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An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'') is a tuning system constructed by dividing the octave into a number of equal steps.
An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps. It is a type of equal temperament.


The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.
The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.


An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.
An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties, as will one with the same number of notes as the MOS has L steps. These two edos form the boundaries of how the MOS can be tuned.


The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').
The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').
== Uses ==
Edos are the most common type of tuning system in contemporary xenharmony. Unlike other types such as rank-2 temperaments and just intonation scales, equal temperaments allow for free modulation and transposition due to their uniform step size. That is, every n-step interval is the same as every other n-step interval. This comes at the expense of less freedom in approximating target intervals. It also encourages a less structured approach to composition where pitch shifts and interval quality changes can happen without much deeper meaning.


== List of edos ==
== List of edos ==
Line 17: Line 21:
!Fifth (¢)
!Fifth (¢)
!Edostep interpretation
!Edostep interpretation
! colspan="2" |Example basic (in 2...23) and [[erac]] groups
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
|-
| rowspan="2" |1
| rowspan="2" |1
Line 23: Line 27:
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |
| rowspan="2" |2/1
|2
|2
|-
|-
Line 32: Line 36:
| rowspan="2" |600, 1200
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |600
| rowspan="2" |
| rowspan="2" |''none available in basic subgroup''
|2.11.23?
|2
|-
|-
|2.<3.>>5.>>7?
|2.<3.>>5.>>7.<17
|-
|-
| rowspan="2" |3
| rowspan="2" |3
| rowspan="2" |A 12edo augmented triad.
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |800
| rowspan="2" |
| rowspan="2" |5/4
|2.5.13?
|2.5
|-
|-
|2.>3.5?
|2.>3.5.>19?
|-
|-
| rowspan="2" |4
| rowspan="2" |4
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| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |600
| rowspan="2" |
| rowspan="2" |19/16
|2.19.23?
|2.19
|-
|-
|2.<3.<5.<7?
|2.<3.<5.<7.<17
|-
|-
| rowspan="2" class="thl" |5
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |720
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|-
|-
| rowspan="2" |6
| rowspan="2" |6
| rowspan="2" |Subset of 12edo. Good approximation of Didacus temperament.
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo. Good approximation of 2.5.7 for its size.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |600, 800
| rowspan="2" |
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5.7.23
|2.9.5
|-
|-
|2.xx3.5.7.<11.>13.23
|2.9.5.>7
|-
|-
| rowspan="2" class="thl" |7
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.13
|2.3.5.11.13
|-
|-
|2.<3.<<5.>13
|2.<3.<<5.>13
|-
|-
| rowspan="2" |8
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Minimal version of Ammonite temperament.
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |750
| rowspan="2" |
| rowspan="2" |''none available in basic subgroup''
|2.19.23
|2.19
|-
|-
|2.x3.x5.x7.x11.x13
|2.x3.x5.x7.x11.x13
|-
|-
| rowspan="2" |9
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale and temperaments like [[Mabilic|Semabila]], loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).  
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).  
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |666.7
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|2.<<3.>5.<<7
|2.<<3.>5.<<7
|-
|-
| rowspan="2" |10
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
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|-
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Good example of Orgone.
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |654.5, 763.6
| rowspan="2" |
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11.15
|2.9.7.11
|-
|-
|2.x3.x5.7.11
|2.x3.x5.7.11
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|{{First 12 edo intervals|edo=13}}
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|646.2, 738.5
|
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17
|2.5.11.13.17.19.23
|-
|-
|[[14edo|14]]
|[[14edo|14]]
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|-
|-
| class="thl" |[[15edo|15]]
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting Porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|{{First 12 edo intervals|edo=15}}
|720
|720
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|{{First 12 edo intervals|edo=16}}
|{{First 12 edo intervals|edo=16}}
|675, 750
|675, 750
|
