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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|13}}
{{ED intro}}
 
== Theory ==
== 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 2.9.5.21.11.13.17.19 subgroup being a particularly good example. In this subgroup, all 21-odd-limit intervals have less than 25% relative error (23.1{{c}}), except for 22/19 and its [[octave complement]], which barely miss with 25.045% relative error. 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 (135/128 is a [[Wikipedia:Continued_fraction|semiconvergent]] to 2<sup>1/13</sup>).
 
In 13edo, the steps less than 600{{c}} 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 ===
=== Odd harmonics ===
{{Harmonics in equal|13}}
{{Harmonics in equal|13}}


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13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].
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 the proximity of 135/128 to 1\13 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 ==
== Intervals ==
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! #
! #
! Cents
! Cents
! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>
! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.9.5.21.11.13.17.19 subgroup temperament; other approaches are possible.</ref>
![[Erv Wilson's Linear Notations|Erv Wilson]]
![[Erv Wilson's Linear Notations|Erv Wilson]]
! Archaeotonic
! Archaeotonic
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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.
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"
{| class="wikitable center-all right-2"
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! #
! #
! Cents
! 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
| 0
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| 185
| 185
| downmajor 2nd, (minor 3rd)
| downmajor 2nd, (minor 3rd)
| ^m2, (m3)
| vM2, (m3)
| ^E, (F)
| vE, (F)
|-
|-
| 3
| 3
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| D
| 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) ===
=== Heptatonic 5th-generated (narrow fifth) ===
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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 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"
{| class="wikitable center-all right-2"
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! #
! #
! Cents
! 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
! colspan="3" | Up/down notation using the narrow 5th of 7\13, <br> with major narrower than minor
|-
|-
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Keyboard: '''D * F * * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)
Keyboard: '''D * F * * G * * A * C * * D''' (generator = wide 3/2 = 8\13 = perfect 5thoid)


Enharmonic interval: dds3
Enharmonic unison: dds3
{| class="wikitable"
{| class="wikitable"
|+notes/intervals in melodic order (s = sub-, d = -oid)
|+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)  
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"
{| class="wikitable"
|+notes/intervals in melodic order
|+notes/intervals in melodic order
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Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)
Keyboard: '''D * E * F * G A * B * C * D''' (generator = 2\13 = perfect 2nd)


Enharmonic interval: dd2
Enharmonic unison: dd2
{| class="wikitable"
{| class="wikitable"
|+notes/intervals in melodic order
|+notes/intervals in melodic order
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Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)
Keyboard: '''D E * * F G * * A B * * C D''' (generator = 4\13 = perfect 3rd)


Enharmonic intervals: vvA1, vm2
Enharmonic unisons: vvA1, vm2
{| class="wikitable"
{| class="wikitable"
|+notes/intervals in melodic order
|+notes/intervals in melodic order
Line 933: Line 945:
Keyboard: '''D * E * * * * * * * * C * D'''  or  '''* * * F * G * A * B * * * *''' (generator = 4\26 = 2\13 = major 2nd)
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"
{| class="wikitable"
|+notes/intervals in melodic order
|+notes/intervals in melodic order
Line 1,084: Line 1,096:
|}
|}


== JI approximation ==
====Sagittal notation====
[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]
This notation is a subset of the notations for EDOs [[26edo#Sagittal notation|26]] and [[52edo#Sagittal notation|52]].
=====Evo flavor=====
 
{{Sagittal chart|Evo}}
 
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
=====Revo flavor=====
 
{{Sagittal chart}}


== Tuning by ear ==
== Notational and compositional approaches ==
13edo can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size.
13edo has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.


== Phi vibes ==
=== The Cryptic Ruse Methods ===
13edo offers two main candidates for diatonic-like scales: the 6L 1s heptatonic MOS generated by 2\13, and the 5L 3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L 1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L 3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.


=== Acoustic phi ===
To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.
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.


See also: [[9edϕ]]
==== Modes and harmony in the archaeotonic scale ====
The 2\13-based heptatonic has been named '''archaeotonic''' after the "Old Ones" that rule the Dreamlands. Modes of the archaeotonic are named after the individual Old Ones.


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


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.
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.


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.
There may be other concordant harmonies possible in this scale that do not represent segments of the harmonic series; further exploration is pending.


{| class="wikitable center-all"
==== Modes and harmony in the oneirotonic scale ====
|+Direct mapping
The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.
|-
 
! Interval
Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.
! Error (abs, [[Cent|¢]])
 
|-
[[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]]
 
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.
 
== Approximation to JI ==
=== Selected 13-odd-limit intervals ===
[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]]
 
=== Local zeta peak and octave stretch ===
{{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 ===
13edo can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size.
 
