EDO
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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.
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.
The notation m\n denotes m steps of n-edo, i.e. the frequency ratio 2^(m/n).
List of edos
Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.
| Edo | Description | First twelve steps (¢) | Fifth (¢) | Edostep interpretation | Example basic (in 2...23) and erac groups | |
|---|---|---|---|---|---|---|
| 1 | Equivalent to the 2-limit. | 1200 | 1200 | 2 | ||
| 2 | ||||||
| 2 | Just a 12edo tritone. | 600, 1200 | 600 | 2.11.23? | ||
| 2.<3.>>5.>>7? | ||||||
| 3 | A 12edo augmented triad. | 400, 800, 1200 | 800 | 2.5.13? | ||
| 2.>3.5? | ||||||
| 4 | A diminished tetrad. | 300, 600, 900, 1200 | 600 | 2.19.23? | ||
| 2.<3.<5.<7? | ||||||
| 5 | 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 | Subset of 12edo. Good approximation of Didacus temperament. | 200, 400, 600, 800, 1000, 1200 | 600, 800 | 2.9.5.7.23 | ||
| 2.xx3.5.7.<11.>13.23 | ||||||
| 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 | 2.3.5.13 | |
| 2.<3.<<5.>13 | ||||||
| 8 | Minimal version of Ammonite temperament. | 150, 300, 450, 600, 750, 900, 1050, 1200 | 750 | 2.19.23 | ||
| 2.x3.x5.x7.x11.x13 | ||||||
| 9 | The first edo to support the antidiatonic scale and temperaments like Semabila, 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. Good example 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 | 2.9.7.11.15 | ||
| 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 | 2.5.11.13.17 | ||
| 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 | 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 | 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 | 2.9.5.7.13 | ||
| 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 | 2.3.19.23 | ||
| 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 | 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 | 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), or a straddle-7 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 | ||
| ... | ||||||
| 87 | A good 17-limit system, which shares 29edo's 3-limit.
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 | ||
| ... | ||||||
| 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 | - | ||
| ... | ||||||
| 140 | A significant edo for interval categorization, as the next resolution level up from 58edo. | |||||
| ... | ||||||
| 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 | - | ||
| ... | ||||||
| 200 | Notable for its extremely good approximation of 3/2, and also for being a schismic and gamelic 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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| 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 | - | ||