|20/19, 133/128, 26/25
|2.5.7.13.19
|2.5.7.13.19
|-
|-
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|{{First 12 edo intervals|edo=17}}
|{{First 12 edo intervals|edo=17}}
|705.9
|705.9
|
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|2.3.13.23
|-
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[18edo|18]]
| rowspan="2" |Straddle-3 version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.  
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.  
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |666.6, 733.3
| rowspan="2" |
| rowspan="2" |
|2.9.5.7.13
|2.9.5.21.13
|-
|-
|2.xx3.>5.>>7.<11.<13
|2.xx3.>5.>>7.<11.<13
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|2.3.5.7.11.17
|2.3.5.7.11.17
|-
|-
|2.3.5.7>.11
|2.>3.<5.>>7.<11
|-
|-
|23
|23
Line 224: Line 228:
|2.5.7.19
|2.5.7.19
|-
|-
|26
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|{{First 12 edo intervals|edo=26}}
Line 274: Line 278:
|-
|-
|33
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports Semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|{{First 12 edo intervals|edo=33}}
|690.9
|690.9
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|-
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an Archy 3, as in Porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |681.1, 713.5
Line 314: Line 318:
|-
|-
|38
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|19edo with neutrals. Functions as a tuning of Mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|{{First 12 edo intervals|edo=38}}
|694.7
|694.7
Line 327: Line 331:
|2.3.11
|2.3.11
|-
|-
|40
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|{{First 12 edo intervals|40|edo=40}}
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|-
|-
| class="thl" |[[41edo|41]]
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|The first reasonably accurate [[Hemifamity]] edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|{{First 12 edo intervals|edo=41}}
|702.4
|702.4
|
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|2.3.5.7.11.13.19
|-
|-
|42
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma Archy.
|{{First 12 edo intervals|edo=42}}
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|685.7, 714.3
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|-
|-
|43
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|Huygens]]". Like all Meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|{{First 12 edo intervals|edo=43}}
|697.7
|697.7
Line 363: Line 367:
|-
|-
|45
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|A nearly optimal tuning of Flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|693.3, 720
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|-
|-
| class="thl"|46
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|The second reasonably accurate [[Hemifamity]] edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|{{First 12 edo intervals|edo=46}}
|704.3
|704.3
Line 377: Line 381:
|-
|-
|47
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|The first edo with two distinct mosdiatonic scales. Supports Magic and Archy with its sharp fifth and Deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of Schismic analogue of Didacus.
|{{First 12 edo intervals|edo=47}}
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|689.4, 714.9
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|-
|-
|48
|48
|Four times 12edo, associated with buzzard temperament.
|Four times 12edo, associated with [[Buzzard]] temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|700
Line 391: Line 395:
|-
|-
|49
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|A nearly optimal tuning of Archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 Meantone (or, more conventionally, Didacus).
|{{First 12 edo intervals|edo=49}}
|{{First 12 edo intervals|edo=49}}
|710.2
|710.2
Line 398: Line 402:
|-
|-
|50
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|Approaches golden Meantone, and serves as a definitive tuning of Meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|{{First 12 edo intervals|edo=50}}
|696
|696
Line 412: Line 416:
|-
|-
|52
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|Doubles 26edo, adding a sharp Archy fifth and a more accurate 5/4 which support Porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|692.3, 715.4
Line 422: Line 426:
|{{First 12 edo intervals|edo=53}}
|{{First 12 edo intervals|edo=53}}
|701.9
|701.9
|
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|2.3.5.7.13.19
|-
|-
|54
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|Double 27edo, and the sharper end of the Pajara tuning range. Can alternatively be used as a very flat Deeptone system or combining the fifths as a straddle-fifth system.
|
|
|688.9, 711.1
|688.9, 711.1
Line 433: Line 437:
|-
|-
|55