== Approximation to irrational intervals ==
=== Golden ratio ===
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]]
 
{| class="wikitable center-all"
|+Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| 2<sup>ϕ</sup> / ϕ
| 2<sup>ϕ</sup> / ϕ
| 0.858
| 0.858
Line 1,122: Line 1,170:
|}
|}


== Local zeta peak ==
== Harmony in 13edo ==
{{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 ==
{{Main | 13edo scales }}
 
Important [[mos]]ses (values in parentheses are (''period'', ''generator'')):
* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)
* archeotonic [[6L 1s]] 2222221 (2\13, 1\1)
* [[Chromatic_pairs#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)
* [[Chromatic_pairs#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.
 
[[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]]
 
[[:File:13edo_horograms.pdf|13edo horograms.pdf]]
 
~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson
 
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 ==
Contrary to popular belief, consonant harmony is possible in 13edo, but it requires a radically different approach than that used in 12edo (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12edo within 13edo is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13edo. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12edo, since the strongest dissonances in 13edo are near the middle of the octave (<u>[[13edo#top|degree]]s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13edo is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N_subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26edo.
Contrary to popular belief, consonant harmony is possible in 13edo, but it requires a radically different approach than that used in 12edo (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12edo within 13edo is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13edo. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12edo, since the strongest dissonances in 13edo are near the middle of the octave (<u>[[13edo#top|degree]]s</u> 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13edo is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N_subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26edo.


By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.
By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.


The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrads, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.


Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.
Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.
Line 1,176: Line 1,195:
[[File:13_edo_45921_chord.mp3]]
[[File:13_edo_45921_chord.mp3]]


== Notational and compositional approaches ==
== Regular temperament properties ==
13edo has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.
=== Uniform maps ===
{{Uniform map|edo=13}}


=== The Cryptic Ruse Methods ===
=== Commas ===
13edo offers two main candidates for diatonic-like scales: the 6L 1s heptatonic MOS generated by 2\13, and the 5L 3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L 1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L 3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.
13et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)


To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.
{| class="commatable wikitable center-1 center-2 right-4 center-5"
 
|-
==== Modes and harmony in the archeotonic scale ====
! [[Harmonic limit|Prime<br>limit]]
The 2\13-based heptatonic has been named '''archeotonic''' after the "Old Ones" that rule the Dreamlands. Modes of the archeotonic are named after the individual Old Ones.
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
 
! [[Monzo]]
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.
! [[Cent]]s
 
! [[Color name]]
[[File:Archeotonic.png|alt=Archeotonic.png|Archeotonic.png]]
! Name(s)
 
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.
 
There may be other concordant harmonies possible in this scale that do not represent segments of the harmonic series; further exploration is pending.
 
==== Modes and harmony in the oneirotonic scale ====
The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.
 
Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.
 
[[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]]
 
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.
 
== Mapping to standard keyboards ==
The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
 
{| class="wikitable"
|-
|-
| 1
| 6
| 11
| 3
| 8
| (13)
| 5
| 5
| 10
| [[2109375/2097152|(14 digits)]]
| 2
| {{monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]], Fokker comma
|-
| 7
| 7
| 12
| [[1029/1000]]
| 4
| {{monzo| -3 1 -3 3 }}
| 9
| 49.49
| 1
| Trizogu
| Place in Chain of 738.5 cent intervals
| Keega
|-
|-
| X
| 7
| *
| [[525/512]]
|  
| {{monzo| -9 1 2 1 }}
| *
| 43.41
| *
| Lazoyoyo
|
| Avicennma, Avicenna's enharmonic diesis
| *
|
| *
| *
|
| *
|  
| X
| Marked are the octatonic scales (X=Sarnathian)
|-
|-
|  
| 7
| *
| [[64/63]]
|  
| {{monzo| 6 -2 0 -1 }}
| *
| 27.26
| *
| Ru
|  
| Septimal comma, Archytas' comma, Leipziger Komma
| *
|-
|  
| 7
| X
| [[64827/64000]]
| *
| {{monzo| -9 3 -3 4 }}
|  
| 22.23
| *
| Laquadzo-atrigu
| *
| Squalentine comma
|  
|  
|-
|-
|  
| 7
| *
| [[3125/3087]]
|  
| {{monzo| 0 -2 5 -3 }}
| X
| 21.18
| *
| Triru-aquinyo
|
| Gariboh comma
| *
| *
|
| *
|
| *
| *
|  
|  
|-
|-
|  
| 7
| *
| [[3136/3125]]
| *
| {{monzo| 6 0 -5 2 }}
|  
| 6.08
| *
| Zozoquingu
|  
| Hemimean comma
| *
|-
| *
| 11
|  
| [[56/55]]
| *
| {{monzo| 3 0 -1 1 -1 }}
|  
| 31.19
| X
| Luzogu
| *
| Undecimal diesis
|  
|  
|-
|-
|  
| 11
| *
| [[121/120]]
| *
| {{monzo| -3 -1 -1 0 2 }}
|  
| 14.37
| *
| Lologu
|  
| Biyatisma
| X
|-
| *
| 11
|  
| [[441/440]]
| *
| {{monzo| -3 2 -1 2 -1 }}
| *
| 3.93
|  
| Luzozogu
| *
| Werckisma
|  
|  
|-
|-
| '''D'''
| 13
| Eb
| [[40/39]]
| E
| {{monzo| 3 -1 1 0 0 -1 }}
| '''F'''
| 43.83
| Gb
| Thuyo
|  
| Tridecimal minor diesis
| '''G'''
|-
| Ab
| 13
| '''A'''
| [[105/104]]
| Bb
| {{monzo| -3 1 1 1 0 -1 }}
| B
| 16.57
| '''C'''
| Thuzoyo
| Db
| Animist comma
| '''D'''
| Keeps the pentatonic scale on the white keys
|-
|-
| '''A'''
| 13
| Bb
| [[169/168]]
| B
| {{monzo| -3 -1 0 -1 0 2 }}
| '''C'''
| 10.27
| Db
| Thothoru
|  
| Buzurgisma
| '''D'''
|}
| Eb
<references/>
| '''E'''
 
