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=== Why would I want to use an EDO? ===
=== Why would I want to use an EDO? ===


If you are a guitarist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, ukelele, banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings. Speaking of string instruments fretted for EDOs, since ascending through the EDOs will crowd a fretboard relatively quickly, especially as one approaches the 30-something edos, [[ed4|Equal divisions of the double octave]] (or higher multiple of the octave) are a relatively tidy compromise solution to the problem of laying out high-edo fretboards.
If you are a guitarist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, ukelele, banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings. Speaking of string instruments fretted for EDOs, since ascending through the EDOs will crowd a fretboard relatively quickly, especially as one approaches the 30-something EDOs, [[ed4|Equal divisions of the double octave]] (or higher multiple of the octave) are a relatively tidy compromise solution to the problem of laying out high-EDO fretboards.


More generally, EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless. This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones). For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.
More generally, EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless. This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones). For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.
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If you're a classically-trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to [[3/2]] (the perfect 5th) can be rewarding. These include {{EDOs| 12, 17, 19, 22, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}. All of these can be notated with some variant on the [[Nominal-Accidental_Chains#A-G circle-of-fifths notation|A-G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51 }} involve more than one such circle.
If you're a classically-trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to [[3/2]] (the perfect 5th) can be rewarding. These include {{EDOs| 12, 17, 19, 22, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}. All of these can be notated with some variant on the [[Nominal-Accidental_Chains#A-G circle-of-fifths notation|A-G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51 }} involve more than one such circle.


Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25}}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.
Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.


EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[RHM Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:
EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[RHM Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:


* '''superflat''' edos (9, 11, 13b, 16, 18b & 23) have a fifth narrower than four-sevenths of an octave = 4\7 = 686¢
* '''superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b & 23 }}) have a fifth narrower than four-sevenths of an octave = 4\7 = 686¢
* '''perfect''' edos (7, 14, 21, 28 & 35) have a fifth of 4\7 = 686¢
* '''perfect''' EDOs ({{EDOs| 7, 14, 21, 28 & 35 }}) have a fifth of 4\7 = 686¢
* '''diatonic''' edos (12, 17, 19, 22, 24, etc.) have a fifth between 686¢ and 720¢
* '''diatonic''' EDOs ({{EDOs| 12, 17, 19, 22, 24, etc. }}) have a fifth between 686¢ and 720¢
* '''pentatonic''' edos (5, 10, 15, 20, 25 & 30) have a fifth of three-fifths of an octave = 3\5 = 720¢
* '''pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25 & 30 }}) have a fifth of three-fifths of an octave = 3\5 = 720¢
* '''supersharp''' edos (8, 13 & 18) have a fifth wider than 3\5 = 720¢
* '''supersharp''' EDOs ({{EDOs| 8, 13 & 18 }}) have a fifth wider than 3\5 = 720¢
* '''trivial''' edos (1, 2, 3, 4 and 6) have a fifth about 100¢ from just, and are contained in 12-edo
* '''trivial''' EDOs ({{EDOs| 1, 2, 3, 4 and 6 }}) have a fifth about 100¢ from just, and are contained in 12 EDO


=== Non-tuning properties ===
=== Non-tuning properties ===
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=== Adding EDOs ===
=== Adding EDOs ===


Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written <12|, saying that twelve steps maps to 2, but the 3-limit val for 12 is <12 19|, telling us that 19 steps maps to 3, and the 5-limit val is <12 19 28|, telling us that 28 steps maps to 5.
Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12 EDO can be written <12|, saying that twelve steps maps to 2, but the 3-limit val for 12 is <12 19|, telling us that 19 steps maps to 3, and the 5-limit val is <12 19 28|, telling us that 28 steps maps to 5.


If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; <12 19 28| + <19 30 44| = <31 49 72|. The relative error in terms of [[relative cents]] is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo!
If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; <12 19 28| + <19 30 44| = <31 49 72|. The relative error in terms of [[relative cent]]s is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12 EDO for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19 EDO is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth for 31 EDO is not much increased from 19 EDO, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15 EDO and 16 EDO, the sum (again 31 EDO) can have a much smaller error due to the cancellation. 24 EDO's flat fifth and 29 EDO's sharp fifth can be added to form 53 EDO!


We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-EDO MOS, where both of these are generators for the same linear temperament, then the mediant, (a+b)\(n+m), will be a generator for a MOS for the same temperament, this time in (n+m)-edo. A visual way of putting this is that through this addition of n and m, one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.
We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-EDO MOS, where both of these are generators for the same linear temperament, then the mediant, (a+b)\(n+m), will be a generator for a MOS for the same temperament, this time in (n+m)-edo. A visual way of putting this is that through this addition of n and m, one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.
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=== Size of an EDO ===
=== Size of an EDO ===


When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal edo]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. [[5edo|5]], [[7edo|7]] and [[9edo|9]] have arguably been used in various kinds of musical traditions in different parts of the world. i[https://soundcloud.com/scottthompson-3/the-13-edos-of-xmas soundcloud.com/scottthompson-3/the-13-edos-of-xmas] s a humorous demonstration of edos 1-13.
When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal EDO]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. [[5edo|5]], [[7edo|7]] and [[9edo|9]] have arguably been used in various kinds of musical traditions in different parts of the world. i[https://soundcloud.com/scottthompson-3/the-13-edos-of-xmas soundcloud.com/scottthompson-3/the-13-edos-of-xmas] s a humorous demonstration of EDOs 1-13.