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|A very sharp Meantone tuning, which is so sharp that it does not even support Septimal Meantone, and is best interpreted as Mohajira as it pertains to Meantone extensions.
|
|
|698.2
|698.2
Line 461: Line 465:
|-
|-
|59
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports Porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
Line 468: Line 472:
|-
|-
|60
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|5 sets of 12edo, supporting Magic temperament and having 10edo's 7 and 13, also supporting 7-limit Compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
Line 496: Line 500:
|-
|-
|64
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of Flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|693.8, 712.5
Line 503: Line 507:
|-
|-
|65
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|A non-Garibaldi Schismic system (in fact, it supports Sensi), and a straddle-7 and -13 system.
|{{First 12 edo intervals|edo=65}}
|{{First 12 edo intervals|edo=65}}
|701.5
|701.5
Line 517: Line 521:
|-
|-
|67
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|Approximate 1/6-comma Meantone and Slendric edo, which also supports [[Orgone]].
|{{First 12 edo intervals|edo=67}}
|{{First 12 edo intervals|edo=67}}
|698.5
|698.5
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|-
|-
|68
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting Sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|{{First 12 edo intervals|edo=68}}
|705
|705
Line 531: Line 535:
|-
|-
|69
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|A nice tuning that is approximately 2/7-comma Meantone, somewhat between standard Septimal Meantone and 19edo. As a result, it is a Mohajira system (setting 7/4 to the semiflat minor seventh) but not a Septimal Meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports Archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|695.7, 713
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|-
|-
|70
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a [[Hemifamity]] system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|{{First 12 edo intervals|edo=70}}
|702.9
|702.9
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|-
|-
|71
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|A dual-fifth system. The sharp fifth is within the Superpyth tuning range (and produces the same mapping for 5 as Superpyth), despite not supporting Archy. The flat fifth, analogously, produces Flattone's mapping for 7 and is well-tuned for Flattone, but does not support Flattone.
|{{First 12 edo intervals|edo=71}}
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|693. 709.9
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|-
|-
|72
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|A multiple of 12edo and a very good Miracle and Compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|{{First 12 edo intervals|edo=72}}
|700
|700
|
|
|2.3.5.7.11.17.19.23
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support Meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
|-
| colspan="6" |...
| colspan="6" |...
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|-
|-
|93
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.  
|The triple of 31edo. As a Meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.  
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|{{First 12 edo intervals|edo=93}}
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|-
|-
|94
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of garibaldi.  
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.  
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|{{First 12 edo intervals|edo=94}}
|702.1
|702.1
|
|
| -
|-
| colspan="6" |...
|-
|99
|Perhaps the strongest 7-limit edo below 100. Supports [[Hemifamity]], [[Didacus]], and [[Ennealimmal]].
|{{First 12 edo intervals|edo=99}}
|703.0
|126/125, 225/224, kleisma
| -
| -
|-
|-
Line 590: Line 628:
|140
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|{{First 12 edo intervals|edo=140}}
|702.9
|
|
|
| -
|
|
|-
|-
| colspan="6" |...
| colspan="6" |...
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|701.9
|701.9
|
|
| -
|-
| colspan="6" |...
|-
|171
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.
|{{First 12 edo intervals|edo=171}}
|701.8
| 225/224, 5120/5103, [[kleisma]]
| -
| -
|-
|-
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|-
|-
|200
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[schismic]] and [[Slendric|gamelic]] system with an 8/7 of exactly 234 cents.
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|{{First 12 edo intervals|edo=200}}
|702
|702
|
|
| -
|-
| colspan="6" |...
|-
|270
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.
|{{First 12 edo intervals|edo=270}}
|702.2
|385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440
| -
| -
|-
|-

