QE<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
+| 15:8<br>8:7 ~ QE<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
 | 9\46
@@ Line 1,183: / Line 1,172: @@
 | [[Vines]]
 | 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[4L&nbsp;6s]] (10-tone)<br>4L&nbsp;10s (14-tone)<br>4L&nbsp;14s (18-tone)<br>4L&nbsp;18s (22-tone)<br>4L&nbsp;22s (26-tone)<br>4L&nbsp;26s (30-tone)<br>4L&nbsp;30s (34-tone)<br>4L&nbsp;34s (38-tone)<br>4L&nbsp;38s (42-tone)
-| 12:11 ~ QE<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
+| 12:11 ~ QE<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
 | 23
@@ Line 1,261: / Line 1,250: @@
 == Instruments ==
-* [[Skip fretting system 46 2 11]]: A skip-fretting system for playing 46-edo on a 23-edo stringed instrument.
+=== Lumatone ===
-* [[Skip fretting system 46 7 11]]
 * [[Lumatone mapping for 46edo]]
+=== Skip fretting ===
+'''[[Skip fretting system 46 2 11]]''' is a [[skip fretting]] system for playing 46-edo on a 23-edo stringed instrument.
+'''Skip fretting system 46 7 11''' is another skip fretting system for 46edo. The examples on this page are for 7-string [[guitar]].
+; Harmonics
+/1: string 2 open
+/1: string 3 fret 5
+/2: not easily accessible
+/4: string 5 fret 4
 == Music ==
 === Modern renditions ===
 ; {{W|Johann Sebastian Bach}}
+* [https://www.youtube.com/watch?v=wMEdFl2puL0 "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
 * [https://www.youtube.com/watch?v=4yetEubmIk0 "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)