| F
== Scales ==
| Gb
{{Main | 13edo scales }}
| '''G'''
 
| Ab
=== Moment of symmetry scales ===
| '''A'''
Important [[mos]]ses (values in parentheses are (''period'', ''generator'')):
|  
* [[oneirotonic]] [[5L 3s]] 22122121 (5\13, 1\1)
|-
* archaeotonic [[6L 1s]] 2222221 (2\13, 1\1)
| '''E'''
* [[No-threes subgroup temperaments#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\13, 1\1)
| F
* [[No-threes subgroup temperaments#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1)
| Gb
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.
| '''G'''
 
| Ab
[[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]]
|
 
| '''A'''
[[:File:13edo_horograms.pdf|13edo horograms.pdf]]
| Bb
 
| '''B'''
~diagram by [[Andrew Heathwaite]], based on [[horogram]]s pioneered by [[Erv Wilson]]
| C
 
| Db
=== Near-12edo scales ===
| '''D'''
 
| Eb
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.
| '''E'''
 
|
=== Animism ===
|-
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:
| '''B'''
 
| C
0 4 5 8 9 13 pentatonic
| Db
 
| '''D'''
and
| Eb
 
|
0 1 3 4 5 8 9 10 12 13 nonatonic
| '''E'''
 
| F
=== Other scales ===
| '''Gb'''
* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]'')
| G
* 461.5cET: 5 5 5... (''nonoctave'')
| Ab
* 738.5cET: 8 8 8... (''nonoctave'')
| '''A'''
 
| Bb
== Instruments ==
| '''B'''
 
|  
=== Lumatone ===
See: [[Lumatone mapping for 13edo]].
 
=== Mapping to standard keyboards ===
The 5L+3s scale (Oneirotonic) can be mapped to the standard piano keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
 
{| class="wikitable"
|-
|-
| C
| 1
| Db
| 6
| D
| 11
| Eb
| 3
| E
| 8
| (13)
| 5
| 10
| 2
| 7
| 12
| 4
| 9
| 1
| Place in Chain of 738.5 cent intervals
|-
| X
| *
|  
|  
| F
| *
| Gb
| *
| G
|  
| Ab
| *
| A
|  
| Bb
| *
| B
| *
| C
|  
| Puts the missing key between a semitone
| *
|
| X
| Marked are the octatonic scales (X=Sarnathian)
|-
|-
| G
| Ab
| A
| Bb
| B
|  
|  
| C
| *
| Db
|  
| D
| *
| Eb
| *
| E
|  
| F
| *
| Gb
|  
| G
| X
| (if that were to be valuable in any way)
| *
|}
|  
 
| *
The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.
| *
 
|
== Regular temperament properties ==
|
=== Uniform maps ===
|-
{{Uniform map|13|12.5|13.5}}
|  
 
| *
=== Commas ===
|  
13edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 13 21 30 36 45 48 }}.)
| X
 