Latest revision as of 00:59, 4 April 2026

An equal division of the octave (EDO or edo, /ˈidoʊ/ EE-doh or /idiˈoʊ/ ee-dee-OH) is a tuning system constructed by dividing the octave into a number of equal steps. It is a type of equal temperament.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain MOS will have crudely similar properties, as will one with the same number of notes as the MOS has L steps. These two edos form the boundaries of how the MOS can be tuned.

The notation m\n denotes m steps of n-edo, i.e. the frequency ratio 2^(m/n).

Uses

Edos are the most common type of tuning system in contemporary xenharmony. Unlike other types such as rank-2 temperaments and just intonation scales, equal temperaments allow for free modulation and transposition due to their uniform step size. That is, every n-step interval is the same as every other n-step interval. This comes at the expense of less freedom in approximating target intervals. It also encourages a less structured approach to composition where pitch shifts and interval quality changes can happen without much deeper meaning.

List of edos

Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.

Popular edos are highlighted. Temperaments are capitalized and can be found in the List of regular temperaments.
Edo Description First twelve steps (¢) Fifth (¢) Edostep interpretation Example basic (in 2...23, primes and 9) and erac groups
1 Equivalent to the 2-limit. 1200 1200 2/1 2
2
2 Just a 12edo tritone. 600, 1200 600 none available in basic subgroup 2
2.<3.>>5.>>7.<17
3 An augmented triad. 400, 800, 1200 800 5/4 2.5
2.>3.5.>19?
4 A diminished tetrad. 300, 600, 900, 1200 600 19/16 2.19
2.<3.<5.<7.<17
5 Equalized pentic, collapsed diatonic, and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size. 240, 480, 720, 960, 1200 720 9/8, 8/7, 7/6 2.3.7
2.>>3.<7
6 Also known as the whole-tone scale, 6edo is a subset of 12edo. Good approximation of 2.5.7 for its size. 200, 400, 600, 800, 1000, 1200 600, 800 9/8, 10/9, 28/25, 8/7 2.9.5
2.9.5.>7
7 Equalized diatonic, and the first edo to (very vaguely) support diatonic functional harmony. 171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200 685.7 9/8, 10/9, 16/15 2.3.5.11.13
2.<3.<<5.>13
8 Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10. 150, 300, 450, 600, 750, 900, 1050, 1200 750 none available in basic subgroup 2.19
2.x3.x5.x7.x11.x13
9 The first edo to support the antidiatonic scale, loosely resembling the pelog scale. It contains approximations to many 7-limit intervals, but not the 7/4 itself (see erac group). 133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200 666.7 9/8, 16/15, 25/24 2.5.11
2.<<3.>5.<<7
10 The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting mosh. 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200 720 16/15, 10/9, 81/80, 36/35 2.3.5.7.13
2.>>3.<7.13
11 Basic smitonic and checkertonic. Simplest reasonable tuning of Orgone. 109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9 654.5, 763.6 128/119, 17/16, 18/17 2.9.7.11
2.x3.x5.7.11
12 The basic tuning of diatonic, and consequently the most widespread EDO. Supports the 5-limit decently well. 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200 700 256/243, chromatic semitone, 16/15, 25/24 2.3.5.17.19
2.3.>5.>>7.17.19
13 Basic oneirotonic, archeotonic, and gramitonic. 92.3, 184.6, 276.9, 369.2, 461.5, 553.8, 646.2, 738.5, 830.8, 923.1, 1015.4, 1107.7 646.2, 738.5 17/16, 18/17, 19/18, 20/19 2.5.11.13.17.19.23
14 Basic semiquartal. 85.7, 171.4, 257.1, 342.9, 428.6, 514.3, 600, 685.7, 771.4, 857.1, 942.9, 1028.6 685.7 28/27, 21/20, 15/14 2.3.7.13
