22edo
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<span style="display: block; text-align: right;">[[xenharmonie/22edo|Deutsch]] - [[22平均律|日本語]] </span> [[toc]] ---- =Theory= In music, //22 equal temperament//, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the [[octave]] into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or 54.55 [[cent]]s. The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the [[Indian|music theory of India]], Bosenquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo|19 equal temperament]], and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''. The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the [[5-limit]] to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak]]. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the [[7-limit|7-]] and [[11-limit]]s to within 3 cents/oct of error. While [[31edo|31 equal temperament]] does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit[[consistent| consistent]]ly. Furthermore, 22-et, unlike 12 and [[19edo|19]], is not a [[Regular Temperaments#meantone|meantone]] system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like. 22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31. 22-et is very close to an extended "quarter-comma superpyth", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma 64:63 instead of the syntonic comma 81:80. Because of this it has nearly pure septimal major thirds (9:7). ==Intervalic Naming Systems== The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents procupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth. ==Intervals by degree (Superpyth/Porcupine)== || Degree || Name and Abbreviation || Cents ||= Approximate Ratios* || || 0 || Natural Unison, 1 || 0 ||= 1/1 || || 1 || s-minor second, sm2 || 54.55 16°<span style="background-color: #ffffff;">21'49"</span> ||= 33/32, 34/33, 32/31 || || 2 || p-diminished second, pd2 || 109.09 <span style="background-color: #ffffff;">32°43'38"</span> ||= 18/17, 17/16, 16/15, 15/14 || || 3 || p-minor second, pm2 || 163.64 49°5'27" ||= 11/10, 10/9, 32/29 || || 4 || (s/p) Major second, M2 || 218.18 <span style="background-color: #ffffff;">65°27'16"</span> ||= 9/8, 8/7, 17/15 || || 5 || s-minor third, sm3 || 272.73 81°<span style="background-color: #ffffff;">49'5"</span> ||= [[7_6|7/6]], [[20_17|20/17]] || || 6 || p-minor third, pm3 || 327.27 <span style="background-color: #ffffff;">98°10'55"</span> ||= 6/5, 17/14, 11/9, 29/24 || || 7 || p-Major third, pM3 || 381.82 114°<span style="background-color: #ffffff;">31'44"</span> ||= 5/4 || || 8 || s-Major third, sM3 || 436.36 <span style="background-color: #ffffff;">130°54'33"</span> ||= 9/7, 14/11, 22/17 || || 9 || Natural Fourth, 4, N4 || 490.91 147°<span style="background-color: #ffffff;">16'22"</span> ||= 4/3 || || 10 || p-Major Fourth, pM4 s-dim fifth || 545.45 <span style="background-color: #ffffff;">163°38'11"</span> ||= 11/8, 15/11 || || 11 || Augmented Fourth, A4, Half-Octave, HO || 600 180° ||= 7/5, 10/7, 17/12, 24/17 || || 12 || p-minor Fifth, pm5 s-aug fourth || 654.55 <span style="background-color: #ffffff;">196°21'49"</span> ||= 16/11, 22/15 || || 13 || Natural Fifth, 5, N5 || 709.09 212°<span style="background-color: #ffffff;">43'38"</span> ||= 3/2 || || 14 || s-minor sixth, sm6 || 763.64 <span style="background-color: #ffffff;">229°5'27"</span> ||= 11/7, 14/9, 17/11 || || 15 || p-minor sixth, pm6 || 818.18 245°<span style="background-color: #ffffff;">16'22"</span> ||= 8/5 || || 16 || p-Major sixth, pM6 || 872.73 <span style="background-color: #ffffff;">261°49'5"</span> ||= 5/3, 18/11, 28/17 || || 17 || s-Major sixth, sM6 || 927.27 278°<span style="background-color: #ffffff;">10'55"</span> ||= [[12_7|12/7]], [[17_10|17/10]] || || 18 || (s/p) minor seventh, m7 || 981.82 <span style="background-color: #ffffff;">294°31'44"</span> ||= 7/4, 16/9, 30/17 || || 19 || p-Major seventh, pM7 || 1036.36 