46edo: Difference between revisions

(22 intermediate revisions by 9 users not shown)

Line 6:

}}

== 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]]. ~~In fact~~, ~~while 41~~ is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak ~~and zeta integral~~ edo]] but ~~not~~ a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap ~~edo]], 46 is zeta gap but not zeta peak or zeta integral, and 53 is a [[strict zeta~~ edo]]. ~~Like 41 and 53, 46~~ 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]]. 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.

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.

Line 29:

Line 31:

! 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]]

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 < −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

|}

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 705:

Line 708:

== Notation ==

=== Ups and downs notation ===

46edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[39edo#Sagittal notation|~~39-EDO~~]].

This notation uses the same sagittal sequence as [[39edo #Sagittal notation|39edo]].

==== Evo flavor ====

Line 729:

Line 738:

default [[File:46-EDO_Revo_Sagittal.svg]]

</imagemap>

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

~~Using [[Helmholtz–Ellis notation]] accidentals, 46edo can also be notated using [[ups and downs notation]]:~~

~~{{Sharpness-sharp5}}~~

Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.

== Approximation to JI ==

Line 754:

Line 756:

| [[10/9]] || 36.1% || Normal || [[Mitonic]] || [[Unidec]], [[hendec]]

|-

| [[31/24]] || 70.2% || Weak || ? || ?

| [[31/24]] || 70.2% || Weak || [[Sensible]], [[sensi]] add-31 || [[Bison]] add-31, [[bisensi]] add-31

|}

Line 775:

Line 777:

|}

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

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

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

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

|-

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

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

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

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

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[214zpi]]~~

~~| 46.0089748051542~~

~~| 26.0818678330031~~

~~| 7.495674~~

~~| 1.356067~~

~~| 17.747832~~

~~| 46edo~~

~~| 1199.76592031814~~

~~| 14~~

~~| 11~~

|}

== Regular temperament properties ==

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

|-

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

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

Line 1,028:

Line 1,000:

| Thothotrigu

|

|-

| 13

| [[351/350]]

| {{Monzo| -1 3 -2 -1 0 1 }}

| 4.94

| Thorugugu

| Ratwolfsma

|-

| 13

| [[352/351]]

| {{monzo| 5 -3 0 0 1 -1 }}

| 4.93

| Thulo

| Minor minthma

|-

| 17

Line 1,083:

Line 1,069:

| [[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

Line 1,095:

Line 1,081:

| [[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

Line 1,101:

Line 1,087:

| [[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

Line 1,156:

Line 1,142:

| [[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

Line 1,168:

Line 1,154:

| [[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

Line 1,186:

Line 1,172:

| [[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

Line 1,227:

Line 1,213:

=== Harmonic scales ===

46edo represents [[harmonic series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. This corresponds to scale steps of 8, 7, 6, 6, 5, 5, 5, 4.

* 8\46 (208.696~~ ~~{{c}}) stands in for frequency ratio [[9/8]] (203.~~910¢~~).

* 8\46 (208.696{{c}}) stands in for frequency ratio [[9/8]] (203.910{{c}}).

* 7\46 (182.609~~ ~~{{c}}) stands in for [[10/9]] (182.~~404¢~~).

* 7\46 (182.609{{c}}) stands in for [[10/9]] (182.404{{c}}).

* 6\46 (156.522~~ ~~{{c}}) stands in for [[11/10]] (165.~~004¢~~) and [[12/11]] (150.~~637¢~~).

* 6\46 (156.522{{c}}) stands in for [[11/10]] (165.004{{c}}) and [[12/11]] (150.637{{c}}).

* 5\46 (130.435~~ ~~{{c}}) stands in for [[13/12]] (138.~~573¢~~), [[14/13]] (128.~~298¢~~) and [[15/14]] (119.~~443¢~~).

* 5\46 (130.435{{c}}) stands in for [[13/12]] (138.573{{c}}), [[14/13]] (128.298{{c}}) and [[15/14]] (119.443{{c}}).

* 4\46 (104.348~~ ~~{{c}}) stands in for [[16/15]] (111.~~731¢~~).

* 4\46 (104.348{{c}}) stands in for [[16/15]] (111.731{{c}}).