15 The basic tuning of Zarlino's intense diatonic, a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting Porcupine temperament and dubitably the 11-limit. 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960 720 81/80, 25/24, 16/15, 33/32, 36/35 2.3.5.7.11.23
16 The most popular antidiatonic edo, which supports Trismegistus and Mavila. 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900 675, 750 20/19, 133/128, 26/25 2.5.7.13.19
17 Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW. 70.6, 141.2, 211.8, 282.4, 352.9, 423.5, 494.1, 564.7, 635.3, 705.9, 776.5, 847.1 705.9 256/243, 24/23, 27/26, 33/32 2.3.13.23
18 Straddle-3 version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric. 66.7, 133.3, 200, 266.7, 333.3, 400, 466.7, 533.3, 600, 666.7, 733.3, 800 666.6, 733.3 2.9.5.21.13
2.xx3.>5.>>7.<11.<13
19 A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament. 63.2, 126.3, 189.5, 252.6, 315.8, 378.9, 442.1, 505.3, 568.4, 631.6, 694.7, 757.9 694.7 25/24, diaschisma, 36/35, 28/27 2.3.5.23
20 Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads. 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720 660, 720 2.7.11.13.19
21 Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads. 57.1, 114.3, 171.4, 228.6, 285.7, 342.9, 400, 457.1, 514.3, 571.4, 628.6, 685.7 685.7, 742.9 2.3.5.7.23
2.x>3.x<5.7.x<11.x<13.23
22 Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy. 54.5, 109.1, 163.6, 218.2, 272.7, 327.3, 381.8, 436.4, 490.9, 545.5, 600, 654.5 709.1 2.3.5.7.11.17
2.>3.<5.>>7.<11
23 The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic. 52.2, 104.3, 156.5, 208.7, 260.9, 313, 365.2, 417.4, 469.6, 521.7, 573.9, 626.1 678.3 2.x3.x5.x7.x11.13.17.23
24 Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval. 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600 700 2.3.11.13.17.19
25 A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification. 48, 96, 144, 192, 240, 288, 336, 384, 432, 480, 528, 576 720 2.5.7.19
26 A simple tuning of Flattone. Has an absurdly accurate 7/4. 46.2, 92.3, 138.5, 184.6, 230.8, 276.9, 323.1, 369.2, 415.4, 461.5, 507.7, 553.8 692.7 2.3.7.11.13
27 A good tuning for Archy and Sensi. It has 3/2 at 16 steps. 44.4, 88.9, 133.3, 177.8, 222.2, 266.7, 311.1, 355.6, 400, 444.4, 488.9, 533.3 711.1 2.3.5.7.13.23
28 A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard oneirotonic scale converging on Buzzard temperament. 42.9, 85.7, 128.6, 171.4, 214.3, 257.1, 300, 342.9, 385.7, 428.6, 471.4, 514.3 685.7, 728.6 2.3.5.7.11
29 Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps. 41.4, 82.8, 124.1, 165.5, 206.9, 248.3, 289.7, 331, 372.4, 413.8, 455.2, 496.6 703.4 2.3.7/5.11/5.13/5.19.23
30 Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8. 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440, 480 680, 720 2.x3.x5.7.11.13
31 The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification. 38.7, 77.4, 116.1, 154.8, 193.5, 232.3, 271, 309.7, 348.4, 387.1, 425.8, 464.5 696.8 2.3.5.7.11.23
32 A standard tuning of Ultrapyth (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic. 37.5, 75, 112.5, 150, 187.5, 225, 262.5, 300, 337.5, 375, 412.5, 450 712.5 2.3.7.11.17.19.23
33 Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports Semaphore with the flat 7/4, which can be interpreted as Barbados temperament in the patent val. 36.4, 72.7, 109.1, 145.5, 181.8, 218.2, 254.5, 290.9, 327.3, 363.6, 400, 436.4 690.9 2.3.11.13.17.19.23
34 An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.

It is the double of 17edo, which it takes its circle of fifths from.

