13edo: Difference between revisions

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== Theory ==

In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.

13edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 cents (in fact, they are both separated from 3/2 by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi|golden generator]] of 741 cents, and so 13edo has the MOS scales 2L 1s, 3L 2s, and 5L 3s and functions as an equalized 8L 5s.

The simplest JI interpretation of 13edo is in the 2.5.11 subgroup, in which it approximates intervals such as 11/10, 121/80, and 64/55. However, it notably has very good approximations to 13, 17, and 19 as well.

Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.

One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]).

In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.

As a temperament of [[21-odd-limit]] [[just intonation]], 13edo has excellent approximations to the 11th and 21st [[harmonic]]s, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.

=== Odd harmonics ===

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13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].

~~One step of 13edo is very close to~~ [[~~135/128~~]] by ~~direct approximation~~, ~~in fact one might stress that it is a~~ [[~~Wikipedia:Continued_fraction|semiconvergent~~]]~~. The 5-limit~~ [[~~aluminium~~]] ~~temperament~~ realizes this proximity through a regular temperament perspective, and ~~EDOs supporting~~ it ~~(for example~~, [[~~494edo~~]] or [[~~1547edo~~]])~~, combine~~ the ~~sound of 13edo~~ with ~~relative simplicities and precision~~ of ~~5-limit JI~~.

The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes this proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.) A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.

== Intervals ==

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13edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.

Enharmonic ~~intervals~~: v⁴A1, ^m2

Enharmonic unisons: v⁴A1, ^m2

{| class="wikitable center-all right-2"

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! #

! Cents

! colspan="3" |[[Ups and ~~Downs Notation~~|Up/down notation]] using the wide 5th of 8\13

! colspan="3" |[[Ups and downs notation|Up/down notation]] using the wide 5th of 8\13

|-

| 0

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| 185

| downmajor 2nd, (minor 3rd)

| ~~^m2~~, (m3)

| vM2, (m3)

| ^E, (F)

| vE, (F)

|-

| 3

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| D

|}

Half-sharps and half-flats can also be used, making the ascending scale:

D E{{Demiflat2}} vE ^F F{{Demisharp2}} G ^G vA A B{{Demiflat2}} vB ^C C{{Demisharp2}} D

=== Heptatonic 5th-generated (narrow fifth) ===

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The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".

The first approach has ~~enharmonic intervals~~ of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.

The first approach has Enharmonic unisons of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.

{| class="wikitable center-all right-2"

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! #

! Cents

! colspan="3" |[[Ups and ~~Downs Notation~~|Up/down notation]] using the narrow 5th of 7\13, with major wider than minor

! colspan="3" |[[Ups and downs notation|Up/down notation]] using the narrow 5th of 7\13, with major wider than minor

! colspan="3" | Up/down notation using the narrow 5th of 7\13, with major narrower than minor

|-

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Keyboard: '''D * F * * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)

Enharmonic ~~interval~~: dds3

Enharmonic unison: dds3

{| class="wikitable"

|+notes/intervals in melodic order (s = sub-, d = -oid)

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Keyboard: '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)

Enharmonic ~~interval~~: d2

Enharmonic unison: d2

{| class="wikitable"

|+notes/intervals in melodic order

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Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)

Enharmonic ~~interval~~: dd2

Enharmonic unison: dd2

{| class="wikitable"

|+notes/intervals in melodic order

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Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)

Enharmonic ~~intervals~~: vvA1, vm2

Enharmonic unisons: vvA1, vm2

{| class="wikitable"

|+notes/intervals in melodic order

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Keyboard: '''D * E * * * * * * * * C * D''' or '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)

Enharmonic ~~interval~~: ddd2

Enharmonic unison: ddd2

{| class="wikitable"

|+notes/intervals in melodic order

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|...

|}

====Sagittal notation====

This notation is a subset of the notations for EDOs [[26edo#Sagittal notation|26]] and [[52edo#Sagittal notation|52]].

=====Evo flavor=====

File:13-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 447 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]

default [[File:13-EDO_Evo_Sagittal.svg]]

</imagemap>

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

=====Revo flavor=====

File:13-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 495 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 495 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]

default [[File:13-EDO_Revo_Sagittal.svg]]

</imagemap>

== Approximation to JI ==

=== Selected 13-odd-limit intervals ===

[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Local zeta peak ===

At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].