310°<span style="background-color: #ffffff;">54'33"</span> ||= 20/11, 9/5, 29/16 || || 20 || p-Augmented Seventh || 1090.91 <span style="background-color: #ffffff;">327°16'22"</span> ||= 15/8, 32/17, 17/9, 28/15 || || 21 || s-Major Seventh, sM7 || 1145.45 343°<span style="background-color: #ffffff;">38'11"</span> ||= 33/17, 64/33, 31/16 || || 22 || Octave, 8 || 1200 360° ||= 2/1 || 22edo intervals can also be notated using [[Ups and Downs Notation|ups and downs]]: The keyboard runs C * * * D * * * E F * * * G * * * A * * * B C. ==Intervals by degree (ups and downs)== ||= Degree ||= Interval Name ||= Abbreviation ||= Cents ||= Example || ||= 0 ||= perfect unison ||= P1 ||= 0 ||= C || ||= 1 ||= minor second ||= m2 ||= 55 ||= Db || ||= 2 ||= upminor 2nd ||= ^m2 ||= 109 ||= Db^ || ||= 3 ||= downmajor 2nd ||= vM2 ||= 164 ||= Dv || ||= 4 ||= major second ||= M2 ||= 218 ||= D || ||= 5 ||= minor third ||= m3 ||= 273 ||= Eb || ||= 6 ||= upminor third ||= ^m3 ||= 327 ||= Eb^ || ||= 7 ||= downmajor third ||= vM3 ||= 382 ||= Ev || ||= 8 ||= major third ||= M3 ||= 436 ||= E || ||= 9 ||= perfect fourth ||= P4 ||= 491 ||= F || ||= 10 ||= up-fourth or dim fifth ||= ^4 or d5 ||= 545 ||= F^ or Gb || ||= 11 ||= downaug fourth or updim fifth ||= vA4 or ^d5 ||= 600 ||= F#v or Gb^ || ||= 12 ||= aug fourth or down-fifth ||= A4 or v5 ||= 655 ||= F# or Gv || ||= 13 ||= perfect fifth ||= P5 ||= 709 ||= G || ||= 14 ||= minor sixth ||= m6 ||= 764 ||= Ab || ||= 15 ||= upminor sixth ||= ^m6 ||= 818 ||= Ab^ || ||= 16 ||= downmajor sixth ||= vM6 ||= 873 ||= Av || ||= 17 ||= major sixth ||= M6 ||= 927 ||= A || ||= 18 ||= minor seventh ||= m7 ||= 982 ||= Bb || ||= 19 ||= upminor seventh ||= ^m7 ||= 1036 ||= Bb^ || ||= 20 ||= downmajor seventh ||= vM7 ||= 1091 ||= Bv || ||= 21 ||= major seventh ||= M7 ||= 1145 ||= B || ||= 22 ||= octave ||= P8 ||= 1200 ||= C || Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D. ==Intervals by degree (Porcupine)== ||= Degree ||= Interval Name ||= Abbreviation ||= Cents ||= Example || ||= 0 ||= perfect unison ||= P1 ||= 0 ||= D || ||= 1 ||= aug unison ||= A1 ||= 55 ||= D# || ||= 2 ||= dim second ||= d2 ||= 109 ||= Eb || ||= 3 ||= perfect second ||= P2 ||= 164 ||= E || ||= 4 ||= aug second ||= A2 ||= 218 ||= E# || ||= 5 ||= dim third ||= d3 ||= 273 ||= Fb || ||= 6 ||= minor third ||= m3 ||= 327 ||= F || ||= 7 ||= major third ||= M3 ||= 382 ||= F# || ||= 8 ||= aug third or dim fourth ||= A3 or d4 ||= 436 ||= Gb || ||= 9 ||= minor fourth ||= m4 ||= 491 ||= G || ||= 10 ||= major fourth ||= M4 ||= 545 ||= G# || ||= 11 ||= aug fourth or dim fifth ||= A4 or d5 ||= 600 ||= Gx or Abb || ||= 12 ||= minor fifth ||= m5 ||= 655 ||= Ab || ||= 13 ||= major fifth ||= M5 ||= 709 ||= A || ||= 14 ||= aug fifth or dim sixth ||= A5 or d6 ||= 764 ||= A# || ||= 15 ||= minor sixth ||= m6 ||= 818 ||= Bb || ||= 16 ||= major sixth ||= M6 ||= 873 ||= B || ||= 17 ||= aug sixth ||= A6 ||= 927 ||= B# || ||= 18 ||= dim seventh ||= d7 ||= 982 ||= Cb || ||= 19 ||= perfect seventh ||= P7 ||= 1036 ||= C || ||= 20 ||= aug seventh ||= A7 ||= 1091 ||= C# || ||= 21 ||= dim octave ||= d8 ||= 1145 ||= Db || ||= 22 ||= octave ||= P8 ||= 1200 ||= D || Yet another notation is pentatonic. This is the only way to use a chain-of-fifths notation without additional accidentals. The pentatonic degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The keyboard runs D * * * E * * * * G * * * A * * * * C * * * D. ==Intervals by degree (pentatonic)== ||= Degree ||= Interval Name ||= Abbreviation ||= Cents ||= Example || ||= 0 ||= perfect unison ||= P1 ||= 0 ||= D || ||= 1 ||= aug unison ||= A1 ||= 55 ||= D# || ||= 2 ||= double-aug unison or double-dim sub3rd ||= AA1 or dds3 ||= 109 ||= Dx or Ebb || ||= 3 ||= dim sub3rd ||= ds3 ||= 164 ||= Eb || ||= 4 ||= minor sub3rd ||= ms3 ||= 218 ||= E || ||= 5 ||= major sub3rd ||= Ms3 ||= 273 ||= E# || ||= 6 ||= aug sub3rd ||= As3 ||= 327 ||= Ex || ||= 7 ||= double-aug sub3rd or double-dim 4thoid ||= AAs3 or dd4d ||= 382 ||= Gbb || ||= 8 ||= dim 4thoid ||= d4d ||= 436 ||= Gb || ||= 9 ||= perfect 4thoid ||= P4d ||= 491 ||= G || ||= 10 ||= aug 4thoid ||= A4d ||= 545 ||= G# || ||= 11 ||= double-aug 4thoid or double-dim 5thoid ||= AA4d or dd5d ||= 600 ||= Gx or Abb || ||= 12 ||= dim 5thoid ||= d5d ||= 655 ||= Ab || ||= 13 ||= perfect 5thoid ||= P5d ||= 709 ||= A || ||= 14 ||= aug 5thoid ||= A5d ||= 764 ||= A# || ||= 15 ||= double-aug 5thoid or double-dim sub7th ||= AA5d or dds7 ||= 818 ||= Ax || ||= 16 ||= dim sub7th ||= ds7 ||= 873 ||= Cbb || ||= 17 ||= minor sub7th ||= ms7 ||= 927 ||= Cb || ||= 18 ||= major sub7th ||= Ms7 ||= 982 ||= C || ||= 19 ||= aug sub7th ||= As7 ||= 1036 ||= C# || ||= 20 ||= double-aug sub7th or double-dim octoid ||= AAs7 or dd8d ||= 1091 ||= Cx or Dbb || ||= 21 ||= dim octoid ||= d8d ||= 1145 ||= Db || ||= 22 ||= octave ||= P8 ||= 1200 ||= D || == == ==Selected just intervals by error== The following table shows how [[Just-24|some prominent just intervals]] are represented in 22edo (ordered by absolute error). || **Interval, complement** || **Error (abs., in [[cent|cents]])** || ||= [[9_7|9/7]], [[14_9|14/9]] ||= 1.280 || ||= [[11_10|11/10]], [[20_11|20/11]] ||= 1.368 || ||= [[16_15|16/15]], [[15_8|15/8]] ||= 2.640 || ||= [[5_4|5/4]], [[8_5|8/5]] ||= 4.496 || ||= [[7_6|7/6]], [[12_7|12/7]] ||= 5.856 || ||= [[11_8|11/8]], [[16_11|16/11]] ||= 5.863 || ||= [[4_3|4/3]], [[3_2|3/2]] ||= 7.136 || ||= [[15_11|15/11]], [[22_15|22/15]] ||= 8.504 || ||= [[15_14|15/14]], [[28_15|28/15]] ||= 10.352 || ||= [[6_5|6/5]], [[5_3|5/3]] ||= 11.631 || ||= [[8_7|8/7]], [[7_4|7/4]] ||= 12.992 || ||= [[12_11|12/11]], [[11_6|11/6]] ||= 12.999 || ||= [[9_8|9/8]], [[16_9|16/9]] ||= 14.272 || ||= [[13_11|13/11]], [[22_13|22/13]] ||= 16.482 || ||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 || ||= [[13_10|13/10]], [[20_13|20/13]] ||= 17.850 || ||= [[18_13|18/13]], [[13_9|13/9]] ||= 17.928 || ||= [[10_9|10/9]], [[9_5|9/5]] ||= 18.767 || ||= [[14_11|14/11]], [[11_7|11/7]] ||= 18.856 || ||= [[14_13|14/13]], [[13_7|13/7]] ||= 19.207 || ||= [[11_9|11/9]], [[18_11|18/11]] ||= 20.135 || ||= [[16_13|16/13]], [[13_8|13/8]] ||= 22.346 || ||= [[15_13|15/13]], [[26_15|26/15]] ||= 24.986 || ||= [[13_12|13/12]], [[24_13|24/13]] ||= 25.064 || [[media type="custom" key="24838814"]] [[file:22ed2-001e.svg]] See also: [[22edo Solfege]], [[22edo Tetrachords]], [[22 EDO Chords]], [[22edo Modes]] ==Properties of 22 equal temperament== Possibly the most striking characteristic of 22-et to those not used to it is that it does **not** "temper out" the syntonic comma of 81/80, and therefore is not a system of [[Regular Temperaments#meantone|meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12-EDO does not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]]. The diatonic scale it produces is instead derived from [[superpyth]] temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22-EDO. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12-equal and meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively. It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22-EDO supports [[porcupine]] temperament. The generator for porcupine is is a flat minor whole tone of [[10_9|10/9]], two of which is a slightly sharp [[6_5|6/5]], and three of which is a slightly flat [[4_3|4/3]], implying the existence of an equal-step tetrachord, which is characteristic of Porcupine. Porcupine is notable for being the 5-limit temperament lowest in [[badness]] which is //not// approximated by the familiar 12-tone equal temperament, and as such represents one excellent point of departure for examining the harmonic properties of 22-EDO. It forms [[MOSScales|MOS]]'s of 7 and 8, which in 22-EDO are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes). The 164¢ "flat minor whole tone" is a key interval in 