35.3, 70.6, 105.9, 141.2, 176.5, 211.8, 247.1, 282.4, 317.6, 352.9, 388.2, 423.5 705.9 2.3.5.13.23
2.3.5.x7.13.x19.23
35 Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system. 34.3, 68.6, 102.9, 137.1, 171.4, 205.7, 240, 274.3, 308.6, 342.9, 377.1, 411.4 685.7, 720.0 2.5.7.11.17
36 Triple 12edo, which functions as an extremely accurate septal Compton and Slendric system. 33.3, 66.7, 100, 133.3, 166.7, 200, 233.3, 266.7, 300, 333.3, 366.7, 400 700 2.3.7.13.17.19.23
37 An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an Archy 3, as in Porcupine. 32.4, 64.9, 97.3, 129.7, 162.2, 194.6, 227, 259.5, 291.9, 324.3, 356.8, 389.2 681.1, 713.5 2.9.5.7.11.13.17.19
2.<x3.5.7.11.13.17.19
38 19edo with neutrals. Functions as a tuning of Mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly. 31.6, 63.2, 94.7, 126.3, 157.9, 189.5, 221.1, 252.6, 284.2, 315.8, 347.4, 378.9 694.7 2.3.5.7.13.17.23
39 Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds. 30.8, 61.5, 92.3, 123.1, 153.8, 184.6, 215.4, 246.2, 276.9, 307.7, 338.5, 369.2 707.7 2.3.11
40 An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system. 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360 690
41 The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. One of two viably small tunings of 11-limit penslen. 29.3, 58.5, 87.8, 117.1, 146.3, 175.6, 204.9, 234.1, 263.4, 292.7, 322, 351.2 702.4 81/80, 64/63, 49/48, 50/49, 55/54, 45/44 2.3.5.7.11.13.19
42 The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma Archy. 28.6, 57.1, 85.7, 114.3, 142.9, 171.4, 200, 228.6, 257.1, 285.7, 314.3, 342.9 685.7, 714.3 2.7.11.17.23
43 A sharp-of-31edo Meantone tuning; its mapping of 11 is "Huygens". Like all Meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral. 27.9, 55.8, 83.7, 111.6, 139.5, 167.4, 195.3, 223.3, 251.2, 279.1, 307, 334.9 697.7 2.3.5.7.11.13.17
44 A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo). 27.3, 54.5, 81.8, 109.1, 136.4, 163.6, 190.9, 218.2, 245.5, 272.7, 300, 327.3 709.1 2.3.5.11.13.17.19.23
45 A nearly optimal tuning of Flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset. 26.7, 53.3, 80, 106.7, 133.3, 160, 186.7, 213.3, 240, 266.7, 293.3, 320 693.3, 720 2.3.7.11.17.19
46 The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. One of two viably small tunings of 11-limit penslen. 26.1, 52.2, 78.3, 104.3, 130.4, 156.5, 182.6, 208.7, 234.8, 260.9, 287, 313 704.3 2.3.5.7.11.13.17.23
47 The first edo with two distinct mosdiatonic scales. Supports Magic and Archy with its sharp fifth and Deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of Schismic analogue of Didacus. 25.5, 51.1, 76.6, 102.1, 127.7, 153.2, 178.7, 204.3, 229.8, 255.3, 280.9, 306.4 689.4, 714.9 2.5.7.13.17
48 Four times 12edo, associated with Buzzard temperament. 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300 700 2.3.7.11.17.19.23
49 A nearly optimal tuning of Archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 Meantone (or, more conventionally, Didacus). 24.5, 49, 73.5, 98, 122.4, 146.9, 171.4, 195.9, 220.4, 244.9, 269.4, 293.9 710.2 2.5.17.19
50 Approaches golden Meantone, and serves as a definitive tuning of Meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts. 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288 696 2.3.5.11.13.23
51 Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset. 23.5, 47.1, 70.6, 94.1, 117.6, 141.2, 164.7, 188.2, 211.8, 235.3, 258.8, 282.4 705.9 2.3.7.13
52 Doubles 26edo, adding a sharp Archy fifth and a more accurate 5/4 which support Porcupine temperament. 23.1, 46.2, 69.2, 92.3, 115.4, 138.5, 161.5, 184.6, 207.7, 230.8, 253.8, 276.9 692.3, 715.4 2.5.7.11.19.23
53 Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament). 22.6, 45.3, 67.9, 90.6, 113.2, 135.8, 158.5, 181.1, 203.8, 226.4, 249.1, 271.7 701.9 81/80, 64/63, 50/49, 65/64, 512/507, 91/90 2.3.5.7.13.19
54 Double 27edo, and the sharper end of the Pajara tuning range. Can alternatively be used as a very flat Deeptone system or combining the fifths as a straddle-fifth system. 688.9, 711.1 2.11.13.17.23
