46edo: Difference between revisions

Line 27:

! #

! Cents

! Approximate ratios<ref name="interval ratios" group="note">{{sg|limit=2.3.5.7.11.13.17.23 subgroup}} ~~This list does not include~~ ratios of 15, ~~since 46edo has intervals involving the 15th harmonic poorly approximated,~~ except for 15/8 and 16/15 themselves. 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>

! 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]]

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

Line 705:

== Notation ==

===Sagittal notation===

=== Sagittal notation ===

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

~~====Evo flavor====~~

==== Evo flavor ====

File:46-EDO_Evo_Sagittal.svg

Line 719:

</imagemap>

====Revo flavor====

==== Revo flavor ====

File:46-EDO_Revo_Sagittal.svg

Line 730:

Line 729:

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

</imagemap>

~~The following table shows [[sagittal notation]] accidentals in one apotome for 46edo.~~

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

=== Ups and downs notation ===

~~! Steps~~

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

~~| 0~~

~~| 1~~

~~| 2~~

~~| 3~~

~~| 4~~

~~| 5~~

|-

~~! Symbol~~

| [[~~File:Sagittal natural.png~~]]

| [[~~File:Sagittal pai.png~~]]

~~| [[File~~:~~Sagittal pakai.png]]~~

~~| [[File:Sagittal jatai.png]]~~

~~| [[File:Sagittal sharp pao.png]]~~

~~| [[File:Sagittal sharp.png]]~~

|}

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

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

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.

Line 1,051:

Line 1,033:

| 78.261

| [[Valentine]]

| 1L 14s (15-tone) 15L 1s (16-tone) 16L 15s (31-tone)

| 1L 14s (15-tone) 15L 1s (16-tone) 16L 15s (31-tone)

| 4:3 ~ [[Maximal evenness|quasi-equal]] 3:1 2:1 ~ QE

|-

Line 1,057:

Line 1,039:

| 130.435

| [[Twothirdtonic]]

| [[1L 8s]] (9-tone) [[9L 1s]] (10-tone) 9L 10s (19-tone) 9L 19s (28-tone) 9L 28s (37-tone)

| [[1L 8s]] (9-tone) [[9L 1s]] (10-tone) 9L 10s (19-tone) 9L 19s (28-tone) 9L 28s (37-tone)

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

|-

Line 1,063:

Line 1,045:

| 182.609

| [[Minortone]] / [[mitonic]]

| [[1L 5s]] (6-tone) [[6L 1s]] (7-tone) 7L 6s (13-tone) 13L 7s (20-tone) 13L 20s (33-tone)

| [[1L 5s]] (6-tone) [[6L 1s]] (7-tone) 7L 6s (13-tone) 13L 7s (20-tone) 13L 20s (33-tone)

| 11:7 7:4 4:3 ~ QE 3:1 2:1 ~ QE

|-

Line 1,069:

Line 1,051:

| 234.783

| [[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)

| [[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

|-

Line 1,075:

Line 1,057:

| 286.957

| [[Amity_family#Gamity|Gamity]]

| [[4L 1s]] (5-tone) [[4L 5s]] (9-tone) 4L 9s (13-tone) 4L 13s (17-tone) 4L 17s (21-tone) 21L 4s (25-tone)

| [[4L 1s]] (5-tone) [[4L 5s]] (9-tone) 4L 9s (13-tone) 4L 13s (17-tone) 4L 17s (21-tone) 21L 4s (25-tone)

| 11:2 9:2 7:2 5:2 3:2 ~ QE, Golden 2:1 ~ QE

|-

Line 1,081:

Line 1,063:

| 339.130

| [[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)

| [[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

|-

Line 1,087:

Line 1,069:

| 391.304

| [[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)

| [[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

|-

Line 1,093:

Line 1,075:

| 443.478

| [[Sensi]]

| [[3L 2s]] (5-tone) [[3L 5s]] (8-tone) [[8L 3s]] (11-tone) 8L 11s (19-tone) 19L 8s (27-tone)

| [[3L 2s]] (5-tone) [[3L 5s]] (8-tone) [[8L 3s]] (11-tone) 8L 11s (19-tone) 19L 8s (27-tone)

| 12:5 7:5 5:2 3:2 ~ QE, Golden 2:1

|-

Line 1,099:

Line 1,081:

| 495.652

| [[Leapday]]

| [[2L 3s]] (5-tone) [[5L 2s]] (7-tone) [[5L 7s]] (12-tone) 12L 5s (17-tone) 17L 12s (29-tone)

| [[2L 3s]] (5-tone) [[5L 2s]] (7-tone) [[5L 7s]] (12-tone) 12L 5s (17-tone) 17L 12s (29-tone)

| 11:8 8:3 5:3 ~ Golden 3:2 ~ QE, Golden 2:1 ~ QE

|-

Line 1,105:

Line 1,087:

| 547.826

| [[Heinz]]

| [[2L 3s]] (5-tone) [[2L 5s]] (7-tone) [[2L 7s]] (9-tone) [[2L 9s]] (11-tone) 11L 2s (13-tone) 11L 13s (24-tone) 11L 24s (35-tone)

| [[2L 3s]] (5-tone) [[2L 5s]] (7-tone) [[2L 7s]] (9-tone) [[2L 9s]] (11-tone) 11L 2s (13-tone) 11L 13s (24-tone) 11L 24s (35-tone)

| 17:4 13:4 9:4 5:4 ~ QE 4:1 3:1 2:1 ~ QE

|-

Line 1,118:

Line 1,100:

| 52.174

| [[Shrutar]]

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[2L 8s]] (10-tone) [[2L 10s]] (12-tone) 2L 12s (14-tone) 2L 14s (16-tone) 2L 16s (18-tone) 2L 18s (20-tone) 2L 20s (22-tone) 22L 2s (24-tone)

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[2L 8s]] (10-tone) [[2L 10s]] (12-tone) 2L 12s (14-tone) 2L 14s (16-tone) 2L 16s (18-tone) 2L 18s (20-tone) 2L 20s (22-tone) 22L 2s (24-tone)

| 21:2 19:2 17:2 15:2 13:2 11:2 9:2 7:2 5:2 3:2 ~ QE, Golden 2:1 ~ QE

|-

Line 1,124:

Line 1,106:

| 78.261

| [[Semivalentine]]

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[2L 8s]] (10-tone) [[2L 10s]] (12-tone) 2L 12s (14-tone) 14L 2s (16-tone) 16L 14s (30-tone)

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[2L 8s]] (10-tone) [[2L 10s]] (12-tone) 2L 12s (14-tone) 14L 2s (16-tone) 16L 14s (30-tone)

| 20:3 17:3 14:3 11:3 8:3 5:3 ~ Golden 3:2 ~ QE, Golden 2:1 ~ QE

|-

Line 1,130:

Line 1,112:

| 104.348

| [[Srutal]] / [[diaschismic]]

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[2L 8s]] (10-tone) [[10L 2s]] (12-tone) 12L 10s (22-tone) 12L 22s (34-tone)

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[2L 8s]] (10-tone) [[10L 2s]] (12-tone) 12L 10s (22-tone) 12L 22s (34-tone)

| 19:4 15:4 11:4 7:4 4:3 ~ QE 3:1 2:1 ~ QE

|-

Line 1,136:

Line 1,118:

| 130.435

|

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[8L 2s]] (10-tone) 8L 10s (18-tone) 18L 10s (28-tone)

| 2L 2s (4-tone) [[2L 4s]] (6-tone) [[2L 6s]] (8-tone) [[8L 2s]] (10-tone) 8L 10s (18-tone) 18L 10s (28-tone)

| 18:5 13:5 8:5 ~ Golden 5:3 ~ Golden 3:2 ~ QE, Golden 2:1 ~ QE

|-

Line 1,142:

Line 1,124:

| 156.522

| [[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

| 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

|-

Line 1,148:

Line 1,130:

| 182.609

| [[Unidec]] / [[hendec]]

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

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

| 16:7 9:7 7:2 5:2 3:2 ~ QE, Golden 2:1 ~ QE

|-

Line 1,154:

Line 1,136:

| 208.696

| [[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)

| 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

|-

Line 1,160:

Line 1,142:

| 234.783

| [[Echidnic]]

| 2L 2s (4-tone) [[4L 2s]] (6-tone) [[6L 4s]] (10-tone) 10L 6s (16-tone) 10L 16s (26-tone) 10L 26s (36-tone)

| 2L 2s (4-tone) [[4L 2s]] (6-tone) [[6L 4s]] (10-tone) 10L 6s (16-tone) 10L 16s (26-tone) 10L 26s (36-tone)

| 14:9 9:5 5:4 ~ QE 4:1 3:1 2:1 ~ QE

|-

Line 1,166:

Line 1,148:

| 260.87

| [[Bamity]]

| 2L 2s (4-tone) [[4L 2s]] (6-tone) [[4L 6s]] (10-tone) 4L 10s (14-tone) 14L 4s (18-tone) 14L 18s (32-tone)

| 2L 2s (4-tone) [[4L 2s]] (6-tone) [[4L 6s]] (10-tone) 4L 10s (14-tone) 14L 4s (18-tone) 14L 18s (32-tone)

| 13:10 10:3 7:3 4:3 ~ QE 3:1 2:1 ~ QE

|-

Line 1,172:

Line 1,154:

| 286.957

| [[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)

| 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

|-

Line 1,184:

Line 1,166:

== Zeta properties ==

===Zeta peak index===

=== Zeta peak index ===

{| class="wikitable"

! colspan="3" |Tuning

|-

! colspan="3" | Tuning

! colspan="3" | Strength

! colspan="2" |Closest EDO

! colspan="2" | Closest EDO

! colspan="2" |Integer limit

! colspan="2" | Integer limit

|-

!ZPI

! ZPI

!Steps per octave

! Steps per octave

!Step size (cents)

! Step size (cents)

!Height

! Height

!Integral

! Integral

!Gap

! Gap

!EDO

! EDO

!Octave (cents)

! Octave (cents)

!Consistent

! Consistent

!Distinct

! Distinct

|-

| [[214zpi]]

|46.0089748051542

| 46.0089748051542

|26.0818678330031

| 26.0818678330031

|7.495674

| 7.495674

|1.356067

| 1.356067

|17.747832

| 17.747832

|46edo

| 46edo

|1199.76592031814

| 1199.76592031814

|14

| 14

|11

| 11

|}

== Scales ==

Line 1,243:

Line 1,227:

=== 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 ~~¢~~) stands in for frequency ratio [[9/8]] (203.910¢).

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

* 7\46 (182.609 ~~¢~~) stands in for [[10/9]] (182.404¢).

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

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

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

* 5\46 (130.435 ~~¢~~) 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¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).

* 4\46 (104.348 ~~¢~~) stands in for [[16/15]] (111.731¢).

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

{| class="wikitable center-all"