22-EDO, in part because it functions as no less than three different consonant ratios in the [[11-limit]]: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22-EDO can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo. 22-EDO also supports Orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms MOS scales with step patterns 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. Harmonically, Orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]] and [[84edo]]. But 22-equal Orwell has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish in 22. Other 5-limit commas 22-EDO tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12-et or 22-et, the [[diatonic tritone]] [[45_32|45/32]], which is a major third above a [[major whole tone]] representing [[9_8|9/8]], is equated to its inverted form, [[64_45|64/45]]. That the magic comma is tempered out means that 22-et is a [[Regular Temperaments#magic|magic]] system, where five major thirds make up a perfect fifth. In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50_49|50/49]], (the [[jubilee comma]]), and [[64_63|64/63]], (the [[septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22-et. As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo]], in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal]], 11 can be notated as every other note of 22. ===Rank Two Temperaments=== [[List of 22et rank two temperaments by badness]] [[List of 22et rank two temperaments by complexity]] [[List of edo-distinct 22et rank two temperaments]] ||~ Periods per octave ||~ Generator ||~ Temperaments || || 1 || 1\22 || [[Sensamagic clan#Sensa|Sensa]]/chromo/ceratitid || || 1 || 3\22 || [[Porcupine]] || || 1 || 5\22 || [[Orson]]/[[orwell]]/blair || || 1 || 7\22 || [[Magic]]/telepathy || || 1 || 9\22 || [[Superpyth]]/[[suprapyth]] || || 2 || 1\22 || [[Shrutar]]/hemipaj/comic || || 2 || 2\22 || [[Srutal]]/[[pajara]]/pajarous || || 2 || 3\22 || [[Porcupine family#Hedgehog|Hedgehog]]/[[echidna]] || || 2 || 4\22 || [[Astrology]]/[[wizard]]/[[antikythera]] || || 2 || 5\22 || [[Doublewide]]/fleetwood || || 11 || 1\22 || [[Hendecatonic]]/undeka || ===Commas=== 22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || ||= 250/243 ||< | 1 -5 3 > ||> 49.17 ||= Maximal Diesis ||= Porcupine Comma ||= || ||= 3125/3072 ||< | -10 -1 5 > ||> 29.61 ||= Small Diesis ||= Magic Comma ||= || ||= 2048/2025 ||< | 11 -4 -2 > ||> 19.55 ||= Diaschisma ||= ||= || ||= 2109375/2097152 ||< | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||= || ||= 9193891/9143623 ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma || ||= || ||= 4758837/4757272 ||< | -53 10 16 > ||> 0.57 ||= Kwazy ||= ||= || ||= 50/49 ||< | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||= || ||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma || ||= 875/864 ||< | -5 -3 3 1 > ||> 21.90 ||= Keema ||= ||= || ||= 2430/2401 ||< | 1 5 1 -4 > ||> 20.79 ||= Nuwell ||= ||= || ||= 245/243 ||< | 0 -5 1 2 > ||> 14.19 ||= Sensamagic ||= ||= || ||= 1728/1715 ||< | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||= || ||= 225/224 ||< | -5 2 2 -1 > ||> 7.71 ||= Septimal Kleisma ||= Marvel Comma ||= || ||= 10976/10935 ||< | 5 -7 -1 3 > ||> 6.48 ||= Hemimage ||= ||= || ||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||= ||= || ||= 65625/65536 ||< | -16 1 5 1 > ||> 2.35 ||= Horwell ||= ||= || ||= 420175/419904 ||< | -6 -8 2 5 > ||> 1.12 ||= Wizma ||= ||= || ||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= ||= || ||= 100/99 ||< | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||= ||= || ||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || || 125/124 || |-4 0 3 0 ... -1> || 13.91 ||= Twizzler || || || ||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= ||= || ||= 896/891 ||< | 7 -4 0 1 -1 > ||> 9.69 ||= Pentacircle ||= ||= || ||= 65536/65219 ||< | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||= ||= || ||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||= ||= || ||= 