11edo: Difference between revisions
No edit summary |
|||
| (9 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
'''11edo''', or 11 equal divisions of the octave (sometimes called '''11-TET''' or '''11-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1. | '''11edo''', or 11 equal divisions of the octave (sometimes called '''11-TET''' or '''11-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1. | ||
11edo lacks a diatonic ([[5L 2s]]) [[perfect fifth|fifth]], or even an [[ | 11edo lacks a diatonic ([[5L 2s]]) [[perfect fifth|fifth]], or even an [[armotonic]] or [[oneirotonic]] fifth, with the 3rd harmonic nearly halfway between its steps. The [[5/4|5th harmonic]], as well, lies right in between its steps. However, the errors cancel to allow 11edo representations of intervals such as [[9/8]], [[5/3]], and [[15/8]]. | ||
11edo can also be considered to approximate the 7th, 11th, and 17th harmonics, with 9\11 approximating [[7/4]], 5\11 approximating [[11/8]], and 1\11 approximating [[17/16]]. In fact, [[22edo]]'s approximations to intervals of 7, 9, 11, 15, and 17 all come from 11edo. Given the logic of 22edo as an [[archy]] system, the same interval represents both 9/8 and [[8/7]]. | 11edo can also be considered to approximate the 7th, 11th, and 17th harmonics, with 9\11 approximating [[7/4]], 5\11 approximating [[11/8]], and 1\11 approximating [[17/16]]. In fact, [[22edo]]'s approximations to intervals of 7, 9, 11, 15, and 17 all come from 11edo. Given the logic of 22edo as an [[archy]] system, the same interval represents both 9/8 and [[8/7]]. | ||
The result of all this is that while 11edo forces harmony to arrange in ways extremely alien to a perspective based on the [[3-limit|3-]] or [[5-limit]], it still retains a breadth of approximation that allows for a complete tonal system to be built around 11edo's properties. Smitonic ( | The result of all this is that while 11edo forces harmony to arrange in ways extremely alien to a perspective based on the [[3-limit|3-]] or [[5-limit]], it still retains a breadth of approximation that allows for a complete tonal system to be built around 11edo's properties. Smitonic (4L 3s), generated by 3\11 (representing [[6/5]] and [[11/9]]), can serve as a useful basis scale for navigating 11edo. The simple [[JI]] interval approximated best by 11edo is [[9/7]], at 1.3{{c}} sharp, while 5/3 and 7/4 can be used as bounding consonances for chords. | ||
== General theory == | == General theory == | ||
| Line 15: | Line 15: | ||
=== Edostep interpretations === | === Edostep interpretations === | ||
One step of 11edo can be interpreted in the 2.9.15.7.17 subgroup as | One step of 11edo can be interpreted in the 2.9.15.7.17 subgroup as: | ||
* [[18/17]] (the interval between 17/16 and 9/8, or [[17/14]] and 9/7) | * [[18/17]] (the interval between 17/16 and 9/8, or [[17/14]] and 9/7) | ||
* 17/16 (the octave-reduced 17th harmonic) | * 17/16 (the octave-reduced 17th harmonic) | ||
| Line 34: | Line 34: | ||
In 22edo, as the diatonic MOS has [[hardness]] 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C. | In 22edo, as the diatonic MOS has [[hardness]] 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C. | ||
This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is smitonic (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance. | This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is [[smitonic]] (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance. | ||
For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system. | For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system. This is in accordance with KISS notation. | ||