On the other hand, if you use the edo to tune a scale or [[Regular_Temperaments|regular temperament]], the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the edos which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these edos can be regarded as automatically tempering it to the corresponding regular temperament.
On the other hand, if you use the edo to tune a scale or [[Regular_Temperaments|regular temperament]], the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.


All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.
All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.
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=== 200...299 ===
=== 200...299 ===
''(some pages do not exist yet)''
''(some pages do not exist yet)''


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== The Scale Tree ==
== The Scale Tree ==


The scale tree, or Stern-Brocot tree, provides a visual map of the world of edos, based on fifth size.
The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.


[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]]
[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]]


The regular edos, up to 72edo:
The regular EDOs, up to 72 EDO:


[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]]
[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]]


== Pergens ==
== Pergens ==
[[pergen|Pergens]] provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for edos 5-24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as P = 6\12, G = 4\12 are marked as "-".


{{todo|cleanup <!-- the simplify wiki markup in following table -->}}
[[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5-24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as P = 6\12, G = 4\12 are marked as "-".
{| class="wikitable"
 
{| class="wikitable center-all"
|-
|-
! | EDO
! EDO
! | Period
! Period
! colspan="11" | Generator in edosteps
! colspan="11" | Generator in EDO steps
|-
|-
! |
!  
! | in edosteps
! in EDO steps
! | 1
! 1
! | 2
! 2
! | 3
! 3
! | 4
! 4
! | 5
! 5
! | 6
! 6
! | 7
! 7
! | 8
! 8
! | 9
! 9
! | 10
! 10
! | 11
! 11
|-
|-
! | 5
! 5
! | 5 = P8
! 5 = P8
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 6
! 6
! | 6 = P8
! 6 = P8
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/2
! 3 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 7
! 7
! | 7 = P8
! 7 = P8
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 8
! 8
! | 8 = P8
! 8 = P8
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 4 = P8/2
! 4 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
|-
|-
! | 9
! 9
! | 9 = P8
! 9 = P8
| style="text-align:center;" | P4/4
| P4/4
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/3
! 3 = P8/3
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 10
! 10
! | 10 = P8
! 10 = P8
| style="text-align:center;" | P4/4
| P4/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 5 = P8/2
! 5 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 11
! 11
! | 11 = P8
! 11 = P8
| style="text-align:center;" | P4/5
| P4/5
| style="text-align:center;" | P5/3
| P5/3
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | P11/4
| P11/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 12
! 12
! | 12 = P8
! 12 = P8
| style="text-align:center;" | P4/5
| P4/5
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 6 = P8/2
! 6 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 4 = P8/3
! 4 = P8/3
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/4
! 3 = P8/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
|-
|-
! | 13b
! 13b
! | 13 = P8
! 13 = P8
| style="text-align:center;" | P4/6
| P4/6
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | P12/5
| P12/5
| style="text-align:center;" | P12/4
| P12/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
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| style="text-align:center;" |  
|  
|-
|-
! | 14
! 14
! | 14 = P8
! 14 = P8
| style="text-align:center;" | P4/6
| P4/6
| style="text-align:center;" | -
| -
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | P11/4
| P11/4
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 7 = P8/2
! 7 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 15
! 15
! | 15 = P8
! 15 = P8
| style="text-align:center;" | P4/6
| P4/6
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | -
| -
| style="text-align:center;" | P12/6
| P12/6
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | P11/3
| P11/3
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 5 = P8/3
! 5 = P8/3
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/5
! 3 = P8/5
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 16
! 16
! | 16 = P8
! 16 = P8
| style="text-align:center;" | P4/7
| P4/7
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/3
| P5/3
| style="text-align:center;" | -
| -
| style="text-align:center;" | P12/5
| P12/5
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 8 = P8/2
! 8 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/3
| P5/3
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 4 = P8/4
! 4 = P8/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 17
! 17
! | 17 = P8
! 17 = P8
| style="text-align:center;" | P4/7
| P4/7
| style="text-align:center;" | P5/5
| P5/5
| style="text-align:center;" | P11/8
| P11/8
| style="text-align:center;" | P11/6
| P11/6
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | P11/4
| P11/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P11/3
| P11/3
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 18b
! 18b
