13edo: Difference between revisions
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{{interwiki | {{interwiki | ||
| de = 13-EDO | | de = 13-EDO | ||
| en = | | en = 13-EDO | ||
| es = | | es = | ||
| ja = 13平均律 | | ja = 13平均律 | ||
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| Prime factorization = 13 is a prime | | Prime factorization = 13 is a prime | ||
| Step size = 92.30769¢ | | Step size = 92.30769¢ | ||
| Fifth = 8\13 | | Fifth = 8\13 (738¢) | ||
| Major 2nd = 3\13 | | Major 2nd = 3\13 (277¢) | ||
| Minor 2nd = -1\13 | | Minor 2nd = -1\13 (-92¢) | ||
| Augmented 1sn = 4\13 | | Augmented 1sn = 4\13 (369¢) | ||
}} | }} | ||
''' | '''13-EDO''' is a tuning system which divides the [[octave]] into 13 equal parts of approximately 92.3 [[cent]]s each. It is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]]. 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. | ||
== Theory == | == Theory == | ||
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|} | |} | ||
As a temperament of 21-odd-limit Just Intonation, | As a temperament of 21-odd-limit Just Intonation, 13-EDO has excellent approximations to the 11th and 21st harmonics, 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 13-EDO 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 12-EDO. 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. | ||
{| class="wikitable center-all right-2" | {| class="wikitable center-all right-2" | ||
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! Archaeotonic | ! Archaeotonic | ||
! Oneirotonic | ! Oneirotonic | ||
! [[ | ! [[26-EDO]] names | ||
! Fox-Raven Notation (J = 360Hz) | ! Fox-Raven Notation (J = 360Hz) | ||
! Pseudo-Diatonic Category | ! Pseudo-Diatonic Category | ||
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<references/> | <references/> | ||
13-EDO can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. | |||
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 | 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 12-EDO music can be directly translated to 13-EDO "on the fly". | ||
{| class="wikitable center-all right-2" | {| class="wikitable center-all right-2" | ||
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genchain of seconds: ...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3... | genchain of seconds: ...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3... | ||
[[:File:13edo-chromatic-scale.mid|13 | [[:File:13edo-chromatic-scale.mid|13 EDO chromatic ascending and descending scale on C (MIDI)]] | ||
[[File:13_Edo_chromatic_scale_on_J.mp3]] | [[File:13_Edo_chromatic_scale_on_J.mp3]] | ||
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== Tuning by ear == | == Tuning by ear == | ||
13-EDO 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 +10.526432c, or +11% of 13-EDO's step size. | |||
== Scales in | == Scales in 13-EDO == | ||
:''Main article: [[13edo scales]]'' | :''Main article: [[13edo scales]]'' | ||
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* [[Chromatic_pairs#Lovecraft|lovecraft]] [[4L 5s]] 212121211 (3\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) | * [[Chromatic_pairs#Sephiroth|Sephiroth]] [[3L 4s]] 3131311 (4\13, 1\1) | ||
Due to the prime character of the number 13, | Due to the prime character of the number 13, 13-EDO 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 13-EDO), 3\13, 4\13, 5\13, & 6\13, respectively. | ||
[[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]] | [[File:13edo_horograms.jpg|alt=13edo_horograms.jpg|13edo_horograms.jpg]] | ||
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2 1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS | 2 1 1 1 1 1 1 1 1 1 1 1 [[1L 11s]] MOS | ||
== Harmony in | == Harmony in 13-EDO == | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO 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 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO 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 13-EDO 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 26et. | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO 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 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO 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 13-EDO 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 26et. | ||
By this, we can assume that the major ninth of | By this, we can assume that the major ninth of 13-EDO can be thought of as analogous to the perfect fifth in 12-EDO and other meantone EDOs.This means that the major second or major ninth is the most consonant interval next to 2/1 in 13-EDO 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 13-EDO. | ||
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 (6L1s) 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 13 | 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 (6L1s) 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 13-EDO 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. | ||