Interval | "Interval categories" are based on 22edo usage of the [[ADIN]] system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and [[7-limit]] qualities. | ||
JI approximations (within the 25-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded. | JI approximations (within the 25-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded. | ||
| Line 56: | Line 56: | ||
|- | |- | ||
|1 | |1 | ||
|109 | |109.1 | ||
|[16/15], 15/14, 18/17, ['''17/16'''] | |[16/15], 15/14, 18/17, ['''17/16'''] | ||
|vC#, ^Db | |vC#, ^Db | ||
| Line 64: | Line 64: | ||
|- | |- | ||
|2 | |2 | ||
|218 | |218.2 | ||
|8/7, '''9/8''', [17/15], [25/22] | |8/7, '''9/8''', [17/15], [25/22] | ||
|D | |D | ||
| Line 72: | Line 72: | ||
|- | |- | ||
|3 | |3 | ||
|327 | |327.3 | ||
|6/5, 11/9, 17/14 | |6/5, 11/9, 17/14 | ||
|vD#, ^Eb | |vD#, ^Eb | ||
| Line 80: | Line 80: | ||
|- | |- | ||
|4 | |4 | ||
|436 | |436.4 | ||
|[9/7], 14/11, 32/25 | |[9/7], 14/11, 32/25 | ||
|E | |E | ||
| Line 88: | Line 88: | ||
|- | |- | ||
|5 | |5 | ||
|545 | |545.5 | ||
|'''11/8''', 15/11 | |'''11/8''', 15/11 | ||
|^F, Gb | |^F, Gb | ||
| Line 96: | Line 96: | ||
|- | |- | ||
|6 | |6 | ||
| | |654.5 | ||
|16/11, 22/15 | |16/11, 22/15 | ||
|F#, vG | |F#, vG | ||
| Line 104: | Line 104: | ||
|- | |- | ||
|7 | |7 | ||
| | |763.6 | ||
|[14/9], 11/7, '''25/16''' | |[14/9], 11/7, '''25/16''' | ||
|^G, Ab | |^G, Ab | ||
| Line 112: | Line 112: | ||
|- | |- | ||
|8 | |8 | ||
| | |872.7 | ||
|5/3, 18/11, 28/17 | |5/3, 18/11, 28/17 | ||
|G#, vA | |G#, vA | ||
| Line 120: | Line 120: | ||
|- | |- | ||
|9 | |9 | ||
| | |981.8 | ||
|'''7/4''', 16/9, [30/17], [44/25] | |'''7/4''', 16/9, [30/17], [44/25] | ||
|^A, Bb | |^A, Bb | ||
| Line 128: | Line 128: | ||
|- | |- | ||
|10 | |10 | ||
| | |1090.9 | ||
|['''15/8'''], 28/15, 17/9, [32/17] | |['''15/8'''], 28/15, 17/9, [32/17] | ||
|A#, vB | |A#, vB | ||
| Line 146: | Line 146: | ||
== Compositional theory == | == Compositional theory == | ||
=== Chords === | === Chords === | ||
11edo has numerous xenharmonic sounding chords. | 11edo has numerous xenharmonic sounding chords. | ||
One approach to harmony in 11edo is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. If 3\11 (the nearminor third) is chosen as the third in question, these chords are present in the smitonic scale, for which 3\11 serves as the generator, and the chords that this begets are [0 3 6] and [0 3 7]\11. Notably, these can be inverted to [0 3 8], [0 5 8], and [0 4 8]\11, all of which are triads bounded by 8\11, which represents the consonance 5/3. The last of these also constitutes a stack of the very well-tuned interval 9/7. | |||
Another is | Another important bounding interval in 11edo is 7/4, represented by 9\11, and important chords bounded by it include [0 5 9] and [0 4 9]\11, which represent 8:11:14 and its complement. | ||
=== Scales === | === Scales === | ||
As a subset of 22edo, 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset. Firstly, you may note that in 11edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo's patent val.) | |||
==== 11edo scales ==== | |||
MOS scales generated from 11edo temperaments. | |||
{| class="wikitable" | |||
!Temperament | |||
!Name | |||
!Chart | |||
!Notes | |||
|- | |||
| rowspan="2" |Sensamagic | |||
|antipentic | |||
|{{Interval ruler|22|0, 330, 430, 770, 870, 1200}} | |||
|Has swapped interval sizes relative to the normal pentatonic scale. | |||
|- | |||
|checkertonic | |||
|{{Interval ruler|22|0, 100, 330, 430, 530, 770, 870, 970, 1200}} | |||