== Tuning by ear ==

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== Approximation to irrational intervals ==

=== ~~Acoustic phi~~ ===

=== Golden ratio ===

13edo has a very ~~close~~ approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] ~~are even closer~~, ~~but unlike all closer EDOs, 13-EDO~~ has ~~no other interval that represents any ratio from the Fibonacci sequence~~ (~~3/2, 5/3, 8/5, 13/8,~~ 21/13, ~~etc~~.~~) except~~ of ~~course for 1/1 and 2/1~~. In a ~~way, one could say that 13-EDO is the only EDO that tempers the ratios~~ of ~~the Fibonacci sequence into a single interval~~.

13edo has a very good approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. The next better approximations are in [[23edo]] and [[36edo]]. As a coincidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. Logarithmic phi has some interesting applications in [[Metallic MOS]].

Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.

See also: [[9edϕ]]

~~=== Logarithmic phi ===~~

~~As a coïncidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error.~~

~~Not until [[144edo|144]] do we find a better EDO in terms of relative error on these two intervals.~~

However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2ϕ ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant ϕ ≃ 1.6180.

~~That being said, logarithmic phi has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales.~~

{| class="wikitable center-all"

|+Direct ~~mapping~~

|+Direct approximation

|-

! Interval

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| 3.179

|}

~~== Local zeta peak ==~~

~~{{Main | 13edo and optimal octave stretching }}~~

~~At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].~~

== Scales ==

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* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)

* archaeotonic [[6L 1s]] 2222221 (2\13, 1\1)

* [[~~Chromatic_pairs~~#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)

* [[No-threes subgroup temperaments#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)

* [[~~Chromatic_pairs~~#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)

* [[No-threes subgroup temperaments#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)

Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two [[degree]]s of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively.

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Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.

~~=== Pathological Modes ===~~

~~2 1 1 1 1 2 1 1 1 1 1 [[2L 9s]] MOS~~

~~3 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS~~

~~2 1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS~~

== Harmony in 13edo ==

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A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.

[[File:Archaeotonic.png|~~alt=~~Archaeotonic.png|Archaeotonic.png]]

[[File:Archaeotonic.png|Archaeotonic.png|link=Special:FilePath/Archaeotonic.png]]

Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.

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== Regular temperament properties ==

=== Uniform maps ===

=== Commas ===

~~13edo~~ [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)

13et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

! [[Monzo]]

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| 43.41

| Lazoyoyo

| Avicennma, Avicenna's ~~Enharmonic Diesis~~

| Avicennma, Avicenna's enharmonic diesis

|-

| 7

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| 27.26

| Ru

| Septimal ~~Comma~~, Archytas' ~~Comma~~, Leipziger Komma

| Septimal comma, Archytas' comma, Leipziger Komma

|-

| 7

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| 22.23

| Laquadzo-atrigu

| Squalentine

| Squalentine comma

|-

| 7

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| 21.18

| Triru-aquinyo

| Gariboh

| Gariboh comma

|-

| 7

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| 16.57

| Thuzoyo

| Animist

| Animist comma

|-

| 13

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=== Animism ===

The animist comma, 105/104, appears whenever 3*5*7 = 2^3*13~~...~~ 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:

The animist comma, 105/104, appears whenever {{nowrap| ~3 × ~5 × ~7 = ~23 × ~13 }}… 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:

0 4 5 8 9 13 pentatonic

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== See also ==

* [[Kentaku's_Approach_to_13EDO|William Lynch's 13 EDO octaton approach]]

* [[13EDO Scales and Chords for Guitar]]

* [[Lumatone mapping for 13edo]]

* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo keyboard pieces in a variety of keys and modes)

* Approaches:

** [[Kentaku's Approach to 13EDO|William Lynch's approach]]

** [[User:Inthar/13edo|Inthar's approach]]

* [[Fendo family]] - temperaments closely related to 13edo

== Further reading ==

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|13}}
+{{ED intro}}
 == Theory ==
-In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.
+edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 cents (in fact, they are both separated from 3/2 by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi|golden generator]] of 741 cents, and so 13edo has the MOS scales 2L 1s, 3L 2s, and 5L 3s and functions as an equalized 8L 5s.
+The simplest JI interpretation of 13edo is in the 2.5.11 subgroup, in which it approximates intervals such as 11/10, 121/80, and 64/55. However, it notably has very good approximations to 13, 17, and 19 as well.
+Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
+One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction|semiconvergent]]).
+In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.
-As a temperament of [[21-odd-limit]] [[just intonation]], 13edo has excellent approximations to the 11th and 21st [[harmonic]]s, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
 === Odd harmonics ===
 {{Harmonics in equal|13}}
@@ Line 18: / Line 26: @@
 edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].
-One step of 13edo is very close to [[135/128]] by direct approximation, in fact one might stress that it is a [[Wikipedia:Continued_fraction|semiconvergent]]. The 5-limit [[aluminium]] temperament realizes this proximity through a regular temperament perspective, and EDOs supporting it (for example, [[494edo]] or [[1547edo]]), combine the sound of 13edo with relative simplicities and precision of 5-limit JI.
+The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes this proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.)  A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.
 == Intervals ==
@@ Line 210: / Line 218: @@
 edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.
-Enharmonic intervals: v⁴A1, ^m2
+Enharmonic unisons: v⁴A1, ^m2
 {| class="wikitable center-all right-2"
@@ Line 216: / Line 224: @@
 ! #
 ! Cents
-! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the wide 5th of 8\13
+! colspan="3" |[[Ups and downs notation|Up/down notation]] using the wide 5th of 8\13
 |-
 | 0
@@ Line 233: / Line 241: @@
 | 185
 | downmajor 2nd, (minor 3rd)
-| ^m2, (m3)
+| vM2, (m3)
-| ^E, (F)
+| vE, (F)
 |-
 | 3
@@ Line 302: / Line 310: @@
 | D
 |}
+{{Sharpness-sharp4}}
+Half-sharps and half-flats can also be used, making the ascending scale:
+D E{{Demiflat2}} vE ^F F{{Demisharp2}} G ^G vA A B{{Demiflat2}} vB ^C C{{Demisharp2}} D
 === Heptatonic 5th-generated (narrow fifth) ===
@@ Line 310: / Line 321: @@
 The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".
-The first approach has enharmonic intervals of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.
+The first approach has Enharmonic unisons of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.
 {| class="wikitable center-all right-2"
@@ Line 316: / Line 327: @@
 ! #
 ! Cents
-! colspan="3" |[[Ups and Downs Notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor
+! colspan="3" |[[Ups and downs notation|Up/down notation]] using the narrow 5th of 7\13, <br> with major wider than minor
 ! colspan="3" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor
 |-
@@ Line 451: / Line 462: @@
 Keyboard: '''D * F * * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)
-Enharmonic interval: dds3
+Enharmonic unison: dds3
 {| class="wikitable"
 |+notes/intervals in melodic order (s = sub-, d = -oid)
@@ Line 571: / Line 582: @@
 Keyboard: '''A * B C * D * E F * G H * A''' (generator = wide 3/2 = 8\13 = perfect 6th)
-Enharmonic interval: d2
+Enharmonic unison: d2
 {| class="wikitable"
 |+notes/intervals in melodic order
@@ Line 692: / Line 703: @@
 Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)
-Enharmonic interval: dd2
+Enharmonic unison: dd2
 {| class="wikitable"
 |+notes/intervals in melodic order
@@ Line 810: / Line 821: @@
 Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)
-Enharmonic intervals: vvA1, vm2
+Enharmonic unisons: vvA1, vm2
 {| class="wikitable"
 |+notes/intervals in melodic order
@@ Line 934: / Line 945: @@
 Keyboard: '''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)
-Enharmonic interval: ddd2
+Enharmonic unison: ddd2
 {| class="wikitable"
 |+notes/intervals in melodic order
@@ Line 1,084: / Line 1,095: @@
 |...
 |}
+====Sagittal notation====
+This notation is a subset of the notations for EDOs [[26edo#Sagittal notation|26]] and [[52edo#Sagittal notation|52]].
+=====Evo flavor=====
+<imagemap>
+File:13-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 447 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]
+default [[File:13-EDO_Evo_Sagittal.svg]]
+</imagemap>
+Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
+=====Revo flavor=====
+<imagemap>
+File:13-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 495 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 495 106 [[26-EDO#Sagittal_notation | 26-EDO notation]]