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[[File:13_edo_45921_chord.mp3]] | [[File:13_edo_45921_chord.mp3]] | ||
== Notational and Compositional Approaches to | == Notational and Compositional Approaches to 13-EDO == | ||
13-EDO 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. | |||
=== The Cryptic Ruse Methods === | === The Cryptic Ruse Methods === | ||
13-EDO offers two main candidates for diatonic-like scales: the 6L1s heptatonic MOS generated by 2\13, and the 5L3s octatonic MOS. Both of these scales are [[Rothenberg propriety|Rothenberg proper]], and bear a slightly-twisted resemblance to the 12-EDO diatonic scale. Specifically, the 6L1s scale resembles the 12-EDO diatonic with one of its semitones replaced with a whole-tone, while the 5L3s scale resembles the 12-EDO diatonic with an extra semitone inserted between two adjacent whole-tones. | |||
To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. | To facilitate discussion of these scales, [[Cryptic Ruse]] has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. | ||
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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. | 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. | ||
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 | 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:Archeotonic.png|alt=Archeotonic.png|Archeotonic.png]] | [[File:Archeotonic.png|alt=Archeotonic.png|Archeotonic.png]] | ||
Treating | Treating 13-EDO 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 overtone series; further exploration is pending. | There may be other concordant harmonies possible in this scale that do not represent segments of the overtone series; further exploration is pending. | ||
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The 5\13-based octatonic has been named '''[[oneirotonic]]''' after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands. | 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 | 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 13-EDO. | ||
[[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]] | [[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]] | ||
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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. | 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. | ||
=== Fox and Inthar's approach === | === Fox and Inthar's approach === | ||
Our approach is based on [[5L 3s]]. In fact, we have an absolute pitch notation for 5L 3s | Our approach is based on [[5L 3s]]. In fact, we have an absolute pitch notation for 5L 3s EDOs called the [[User:Inthar/Fox-Raven notation|Fox-Raven notation]]. | ||
== Mapping to Standard Keyboards == | == Mapping to Standard Keyboards == | ||
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=== Animism === | === Animism === | ||
The animist comma, 105/104, appears whenever 3*5*7=13... | The animist comma, 105/104, appears whenever 3*5*7=13... 13-EDO does not approximate 3 and 7 individually (26-EDO does), but 13-EDO has 21/16 (=3*7) and is also an animist temperament. In 13-EDO, 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 | 0 4 5 8 9 13 pentatonic | ||
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== Introductory Materials == | == Introductory Materials == | ||
* [[File:13edo_1MC.mp3|270px]] 13- | * [[File:13edo_1MC.mp3|270px]] 13-EDO example composition by [[User:IlL|Inthar]] ([[File:13edo_1MC_score.pdf|score]]) | ||
* [[File:13edo_Prelude_in_J_Oneirominor.mp3|Flute stop + harpsichord]] F# Oneirominor Prelude ([[:File:13edo_Prelude_in_J_Oneirominor Score.pdf|Score]]) | * [[File:13edo_Prelude_in_J_Oneirominor.mp3|Flute stop + harpsichord]] F# Oneirominor Prelude ([[:File:13edo_Prelude_in_J_Oneirominor Score.pdf|Score]]) | ||
* [[File:13edo_Fugue_in_J_Oneirominor chip.mp3|chiptune]] F# Oneirominor Fugue ([[:File:13edo_Fugue_in_J_Oneirominor Score.pdf|Score]]) | * [[File:13edo_Fugue_in_J_Oneirominor chip.mp3|chiptune]] F# Oneirominor Fugue ([[:File:13edo_Fugue_in_J_Oneirominor Score.pdf|Score]]) | ||
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* [https://soundcloud.com/yinbell/study-in-a-newly-discovered-13-et-scale Study in a newly discovered 13-ET scale] by [[Yin Bell]] | * [https://soundcloud.com/yinbell/study-in-a-newly-discovered-13-et-scale Study in a newly discovered 13-ET scale] by [[Yin Bell]] | ||
==See also== | == See also == | ||
* [[Kentaku's_Approach_to_13EDO|William Lynch's 13 EDO octaton approach]] | * [[Kentaku's_Approach_to_13EDO|William Lynch's 13 EDO octaton approach]] | ||
* [[13EDO Scales and Chords for Guitar]] | * [[13EDO Scales and Chords for Guitar]] | ||
* [[Well-Tempered 13-Tone Clavier|The Well-Tempered 13-Tone Clavier]]: A collab project to create 26 preludes and 26 fugues, one in each | * [[Well-Tempered 13-Tone Clavier|The Well-Tempered 13-Tone Clavier]]: A collab project to create 26 preludes and 26 fugues, one in each 13-EDO key. | ||
[[Category:13-tone]] | [[Category:13-tone]] | ||