|Can be considered the "basic" 8-form scale of sensamagic harmony. The basic 1/1-9/8-5/3 chord exists on almost all keys, and inversions thereof can be considered as the basic chords of the system. Has similar step sizes to 12edo diatonic, allowing for familiar kinds of melodic shapes to be used against entirely novel harmony | |||
|- | |||
| rowspan="2" |Machine | |||
|pedal | |||
|{{Interval ruler|22|0, 220, 440, 660, 880, 1200}} | |||
| | |||
|- | |||
|machinoid | |||
|{{Interval ruler|22|0, 220, 440, 660, 880, 1100, 1200}} | |||
|Acts as one possible 22edo counterpart to the "whole tone scale". Highlights the chords used by antidiatonic, but now features both simultaneously on 3 of its degrees. | |||
|- | |||
|Orgone | |||
|smitonic | |||
|{{Interval ruler|22|0, 100, 330, 430, 660, 760, 990, 1200}} | |||
|Acts as an altered version of 12edo diatonic, and can be seen as an 11edo analog to diatonic, being generated by its primary consonance (5/3 in this case). It reaches 9/7 after five scale steps and places it on the same degree as 11/8. | |||
|- | |||
| rowspan="3" |Joan | |||
|pentic | |||
|{{Interval ruler|22|0, 100, 550, 650, 760, 1200}} | |||
|One of two tunings of pentic available in 22edo. | |||
|- | |||
|antidiatonic | |||
|{{Interval ruler|22|0, 100, 200, 550, 650, 760, 870, 1200}} | |||
|Has swapped interval sizes relative to diatonic, and therefore diatonic chord/harmony logic can be loosely "translated" into antidiatonic. This can be seen as a system generated by wolf fifths. A fifth tuned flat enough to generate a hard antidiatonic scale is called a "zavala" fifth. The "third" in this system is either 8/7 or 9/7. The "fifth" is 16/11, meaning that the counterparts of diatonic major and minor chords are "essentially tempered", because they are bounded by 16/11 yet composed of only 7-limit intervals. Additionally, this system allows /7 harmony, with 7:9:11 being an available chord on some degrees. | |||
|- | |||
|balzano | |||
|{{Interval ruler|22|0, 100, 200, 330, 550, 650, 760, 870, 980, 1200}} | |||
|Acts as an expanded version of antidiatonic with additional notes. Many of the same harmonic rules apply, except that in this case 8/7 and 9/7 fall on different degrees (both shared with 6/5), so two possible "tertian" chords exist on a single degree. | |||
|} | |||
Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 1-2-1-2-1-2-2.{{Navbox EDO}} | |||
{{Cat|Edos}} | {{Cat|Edos}} | ||
Latest revision as of 20:45, 16 May 2026
11edo, or 11 equal divisions of the octave (sometimes called 11-TET or 11-tone equal temperament), is the equal tuning featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1.
11edo lacks a diatonic (5L 2s) fifth, or even an armotonic or oneirotonic fifth, with the 3rd harmonic nearly halfway between its steps. The 5th harmonic, as well, lies right in between its steps. However, the errors cancel to allow 11edo representations of intervals such as 9/8, 5/3, and 15/8.
11edo can also be considered to approximate the 7th, 11th, and 17th harmonics, with 9\11 approximating 7/4, 5\11 approximating 11/8, and 1\11 approximating 17/16. In fact, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 all come from 11edo. Given the logic of 22edo as an archy system, the same interval represents both 9/8 and 8/7.
The result of all this is that while 11edo forces harmony to arrange in ways extremely alien to a perspective based on the 3- or 5-limit, it still retains a breadth of approximation that allows for a complete tonal system to be built around 11edo's properties. Smitonic (4L 3s), generated by 3\11 (representing 6/5 and 11/9), can serve as a useful basis scale for navigating 11edo. The simple JI interval approximated best by 11edo is 9/7, at 1.3¢ sharp, while 5/3 and 7/4 can be used as bounding consonances for chords.