| *
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|  
| *
| *
|
| *
|
| *
| *
|
|  
|-
|-
! [[Harmonic limit|Prime <br> limit]]
|  
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
| *
! [[Monzo]]
| *
! [[Cent]]s
|
! [[Color name]]
| *
! Name(s)
|
| *
| *
|
| *
|
| X
| *
|
|
|-
|-
| 5
|  
| [[2109375/2097152|(14 digits)]]
| *
| {{monzo| -21 3 7 }}
| *
| 10.06
|
| Lasepyo
| *
| [[Semicomma]], Fokker comma
|
| X
| *
|
| *
| *
|  
| *
|  
|  
|-
|-
| 7
| '''D'''
| [[1029/1000]]
| Eb
| {{monzo| -3 1 -3 3 }}
| E
| 49.49
| '''F'''
| Trizogu
| Gb
| Keega
|  
|-
| '''G'''
| 7
| Ab
| [[525/512]]
| '''A'''
| {{monzo| -9 1 2 1 }}
| Bb
| 43.41
| B
| Lazoyoyo
| '''C'''
| Avicennma, Avicenna's Enharmonic Diesis
| Db
| '''D'''
| Keeps the pentatonic scale on the white keys
|-
|-
| 7
| '''A'''
| [[64/63]]
| Bb
| {{monzo| 6 -2 0 -1 }}
| B
| 27.26
| '''C'''
| Ru
| Db
| Septimal Comma, Archytas' Comma, Leipziger Komma
|  
|-
| '''D'''
| 7
| Eb
| [[64827/64000]]
| '''E'''
| {{monzo| -9 3 -3 4 }}
| F
| 22.23
| Gb
| Laquadzo-atrigu
| '''G'''
| Squalentine
| Ab
| '''A'''
|  
|-
|-
| 7
| '''E'''
| [[3125/3087]]
| F
| {{monzo| 0 -2 5 -3 }}
| Gb
| 21.18
| '''G'''
| Triru-aquinyo
| Ab
| Gariboh
|  
|-
| '''A'''
| 7
| Bb
| [[3136/3125]]
| '''B'''
| {{monzo| 6 0 -5 2 }}
| C
| 6.08
| Db
| Zozoquingu
| '''D'''
| Hemimean comma
| Eb
| '''E'''
|  
|-
|-
| 11
| '''B'''
| [[56/55]]
| C
| {{monzo| 3 0 -1 1 -1 }}
| Db
| 31.19
| '''D'''
| Luzogu
| Eb
| Undecimal diesis
|  
|-
| '''E'''
| 11
| F
| [[121/120]]
| '''Gb'''
| {{monzo| -3 -1 -1 0 2 }}
| G
| 14.37
| Ab
| Lologu
| '''A'''
| Biyatisma
| Bb
| '''B'''
|  
|-
|-
| 11
| C
| [[441/440]]
| Db
| {{monzo| -3 2 -1 2 -1 }}
| D
| 3.93
| Eb
| Luzozogu
| E
| Werckisma
|
| F
| Gb
| G
| Ab
| A
| Bb
| B
| C
| Puts the missing key between a semitone
|-
|-
| 13
| G
| [[40/39]]
| Ab
| {{monzo| 3 -1 1 0 0 -1 }}
| A
| 43.83
| Bb
| Thuyo
| B
| Tridecimal minor diesis
|  
|-
| C
| 13
| Db
| [[105/104]]
| D
| {{monzo| -3 1 1 1 0 -1 }}
| Eb
| 16.57
| E
| Thuzoyo
| F
| Animist
| Gb
|-
| G
| 13
| (if that were to be valuable in any way)
| [[169/168]]
|}
| {{monzo| -3 -1 0 -1 0 2 }}
 
| 10.27
The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.
| Thothoru
 
| Buzurgisma
== Music ==
|}
{{Main| 13edo/Music}}
<references/>
{{Catrel|13edo tracks}}
 
=== 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:
 
0 4 5 8 9 13 pentatonic
 
and
 
0 1 3 4 5 8 9 10 12 13 nonatonic


== Introductory materials ==
== Introductory materials ==
Line 1,526: Line 1,552:
* [[File:13edo_1MC.mp3|270px]] 13edo example composition ([[File:13edo_1MC_score.pdf|score]])
* [[File:13edo_1MC.mp3|270px]] 13edo example composition ([[File:13edo_1MC_score.pdf|score]])


==== Oneirotonic Modal Studies ====
''Oneirotonic Modal Studies''
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
Line 1,535: Line 1,561:
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
== Music ==
{{Main| 13edo/Music}}
{{Catrel|13edo tracks}}


== See also ==
== See also ==
* [[Kentaku's_Approach_to_13EDO|William Lynch's 13 EDO octaton approach]]
* [[13EDO Scales and Chords for Guitar]]
* [[13EDO Scales and Chords for Guitar]]
* [[Lumatone mapping for 13edo]]
* [[Lumatone mapping for 13edo]]
* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo keyboard pieces in a variety of keys and modes)
* [[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 ==
== Further reading ==
Retrieved from "https://en.xen.wiki/w/13edo"