+default [[File:13-EDO_Revo_Sagittal.svg]]
+</imagemap>
 == Approximation to JI ==
 === Selected 13-odd-limit intervals ===
 [[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]
+=== Local zeta peak ===
+{{Main | 13edo and optimal octave stretching }}
+At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].
 == Tuning by ear ==
@@ Line 1,093: / Line 1,133: @@
 == Approximation to irrational intervals ==
-=== Acoustic phi ===
+=== Golden ratio ===
-edo has a very close approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] are even closer, but unlike all closer EDOs, 13-EDO has no other interval that represents any ratio from the Fibonacci sequence (3/2, 5/3, 8/5, 13/8, 21/13, etc.) except of course for 1/1 and 2/1. In a way, one could say that 13-EDO is the only EDO that tempers the ratios of the Fibonacci sequence into a single interval.
+edo has a very good approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. The next better approximations are in [[23edo]] and [[36edo]]. As a coincidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. Logarithmic phi has some interesting applications in [[Metallic MOS]].
+Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.
 See also: [[9edϕ]]
-=== Logarithmic phi ===
-As a coïncidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error.
-Not until [[144edo|144]] do we find a better EDO in terms of relative error on these two intervals.
-However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2<sup>ϕ</sup> ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant ϕ ≃ 1.6180.
-That being said, logarithmic phi has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales.
 {| class="wikitable center-all"
-|+Direct mapping
+|+Direct approximation
 |-
 ! Interval
@@ Line 1,122: / Line 1,155: @@
 | 3.179
 |}
-== Local zeta peak ==
-{{Main | 13edo and optimal octave stretching }}
-At the [[13edo and optimal octave stretching|local zeta peak of 13edo]], there is an improvement in both [[acoustic phi]] and [[logarithmic phi]].
 == Scales ==
@@ Line 1,133: / Line 1,162: @@
 * [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)
 * archaeotonic [[6L 1s]] 2222221 (2\13, 1\1)
-* [[Chromatic_pairs#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)
+* [[No-threes subgroup temperaments#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)
-* [[Chromatic_pairs#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)
+* [[No-threes subgroup temperaments#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)
 Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two [[degree]]s of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.
@@ Line 1,144: / Line 1,173: @@
 Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.
-=== Pathological Modes ===
-1 1 1 1 2 1 1 1 1 1 [[2L 9s]] MOS
-1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
-1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS
 == Harmony in 13edo ==
@@ Line 1,190: / Line 1,212: @@
 A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.
-[[File:Archaeotonic.png|alt=Archaeotonic.png|Archaeotonic.png]]
+[[File:Archaeotonic.png|Archaeotonic.png|link=Special:FilePath/Archaeotonic.png]]
 Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.
@@ Line 1,407: / Line 1,429: @@
 == Regular temperament properties ==
 === Uniform maps ===
-{{Uniform map|13|12.5|13.5}}
+{{Uniform map|edo=13}}
 === Commas ===
-edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)
+et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)
 {| class="commatable wikitable center-1 center-2 right-4 center-5"
 |-
-! [[Harmonic limit|Prime <br> limit]]
+! [[Harmonic limit|Prime<br>limit]]
 ! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
 ! [[Monzo]]
@@ Line 1,440: / Line 1,462: @@
 | 43.41
 | Lazoyoyo
-| Avicennma, Avicenna's Enharmonic Diesis
+| Avicennma, Avicenna's enharmonic diesis
 |-
 | 7
@@ Line 1,447: / Line 1,469: @@
 | 27.26
 | Ru
-| Septimal Comma, Archytas' Comma, Leipziger Komma
+| Septimal comma, Archytas' comma, Leipziger Komma
 |-
 | 7
@@ Line 1,454: / Line 1,476: @@
 | 22.23
 | Laquadzo-atrigu
-| Squalentine
+| Squalentine comma
 |-
 | 7
@@ Line 1,461: / Line 1,483: @@
 | 21.18
 | Triru-aquinyo
-| Gariboh
+| Gariboh comma
 |-
 | 7
@@ Line 1,503: / Line 1,525: @@
 | 16.57
 | Thuzoyo
-| Animist
+| Animist comma
 |-
 | 13
@@ Line 1,515: / Line 1,537: @@
 === Animism ===
-The animist comma, 105/104, appears whenever 3*5*7 = 2^3*13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:
+The animist comma, 105/104, appears whenever {{nowrap| ~3 × ~5 × ~7 = ~2<sup>3</sup> × ~13 }}… 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:
 4 5 8 9 13 pentatonic
@@ Line 1,542: / Line 1,564: @@
 == See also ==
-* [[Kentaku's_Approach_to_13EDO|William Lynch's 13 EDO octaton approach]]
 * [[13EDO Scales and Chords for Guitar]]
 * [[Lumatone mapping for 13edo]]
 * [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo keyboard pieces in a variety of keys and modes)
+* Approaches:
+** [[Kentaku's Approach to 13EDO|William Lynch's approach]]
+** [[User:Inthar/13edo|Inthar's approach]]
+* [[Fendo family]] - temperaments closely related to 13edo
 == Further reading ==