{| class="wikitable center-all"

Line 1,264:

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 ===

* [[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 1,279:

Line 1,281:

=== 21st century ===

; [[Bryan Deister]]

* [https://www.youtube.com/watch?v=SCpbbov1fSQ ''music box 46edo''] (2025)

; [[Jake Freivald]] ([https://soundcloud.com/jdfreivald site])

* [https://soundcloud.com/jdfreivald/a-seed-planted-yet-another ''A Seed Planted - (Yet another version: 46 EDO)''] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 play]{{dead link}}

Line 1,315:

Line 1,320:

; [[Tristan Bay]]

* [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)

; vivi mouse

* [https://soundcloud.com/vivi-mouse-emoji/valentines-day ''valentine's day''] (2025)

== Notes ==

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|46}}
+{{ED intro}}
 == 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]]. In fact, while 41 is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta integral edo]] but not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta peak or zeta integral, and 53 is a [[strict zeta edo]]. Like 41 and 53, 46 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]]. 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.
+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 705: / Line 708: @@
 == Notation ==
+=== Ups and downs notation ===
+edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
+{{Sharpness-sharp5a}}
+Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+{{Sharpness-sharp5}}
 === Sagittal notation ===
-This notation uses the same sagittal sequence as [[39edo#Sagittal notation|39-EDO]].
+This notation uses the same sagittal sequence as [[39edo #Sagittal notation|39edo]].
 ==== Evo flavor ====
@@ Line 729: / Line 738: @@
 default [[File:46-EDO_Revo_Sagittal.svg]]
 </imagemap>
-=== Ups and downs notation ===
-Using [[Helmholtz–Ellis notation]] accidentals, 46edo can also be notated using [[ups and downs notation]]:
-{{Sharpness-sharp5}}
-Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.
 == Approximation to JI ==
@@ Line 754: / Line 756: @@
 | [[10/9]] || 36.1% || Normal || [[Mitonic]] || [[Unidec]], [[hendec]]
 |-
-| [[31/24]] || 70.2% || Weak || ? || ?
+| [[31/24]] || 70.2% || Weak || [[Sensible]], [[sensi]] add-31 || [[Bison]] add-31, [[bisensi]] add-31
 |}
@@ Line 775: / Line 777: @@
 |}
-For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see [[Icositritonic]].
+For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see [[icositritonic]].
-=== Zeta peak index ===
-{| class="wikitable center-all"
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest edo
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[214zpi]]
-| 46.0089748051542
-| 26.0818678330031
-| 7.495674
-| 1.356067
-| 17.747832
-| 46edo
-| 1199.76592031814
-| 14
-| 11
-|}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
@@ Line 1,028: / Line 1,000: @@
 | Thothotrigu
 |
+|-
+| 13
+| [[351/350]]
+| {{Monzo| -1 3 -2 -1 0 1 }}
+| 4.94
+| Thorugugu
+| Ratwolfsma
+|-
+| 13
+| [[352/351]]
+| {{monzo| 5 -3 0 0 1 -1 }}
+| 4.93
+| Thulo
+| Minor minthma
 |-
 | 17
@@ Line 1,083: / 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 1,095: / 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 1,101: / 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 1,156: / 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 1,168: / 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 1,186: / 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 1,227: / Line 1,213: @@
 === Harmonic scales ===
 edo represents [[harmonic series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. This corresponds to scale steps of 8, 7, 6, 6, 5, 5, 5, 4.
-* 8\46 (208.696&nbsp;{{c}}) stands in for frequency ratio [[9/8]] (203.910¢).
+* 8\46 (208.696{{c}}) stands in for frequency ratio [[9/8]] (203.910{{c}}).
-* 7\46 (182.609&nbsp;{{c}}) stands in for [[10/9]] (182.404¢).
+* 7\46 (182.609{{c}}) stands in for [[10/9]] (182.404{{c}}).
-* 6\46 (156.522&nbsp;{{c}}) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
+* 6\46 (156.522{{c}}) stands in for [[11/10]] (165.004{{c}}) and [[12/11]] (150.637{{c}}).
-* 5\46 (130.435&nbsp;{{c}}) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
+* 5\46 (130.435{{c}}) stands in for [[13/12]] (138.573{{c}}), [[14/13]] (128.298{{c}}) and [[15/14]] (119.443{{c}}).
-* 4\46 (104.348&nbsp;{{c}}) stands in for [[16/15]] (111.731¢).
+* 4\46 (104.348{{c}}) stands in for [[16/15]] (111.731{{c}}).
 {| class="wikitable center-all"
@@ Line 1,264: / 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 ===
 * [[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)
@@ Line 1,279: / Line 1,281: @@
 === 21st century ===
+; [[Bryan Deister]]
+* [https://www.youtube.com/watch?v=SCpbbov1fSQ ''music box 46edo''] (2025)
 ; [[Jake Freivald]] ([https://soundcloud.com/jdfreivald site])
 * [https://soundcloud.com/jdfreivald/a-seed-planted-yet-another ''A Seed Planted - (Yet another version: 46 EDO)''] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 play]{{dead link}}
@@ Line 1,315: / Line 1,320: @@
 ; [[Tristan Bay]]
 * [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)
+; vivi mouse
+* [https://soundcloud.com/vivi-mouse-emoji/valentines-day ''valentine's day''] (2025)
 == Notes ==