|-

@@ Line 27: / Line 27: @@
 ! &#35;
 ! Cents
-! Approximate ratios<ref name="interval ratios" group="note">{{sg|limit=2.3.5.7.11.13.17.23&nbsp;subgroup}} This list does not include ratios of 15, since 46edo has intervals involving the 15th harmonic poorly approximated, except for 15/8 and 16/15 themselves. 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>
+! 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]]
 ! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
@@ Line 705: / Line 705: @@
 == Notation ==
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as [[39edo#Sagittal notation|39-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:46-EDO_Evo_Sagittal.svg
@@ Line 719: / Line 719: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:46-EDO_Revo_Sagittal.svg
@@ Line 730: / Line 729: @@
 default [[File:46-EDO_Revo_Sagittal.svg]]
 </imagemap>
-The following table shows [[sagittal notation]] accidentals in one apotome for 46edo.
-{| class="wikitable center-all"
+=== Ups and downs notation ===
-! Steps
+Using [[Helmholtz–Ellis notation]] accidentals, 46edo can also be notated using [[ups and downs notation]]:
-| 0
-| 1
-| 2
-| 3
-| 4
-| 5
-|-
-! Symbol
-| [[File:Sagittal natural.png]]
-| [[File:Sagittal pai.png]]
-| [[File:Sagittal pakai.png]]
-| [[File:Sagittal jatai.png]]
-| [[File:Sagittal sharp pao.png]]
-| [[File:Sagittal sharp.png]]
-|}
-=== Ups and downs notation ===
-Using [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] 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.
@@ Line 1,051: / Line 1,033: @@
 | 78.261
 | [[Valentine]]
-| 1L 14s (15-tone)<br>15L 1s (16-tone)<br>16L 15s (31-tone)
+| 1L&nbsp;14s (15-tone)<br>15L&nbsp;1s (16-tone)<br>16L&nbsp;15s (31-tone)
 | 4:3 ~ [[Maximal evenness|quasi-equal]]<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,057: / Line 1,039: @@
 | 130.435
 | [[Twothirdtonic]]
-| [[1L 8s]] (9-tone)<br>[[9L 1s]] (10-tone)<br>9L 10s (19-tone)<br>9L 19s (28-tone)<br>9L 28s (37-tone)
+| [[1L&nbsp;8s]] (9-tone)<br>[[9L&nbsp;1s]] (10-tone)<br>9L&nbsp;10s (19-tone)<br>9L&nbsp;19s (28-tone)<br>9L&nbsp;28s (37-tone)
 | 6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,063: / Line 1,045: @@
 | 182.609
 | [[Minortone]] / [[mitonic]]
-| [[1L 5s]] (6-tone)<br>[[6L 1s]] (7-tone)<br>7L 6s (13-tone)<br>13L 7s (20-tone)<br>13L 20s (33-tone)
+| [[1L&nbsp;5s]] (6-tone)<br>[[6L&nbsp;1s]] (7-tone)<br>7L&nbsp;6s (13-tone)<br>13L&nbsp;7s (20-tone)<br>13L&nbsp;20s (33-tone)
 | 11:7<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,069: / Line 1,051: @@
 | 234.783
 | [[Rodan]]
-| [[1L 4s]] (5-tone)<br>[[1L 5s]] (6-tone)<br>[[5L 6s]] (11-tone)<br>5L 11s (16-tone)<br>5L 16s (21-tone)<br>5L 21s (26-tone)<br>5L 26s (31-tone)<br>5L 31s (36-tone)<br>5L 36s (41-tone)
+| [[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
 |-
@@ Line 1,075: / Line 1,057: @@
 | 286.957
 | [[Amity_family#Gamity|Gamity]]
-| [[4L 1s]] (5-tone)<br>[[4L 5s]] (9-tone)<br>4L 9s (13-tone)<br>4L 13s (17-tone)<br>4L 17s (21-tone)<br>21L 4s (25-tone)
+| [[4L&nbsp;1s]] (5-tone)<br>[[4L&nbsp;5s]] (9-tone)<br>4L&nbsp;9s (13-tone)<br>4L&nbsp;13s (17-tone)<br>4L&nbsp;17s (21-tone)<br>21L&nbsp;4s (25-tone)
 | 11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
 |-
@@ Line 1,081: / Line 1,063: @@
 | 339.130
 | [[Amity]] / [[hitchcock]]
-| [[4L 3s]] (7-tone)<br>[[7L 4s]] (11-tone)<br>7L 11s (18-tone)<br>7L 18s (25-tone)<br>7L 25s (32-tone)<br>7L 32s (39-tone)
+| [[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
 |-
@@ Line 1,087: / Line 1,069: @@
 | 391.304
 | [[Magus]] / [[amigo]]