General theory
JI approximation
11edo does not approximate harmonics 3 or 5 well at all, both ratios falling almost directly between the step sizes; however, this makes their difference tone 5/3 somewhat accurate. Using dual fifths, 11edo can approximate ratios of 9 such as 9/7, as well as ratios of 15 and 25. In addition, 11edo approximates 7/4, 11/8, and 17/16 quite well, making it a 2.9.15.7.11.17 subgroup system.
One may also note that the chord 13:19:23:26 is well-tuned in 11edo, even if the individual primes 13, 19, and 23 are not.
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -47.4 | +50.0 | +13.0 | +14.3 | -5.9 | +32.2 | +2.6 | +4.1 | +29.8 | -34.4 | +26.3 | -9.0 |
| Relative (%) | -43.5 | +45.9 | +11.9 | +13.1 | -5.4 | +29.5 | +2.4 | +3.8 | +27.3 | -31.5 | +24.1 | -8.2 | |
| Steps
(reduced) |
17
(6) |
26
(4) |
31
(9) |
35
(2) |
38
(5) |
41
(8) |
43
(10) |
45
(1) |
47
(3) |
48
(4) |
50
(6) |
51
(7) | |
Edostep interpretations
One step of 11edo can be interpreted in the 2.9.15.7.17 subgroup as:
- 18/17 (the interval between 17/16 and 9/8, or 17/14 and 9/7)
- 17/16 (the octave-reduced 17th harmonic)
- 16/15 (the interval between 9/8 and 6/5)
- 15/14 (the interval between 6/5 and 9/7).
11edo's edostep carries the following additional interpretations in the 2.5/3.7.11.17 subgroup:
- 35/33 (the interval between 6/5 and 14/11)
- 128/121 (the interval between 11/8 and 16/11)
- 128/119 (the interval between 17/16 and 8/7)
- 121/112 (the interval between 14/11 and 11/8).
Intervals and notation
As 11edo does not have a chain of fifths (unless one counts the hard 2L 5s scale generated by 6\11) suitable for notation, the task of notating 11edo proves challenging.
One approach is to treat 11edo as a subset of 22edo. The native-fifths notation for 22edo is derived through stacking 22edo's tempered version of 3/2 (13\22) and assigning names accordingly. Only every other position in the chain of fifths is a note of 11edo, so therefore starting from C, the notes C, D, E, Gb, F#, Ab, G#, Bb, and A# are part of 11edo.
In 22edo, as the diatonic MOS has hardness 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C.
This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is smitonic (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.
For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system. This is in accordance with KISS notation.
"Interval categories" are based on 22edo usage of the ADIN system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and 7-limit qualities.
JI approximations (within the 25-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
| Edostep | Cents | 2.9.15.7.11.17 JI approximation |
Notation | Smitonic interval quality | Interval category (ADIN) | |
|---|---|---|---|---|---|---|
| 22edo subset | Smitonic | |||||
| 0 | 0 | 1/1 | C | 1 | Perfect unison | Perfect unison |
| 1 | 109.1 | [16/15], 15/14, 18/17, [17/16] | vC#, ^Db | 1#, 2b | Minor second | Nearminor second |
| 2 | 218.2 | 8/7, 9/8, [17/15], [25/22] | D | 2 | Major second | Supermajor second |
| 3 | 327.3 | 6/5, 11/9, 17/14 | vD#, ^Eb | 3 | Perfect third | Nearminor third |
| 4 | 436.4 | [9/7], 14/11, 32/25 | E | 3#, 4b | Minor fourth, augmented third | Supermajor third |
| 5 | 545.5 | 11/8, 15/11 | ^F, Gb | 4 | Major fourth | Near fourth |
| 6 | 654.5 | 16/11, 22/15 | F#, vG | 4#, 5b | Minor fifth | Near fifth |
| 7 | 763.6 | [14/9], 11/7, 25/16 | ^G, Ab | 5 | Major fifth, diminished sixth | Subminor sixth |
| 8 | 872.7 | 5/3, 18/11, 28/17 | G#, vA | 6 | Perfect sixth | Nearmajor sixth |
| 9 | 981.8 | 7/4, 16/9, [30/17], [44/25] | ^A, Bb | 6#, 7b | Minor seventh | Subminor seventh |
| 10 | 1090.9 | [15/8], 28/15, 17/9, [32/17] | A#, vB | 7 | Major seventh | Nearmajor seventh |
| 11 | 1200 | 2/1 | C | 1 | Perfect octave | Octave |
Compositional theory
Chords
11edo has numerous xenharmonic sounding chords.