! | 18 = P8
! 18 = P8
| style="text-align:center;" | P4/8
| P4/8
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | P12/4
| P12/4
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 9 = P8/2
! 9 = P8/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P4/4
| P4/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 6 = P8/3
! 6 = P8/3
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/6
! 3 = P8/6
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 19
! 19
! | 19 = P8
! 19 = P8
| style="text-align:center;" | P4/8
| P4/8
| style="text-align:center;" | P4/4
| P4/4
| style="text-align:center;" | P11/9
| P11/9
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | P12/6
| P12/6
| style="text-align:center;" | P12/5
| P12/5
| style="text-align:center;" | WWP5/7
| WWP5/7
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P11/3
| P11/3
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 20
! 20
! | 20 = P8
! 20 = P8
| style="text-align:center;" | P4/8
| P4/8
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/4
| P5/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | P11/4
| P11/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | W<span style="vertical-align: super;">3</span>P5/8
| W<sup>3</sup>P5/8
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 10 = P8/2
! 10 = P8/2
| style="text-align:center;" | M2/4
| M2/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/4
| P5/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 5 = P8/4
! 5 = P8/4
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 4 = P8/5
! 4 = P8/5
| style="text-align:center;" | P5/4
| P5/4
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 21
! 21
! | 21 = P8
! 21 = P8
| style="text-align:center;" | P4/9
| P4/9
| style="text-align:center;" | P5/6
| P5/6
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/3
| P5/3
| style="text-align:center;" | P11/6
| P11/6
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | W<span style="vertical-align: super;">3</span>P4/9
| W<sup>3</sup>P4/9
| style="text-align:center;" | -
| -
| style="text-align:center;" | P11/3
| P11/3
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 7 = P8/3
! 7 = P8/3
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/7
! 3 = P8/7
| style="text-align:center;" | P5/3
| P5/3
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 22
! 22
! | 22 = P8
! 22 = P8
| style="text-align:center;" | P4/9
| P4/9
| style="text-align:center;" | -
| -
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | -
| -
| style="text-align:center;" | P12/7
| P12/7
| style="text-align:center;" | -
| -
| style="text-align:center;" | P12/5
| P12/5
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 11 = P8/2
! 11 = P8/2
| style="text-align:center;" | M2/4
| M2/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P4/3
| P4/3
| style="text-align:center;" | P12/5
| P12/5
| style="text-align:center;" | P12/7
| P12/7
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | 23
! 23
! | 23 = P8
! 23 = P8
| style="text-align:center;" | P4/10
| P4/10
| style="text-align:center;" | P4/5
| P4/5
| style="text-align:center;" | P11/11
| P11/11
| style="text-align:center;" | P12/9
| P12/9
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | P12/6
| P12/6
| style="text-align:center;" | WWP4/8
| WWP4/8
| style="text-align:center;" | WWP4/7
| WWP4/7
| style="text-align:center;" | P12/4
| P12/4
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | P11/3
| P11/3
|-
|-
! | 24
! 24
! | 24 = P8
! 24 = P8
| style="text-align:center;" | P4/10
| P4/10
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | W<span style="vertical-align: super;">4</span>P5/10
| W<sup>4</sup>P5/10
|-
|-
! | "
! "
! | 12 = P8/2
! 12 = P8/2
| style="text-align:center;" | M2/4
| M2/4
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 8 = P8/3
! 8 = P8/3
| style="text-align:center;" | P5/2
| P5/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 6 = P8/4
! 6 = P8/4
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | -
| -
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 4 = P8/6
! 4 = P8/6
| style="text-align:center;" | P4/2
| P4/2
| style="text-align:center;" | -
| -
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! | "
! "
! | 3 = P8/8
! 3 = P8/8
| style="text-align:center;" | P5
| P5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
! |
!  
! |
!  
! | 1
! 1
! | 2
! 2
! | 3
! 3
! | 4
! 4
! | 5
! 5
! | 6
! 6
! | 7
! 7
! | 8
! 8
! | 9
! 9
! | 10
! 10
! | 11
! 11
|}
|}


== Links and Articles ==
== Links and Articles ==


* [[Collection of edo impressions]]
* [[Collection of EDO impressions]]
* [[Chuckles McGee's EDO personalities]]
* [[Chuckles McGee's EDO personalities]]
* [[Macrotonal edos]]
* [[Macrotonal EDO]]
* [[Expression to EDO calculator]]
* [[Expression to EDO calculator]]
* [http://www.webcitation.org/5xZz8RtQB Teen Tunes] by [[Ivor Darreg]]
* [http://www.webcitation.org/5xZz8RtQB Teen Tunes] by [[Ivor Darreg]]
Line 1,141: Line 1,143:
* [[Minimal consistent EDOs]]
* [[Minimal consistent EDOs]]
* [[Consistency levels of small EDOs]]
* [[Consistency levels of small EDOs]]
* [[Relative errors of small edos]]
* [[Relative errors of small EDOs]]
* [[Distinct EDO Scales]]
* [[Distinct EDO Scales]]
* [[List of rank one temperaments by step size]]
* [[List of rank one temperaments by step size]]
Line 1,151: Line 1,153:
[[Category:Table]]
[[Category:Table]]
[[Category:Abbreviation]]
[[Category:Abbreviation]]
[[Category:Todo:discuss title]]
Retrieved from "https://en.xen.wiki/w/EDO"