-| [[1L 2s]] (3-tone)<br>[[3L 1s]] (4-tone)<br>[[3L 4s]] (7-tone)<br>[[3L 7s]] (10-tone)<br>3L 10s (13-tone)<br>3L 13s (16-tone)<br>3L 16s (19-tone)<br>3L 19s (21-tone)<br>3L 21s (24-tone)<br>3L 24s (27-tone)<br>3L 27s (30-tone)<br>3L 30s (33-tone)<br>3L 33s (36-tone)<br>3L 36s (39-tone)<br>3L 39s (42-tone)
+| [[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
 |-
@@ Line 1,093: / Line 1,075: @@
 | 443.478
 | [[Sensi]]
-| [[3L 2s]] (5-tone)<br>[[3L 5s]] (8-tone)<br>[[8L 3s]] (11-tone)<br>8L 11s (19-tone)<br>19L 8s (27-tone)
+| [[3L&nbsp;2s]] (5-tone)<br>[[3L&nbsp;5s]] (8-tone)<br>[[8L&nbsp;3s]] (11-tone)<br>8L&nbsp;11s (19-tone)<br>19L&nbsp;8s (27-tone)
 | 12:5<br>7:5<br>5:2<br>3:2 ~ QE, Golden<br>2:1
 |-
@@ Line 1,099: / Line 1,081: @@
 | 495.652
 | [[Leapday]]
-| [[2L 3s]] (5-tone)<br>[[5L 2s]] (7-tone)<br>[[5L 7s]] (12-tone)<br>12L 5s (17-tone)<br>17L 12s (29-tone)
+| [[2L&nbsp;3s]] (5-tone)<br>[[5L&nbsp;2s]] (7-tone)<br>[[5L&nbsp;7s]] (12-tone)<br>12L&nbsp;5s (17-tone)<br>17L&nbsp;12s (29-tone)
 | 11:8<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
 |-
@@ Line 1,105: / Line 1,087: @@
 | 547.826
 | [[Heinz]]
-| [[2L 3s]] (5-tone)<br>[[2L 5s]] (7-tone)<br>[[2L 7s]] (9-tone)<br>[[2L 9s]] (11-tone)<br>11L 2s (13-tone)<br>11L 13s (24-tone)<br>11L 24s (35-tone)
+| [[2L&nbsp;3s]] (5-tone)<br>[[2L&nbsp;5s]] (7-tone)<br>[[2L&nbsp;7s]] (9-tone)<br>[[2L&nbsp;9s]] (11-tone)<br>11L&nbsp;2s (13-tone)<br>11L&nbsp;13s (24-tone)<br>11L&nbsp;24s (35-tone)
 | 17:4<br>13:4<br>9:4<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,118: / Line 1,100: @@
 | 52.174
 | [[Shrutar]]
-| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[2L 10s]] (12-tone)<br>2L 12s (14-tone)<br>2L 14s (16-tone)<br>2L 16s (18-tone)<br>2L 18s (20-tone)<br>2L 20s (22-tone)<br>22L 2s (24-tone)
+| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[2L&nbsp;8s]] (10-tone)<br>[[2L&nbsp;10s]] (12-tone)<br>2L&nbsp;12s (14-tone)<br>2L&nbsp;14s (16-tone)<br>2L&nbsp;16s (18-tone)<br>2L&nbsp;18s (20-tone)<br>2L&nbsp;20s (22-tone)<br>22L&nbsp;2s (24-tone)
 | 21:2<br>19:2<br>17:2<br>15:2<br>13:2<br>11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
 |-
@@ Line 1,124: / Line 1,106: @@
 | 78.261
 | [[Semivalentine]]
-| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[2L 10s]] (12-tone)<br>2L 12s (14-tone)<br>14L 2s (16-tone)<br>16L 14s (30-tone)
+| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[2L&nbsp;8s]] (10-tone)<br>[[2L&nbsp;10s]] (12-tone)<br>2L&nbsp;12s (14-tone)<br>14L&nbsp;2s (16-tone)<br>16L&nbsp;14s (30-tone)
 | 20:3<br>17:3<br>14:3<br>11:3<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
 |-
@@ Line 1,130: / Line 1,112: @@
 | 104.348
 | [[Srutal]] / [[diaschismic]]
-| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[10L 2s]] (12-tone)<br>12L 10s (22-tone)<br>12L 22s (34-tone)
+| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[2L&nbsp;8s]] (10-tone)<br>[[10L&nbsp;2s]] (12-tone)<br>12L&nbsp;10s (22-tone)<br>12L&nbsp;22s (34-tone)
 | 19:4<br>15:4<br>11:4<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,136: / Line 1,118: @@
 | 130.435
 |
-| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[8L 2s]] (10-tone)<br>8L 10s (18-tone)<br>18L 10s (28-tone)
+| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[8L&nbsp;2s]] (10-tone)<br>8L&nbsp;10s (18-tone)<br>18L&nbsp;10s (28-tone)
 | 18:5<br>13:5<br>8:5 ~ Golden<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
 |-
@@ Line 1,142: / Line 1,124: @@
 | 156.522
 | [[Bison]]
-| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>8L 6s (14-tone)<br>8L 14s (22-tone)<br>8L 22s (30-tone)<br>8L 30s (38-tone
+| 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