540/539 ||< | 2 3 1 -2 -1 > ||> 3.21 ||= Swetisma ||= ||= || ||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||= ||= || ||= 9801/9800 ||< | -3 4 -2 -2 2 > ||> 0.18 ||= Kalisma ||= Gauss' Comma ||= || ||= 91/90 ||< | -1 -2 -1 1 0 1 > ||> 19.13 ||= Superleap ||= ||= || ===How to Notate 22edo in Sagittal=== When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents: [[image:22edo.png]] This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma. The division of the apotome into three syntonic commas also indicates 22's tempering out of the [[250_243|porcupine comma]] (which is equivalent to three syntonic commas minus a Pythagorean apotome). =The Decatonic System= <span style="background-color: #ffffff; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">The decatonic system is an approach of notation based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system bases music into a 10 tone scale rather than 7. </span> <span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">This approach requires an entire re-learning of chords, intervals, and notation but the advantage is that it allows 22 EDO to be notated using only one pair of accidentals, as well as</span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">gives the opportunity to escape a heptatonic thinking pattern.</span> <span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"> ==[[#TOC-Decatonic-Alphabet]]Decatonic Alphabet== </span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">The system is based on two chains of fifths. One represented by latin letters, the other greek. The two chains can be looked at as two juxtaposed pentatonic scales.</span> <span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">Chain 1: C G D A E</span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">Chain 2: <span style="background-color: transparent; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">γ δ α ε β </span></span> <span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">The alphabet is ascending order: C<span style="font-family: helvetica,arial,sans-serif; font-size: 12px;"> δ D ε E γ G α A β C </span></span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"> </span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"><span style="font-family: helvetica,arial,sans-serif; font-size: 12px;">In this alphabet, a chain of fifths is preserved because equivalent greek letters also represent fifths if they are the same as their latin counter parts. For example G D is a fifth as well as </span><span style="color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">γ δ. </span></span> = = || **Mode Name** || Notes || Step Structure || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Keter</span> || <span style="background-color: #ffffff; font-family: helvetica,arial,sans-serif; font-size: 12px;">α A β C δ D ε E</span><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">˥ </span><span style="background-color: #ffffff; font-family: helvetica,arial,sans-serif; font-size: 12px;">γ G α</span> || 2 2 3 2 2 2 3 2 2 2 || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Chokhmah</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Binah</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Chesed</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Gevurah</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Tiferet</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Netzach</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Hod</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Yesod</span> || || || || <span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Malkuth</span> || || || <span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"> </span> ==External links== [[http://lumma.org/tuning/erlich/erlich-decatonic.pdf|Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']] [[http://porcupinemusic.weebly.com/|"Porcupine