One approach to harmony in 11edo is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. If 3\11 (the nearminor third) is chosen as the third in question, these chords are present in the smitonic scale, for which 3\11 serves as the generator, and the chords that this begets are [0 3 6] and [0 3 7]\11. Notably, these can be inverted to [0 3 8], [0 5 8], and [0 4 8]\11, all of which are triads bounded by 8\11, which represents the consonance 5/3. The last of these also constitutes a stack of the very well-tuned interval 9/7.
Another important bounding interval in 11edo is 7/4, represented by 9\11, and important chords bounded by it include [0 5 9] and [0 4 9]\11, which represent 8:11:14 and its complement.
Scales
As a subset of 22edo, 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset. Firstly, you may note that in 11edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the sensamagic category of temperaments, which in 11edo specifically becomes sentry. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo's patent val.)
11edo scales
MOS scales generated from 11edo temperaments.
| Temperament | Name | Chart | Notes |
|---|---|---|---|
| Sensamagic | antipentic | ├─────┴─┴─────┴─┴─────┤ 6 2 6 2 6 | Has swapped interval sizes relative to the normal pentatonic scale. |
| checkertonic | ├─┴───┴─┴─┴───┴─┴─┴───┤ 2 4 2 2 4 2 2 4 | Can be considered the "basic" 8-form scale of sensamagic harmony. The basic 1/1-9/8-5/3 chord exists on almost all keys, and inversions thereof can be considered as the basic chords of the system. Has similar step sizes to 12edo diatonic, allowing for familiar kinds of melodic shapes to be used against entirely novel harmony | |
| Machine | pedal | ├───┴───┴───┴───┴─────┤ 4 4 4 4 6 | |
| machinoid | ├───┴───┴───┴───┴───┴─┤ 4 4 4 4 4 2 | Acts as one possible 22edo counterpart to the "whole tone scale". Highlights the chords used by antidiatonic, but now features both simultaneously on 3 of its degrees. | |
| Orgone | smitonic | ├─┴───┴─┴───┴─┴───┴───┤ 2 4 2 4 2 4 4 | Acts as an altered version of 12edo diatonic, and can be seen as an 11edo analog to diatonic, being generated by its primary consonance (5/3 in this case). It reaches 9/7 after five scale steps and places it on the same degree as 11/8. |
| Joan | pentic | ├─┴───────┴─┴─┴───────┤ 2 8 2 2 8 | One of two tunings of pentic available in 22edo. |
| antidiatonic | ├─┴─┴─────┴─┴─┴─┴─────┤ 2 2 6 2 2 2 6 | Has swapped interval sizes relative to diatonic, and therefore diatonic chord/harmony logic can be loosely "translated" into antidiatonic. This can be seen as a system generated by wolf fifths. A fifth tuned flat enough to generate a hard antidiatonic scale is called a "zavala" fifth. The "third" in this system is either 8/7 or 9/7. The "fifth" is 16/11, meaning that the counterparts of diatonic major and minor chords are "essentially tempered", because they are bounded by 16/11 yet composed of only 7-limit intervals. Additionally, this system allows /7 harmony, with 7:9:11 being an available chord on some degrees. | |
| balzano | ├─┴─┴─┴───┴─┴─┴─┴─┴───┤ 2 2 2 4 2 2 2 2 4 | Acts as an expanded version of antidiatonic with additional notes. Many of the same harmonic rules apply, except that in this case 8/7 and 9/7 fall on different degrees (both shared with 6/5), so two possible "tertian" chords exist on a single degree. |
Another temperament that resides in 11edo is called orgone, and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 1-2-1-2-1-2-2.