 |-
@@ Line 1,148: / Line 1,130: @@
 | 182.609
 | [[Unidec]] / [[hendec]]
-| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>6L 8s (14-tone)<br>6L 14s (20-tone)<br>20L 6s (26-tone)
+| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[6L&nbsp;2s]] (8-tone)<br>6L&nbsp;8s (14-tone)<br>6L&nbsp;14s (20-tone)<br>20L&nbsp;6s (26-tone)
 | 16:7<br>9:7<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
 |-
@@ Line 1,154: / Line 1,136: @@
 | 208.696
 | [[Abigail]]
-| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>6L 8s (14-tone)<br>6L 14s (20-tone)<br>6L 20s (26-tone)<br>6L 26s (32-tone)<br>6L 32s (38-tone)<br>6L 38s (44-tone)
+| 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
 |-
@@ Line 1,160: / Line 1,142: @@
 | 234.783
 | [[Echidnic]]
-| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[6L 4s]] (10-tone)<br>10L 6s (16-tone)<br>10L 16s (26-tone)<br>10L 26s (36-tone)
+| 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[6L&nbsp;4s]] (10-tone)<br>10L&nbsp;6s (16-tone)<br>10L&nbsp;16s (26-tone)<br>10L&nbsp;26s (36-tone)
 | 14:9<br>9:5<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,166: / Line 1,148: @@
 | 260.87
 | [[Bamity]]
-| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[4L 6s]] (10-tone)<br>4L 10s (14-tone)<br>14L 4s (18-tone)<br>14L 18s (32-tone)
+| 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[4L&nbsp;6s]] (10-tone)<br>4L&nbsp;10s (14-tone)<br>14L&nbsp;4s (18-tone)<br>14L&nbsp;18s (32-tone)
 | 13:10<br>10:3<br>7:3<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
 |-
@@ Line 1,172: / Line 1,154: @@
 | 286.957
 | [[Vines]]
-| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[4L 6s]] (10-tone)<br>4L 10s (14-tone)<br>4L 14s (18-tone)<br>4L 18s (22-tone)<br>4L 22s (26-tone)<br>4L 26s (30-tone)<br>4L 30s (34-tone)<br>4L 34s (38-tone)<br>4L 38s (42-tone)
+| 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
 |-
@@ Line 1,184: / Line 1,166: @@
 == Zeta properties ==
-===Zeta peak index===
+=== Zeta peak index ===
 {| class="wikitable"
-! colspan="3" |Tuning
+|-
+! colspan="3" | Tuning
 ! colspan="3" | Strength
-! colspan="2" |Closest EDO
+! colspan="2" | Closest EDO
-! colspan="2" |Integer limit
+! colspan="2" | Integer limit
 |-
-!ZPI
+! ZPI
-!Steps per octave
+! Steps per octave
-!Step size (cents)
+! Step size (cents)
-!Height
+! Height
-!Integral
+! Integral
-!Gap
+! Gap
-!EDO
+! EDO
-!Octave (cents)
+! Octave (cents)
-!Consistent
+! Consistent
-!Distinct
+! Distinct
 |-
 | [[214zpi]]
-|46.0089748051542
+| 46.0089748051542
-|26.0818678330031
+| 26.0818678330031
-|7.495674
+| 7.495674
-|1.356067
+| 1.356067
-|17.747832
+| 17.747832
-|46edo
+| 46edo
-|1199.76592031814
+| 1199.76592031814
-|14
+| 14
-|11
+| 11
 |}
 == Scales ==
 {{main|List of MOS scales in 46edo}}
@@ Line 1,243: / Line 1,227: @@
 === 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;&cent;) stands in for frequency ratio [[9/8]] (203.910¢).
+* 8\46 (208.696&nbsp;{{c}}) stands in for frequency ratio [[9/8]] (203.910¢).
-* 7\46 (182.609&nbsp;&cent;) stands in for [[10/9]] (182.404¢).
+* 7\46 (182.609&nbsp;{{c}}) stands in for [[10/9]] (182.404¢).
-* 6\46 (156.522&nbsp;&cent;) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
+* 6\46 (156.522&nbsp;{{c}}) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
-* 5\46 (130.435&nbsp;&cent;) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
+* 5\46 (130.435&nbsp;{{c}}) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
-* 4\46 (104.348&nbsp;&cent;) stands in for [[16/15]] (111.731¢).
+* 4\46 (104.348&nbsp;{{c}}) stands in for [[16/15]] (111.731¢).
 {| class="wikitable center-all"
 |-