55 A very sharp Meantone tuning, which is so sharp that it does not even support Septimal Meantone, and is best interpreted as Mohajira as it pertains to Meantone extensions. 698.2 2.3.5.11.17.23
56 An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
57 Has 19edo's 5-limit combined with better interpretations of higher limits.
58 Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals. 20.7, 41.4, 62.1, 82.8, 103.4, 124.1, 144.8, 165.5, 186.2, 206.9, 227.6, 248.3 703.4 2.3.7.17
59 Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports Porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
60 5 sets of 12edo, supporting Magic temperament and having 10edo's 7 and 13, also supporting 7-limit Compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
61 Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
62 Doubled 31edo, which shares its mappings through the 11-limit.
63 A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification. 19, 38.1, 57.1, 76.2, 95.2, 114.3, 133.3, 152.4, 171.4, 190.5, 209.5, 228.6 704.8 2.3.5.7.11.13.23
64 An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of Flattone with its flat 3, 5, and 7. 18.8, 37.5, 56.3, 75, 93.8, 112.5, 131.3, 150, 168.8, 187.5, 206.3, 225 693.8, 712.5 2.13.19
65 A non-Garibaldi Schismic system (in fact, it supports Sensi), and a straddle-7 and -13 system. 18.5, 36.9, 55.4, 73.8, 92.3, 110.8, 129.2, 147.7, 166.2, 184.6, 203.1, 221.5 701.5 2.3.5.11.17.19
66 Tripled 22edo, with an improved approximation to 7 that supports Slendric. 18.2, 36.4, 54.5, 72.7, 90.9, 109.1, 127.3, 145.5, 163.6, 181.8, 200, 218.2 709.1, 690.9 2.5.7.11.13.17
67 Approximate 1/6-comma Meantone and Slendric edo, which also supports Orgone. 17.9, 35.8, 53.7, 71.6, 89.6, 107.5, 125.4, 143.3, 161.2, 179.1, 197, 214.9 698.5 2.3.7.11.13.17.23
68 Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting Sensamagic. Additionally, there is a second diatonic fifth. 17.6, 35.3, 52.9, 70.6, 88.2, 105.9, 123.5, 141.2, 158.8, 176.5, 194.1, 211.8 705 2.3.5.7.11.17.19
69 A nice tuning that is approximately 2/7-comma Meantone, somewhat between standard Septimal Meantone and 19edo. As a result, it is a Mohajira system (setting 7/4 to the semiflat minor seventh) but not a Septimal Meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports Archy with the sharp fifth. As a dual-fifth system, it is neogothic. 17.4, 34.8, 52.2, 69.6, 87, 104.3, 121.7, 139.1, 156.5, 173.9, 191.3, 208.7 695.7, 713 2.5.7.11.13.17.19.23
70 Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a Hemifamity system, as is typical with tunings with sharpened fifths. 17.1, 34.3, 51.4, 68.6, 85.7, 102.9, 120, 137.1, 154.3, 171.4, 188.6, 205.7 702.9 2.3.11.13.17
71 A dual-fifth system. The sharp fifth is within the Superpyth tuning range (and produces the same mapping for 5 as Superpyth), despite not supporting Archy. The flat fifth, analogously, produces Flattone's mapping for 7 and is well-tuned for Flattone, but does not support Flattone. 16.9, 33.8, 50.7, 67.6, 84.5, 101.4, 118.3, 135.2, 152.1, 169, 185.9, 202.8 693. 709.9 2.5.7.13.17.23
72 A multiple of 12edo and a very good Miracle and Compton system. It is also the first multiple of 12 to have a second MOS diatonic. 16.7, 33.3, 50, 66.7, 83.3, 100, 116.7, 133.3, 150, 166.7, 183.3, 200 700 2.3.5.7.11.17.19.23
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80 Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium. 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180 705 2.3.5.11.13.17.19.23
81 A convergent to Golden Meantone, and the last one to support Meantone in the patent val. 14.8, 29.6, 44.4, 59.3, 74.1, 88.9, 103.7, 118.5, 133.3, 148.1, 163, 177.8 696.3 2.9.5.11.13.17.19
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84 A tuning system notable for its large number of contorted mappings, and also for its tuning of Orwell. 14.3, 28.6, 42.9, 57.1, 71.4, 85.7, 100, 114.3, 128.6, 142.9, 157.1, 171.4 700 2.3.5.7.13.19.23
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87 A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit Rodan (41 & 46) temperament.

Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.

13.8, 27.6, 41.4, 55.2, 69, 82.8, 96.6, 110.3, 124.1, 137.9, 151.7, 165.5 703.4 2.3.5.7.11.13.17
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93 The triple of 31edo. As a Meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.

Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.

12.9, 25.8, 38.7, 51.6, 64.5, 77.4, 90.3, 103.2, 116.1, 129, 141.9, 154.8 696.8, 709.7 2.5.7.11.13.17.19.23
94 A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.

Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.

12.8, 25.5, 38.3, 51.1, 63.8, 76.6, 89.4, 102.1, 114.9, 127.7, 140.4, 153.2 702.1 -
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99 Perhaps the strongest 7-limit edo below 100. Supports Hemifamity, Didacus, and Ennealimmal. 12.1, 24.2, 36.4, 48.5, 60.6, 72.7, 84.8, 97, 109.1, 121.2, 133.3, 145.5 703.0 126/125, 225/224, kleisma -
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140 A significant edo for interval categorization, as the next resolution level up from 58edo. 8.6, 17.1, 25.7, 34.3, 42.9, 51.4, 60, 68.6, 77.1, 85.7, 94.3, 102.9 702.9 -
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159 Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura. 7.5, 15.1, 22.6, 30.2, 37.7, 45.3, 52.8, 60.4, 67.9, 75.5, 83, 90.6 701.9 -
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171 Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the Schismic and Ennealimmal temperaments. It is also a Neutral temperament, as 11/9 is mapped to exactly half of a perfect fifth. 7, 14, 21.1, 28.1, 35.1, 42.1, 49.1, 56.1, 63.2, 70.2, 77.2, 84.2 701.8 225/224, 5120/5103, kleisma -
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200 Notable for its extremely good approximation of 3/2, and also for being a Schismic and Slendric system with an 8/7 of exactly 234 cents. 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 702 -
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270 Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits. 4.4, 8.9, 13.3, 17.8, 22.2, 26.7, 31.1, 35.6, 40, 44.4, 48.9, 53.3 702.2 385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440 -
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306 Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2. 3.9, 7.8, 11.8, 15.7, 19.6, 23.5, 27.5, 31.4, 35.3, 39.2, 43.1, 47.1 702 -
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311 An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. 3.9, 7.7, 11.6, 15.4, 19.3, 23.2, 27, 30.9, 34.7, 38.6, 42.4, 46.3 702.3 -
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612 Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple). 2, 3.9, 5.9, 7.8, 9.8, 11.8, 13.7, 15.7, 17.6, 19.6, 21.6, 23.5 702 -
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665 Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth. 1.8, 3.6, 5.4, 7.2, 9, 10.8, 12.6, 14.4, 16.2, 18, 19.8, 21.7 702 -
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