Music" - Website Focused on the Development of 22 EDO music ]] ==References== Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951] Bosanquet, R.H.M. [[http://www.webcitation.org/5kjJcrhEx|''On the Hindoo division of the octave, with additions to the theory of higher orders'']], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965 ---- =Music= * [[@https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version|Stephen Weigel · Dose Of Familiarity/Ode to Microtonality]] * [[@http://soonlabel.com/xenharmonic/archives/1145|Canon 2 in 1 upon a ground (22edo)]] by Claudi Meneghin * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3|Tibia]]</span></span> by [[Paul Erlich]] ** Sagittal score of Tibia, [[file:xenharmonic/TIBIA.pdf|in F||\]] or [[file:xenharmonic/tibia in g.pdf|in G]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095|Glassic]]</span></span> by Paul Erlich and [[Ara Sarkissian]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://lumma.org/tuning/erlich/decatonic-swing.mp3|Decatonic Swing]]</span></span> by Paul Erlich and Ara Sarkissian (jazz) * [[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3|12-22hexachordal Dirge]] by [[Joel Grant Taylor]] * [[@https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs|Chord sequence in Paul Erlich's 22 EDO decatonic major]] by [[Jake Freivald]] * [[https://soundcloud.com/jdfreivald/porcupine-comma-pump|Porcupine Comma Pump]] by [[Jake Freivald]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Dragged%20By%20a%20Storm%20Across%20the%20Desert%20Years.mp3|Dragged by a Storm Across the Desert Years]]</span></span> by * [[IgliashonJones|Igliashon Jones]] (synth with electric guitar) * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3|Numerology]]</span></span> by Iglashion Jones (progressive metal) * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Revenge%20of%20the%20Inorganic%20Compounds.mp3|Revenge of the inorganic compounds]]</span></span> by Iglashion Jones (progressive metal) * [[http://chrisvaisvil.com/?p=267|My Crazy Aunt Sophie]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3|play]]</span></span> by [[Chris Vaisvil]]. Blatantly xenharmonic piano. * [[http://soundclick.com/share?songid=8839058|where words are said to mean]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+wherewordsaresaidtomean.mp3|play]]</span></span> by [[Andrew Heathwaite]], a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This & the following pieces by Andrew are for 22-tone guitar & voice. * [[http://soundclick.com/share?songid=9101704|I've come with a bucket of roses]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+ivecomewithabucketofroses.mp3|play]]</span></span> by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2). * [[http://soundclick.com/share?songid=9101705|one drop of rain]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+onedropofrain.mp3|play]]</span></span> by Andrew Heathwaite (orwell-9). * [[http://soundclick.com/share?songid=8839060|being a]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+beinga.mp3|play]]</span></span> by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3). * [[http://soundclick.com/share?songid=8839071|my own house]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+myownhouse.mp3|play]]</span></span> by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7). * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3|Comets Over Flatland 17]]</span></span> by [[Randy Winchester]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3|Night on Porcupine Mountain]]</span></span> Mussorgsky-Smith * <span class="ywp-page-play-pause ywp-page-video ywp-link-hover"><span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link">[[http://www.youtube.com/watch?v=lO5xSjIHyMg|Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity]]</span></span> - youtube * <span class="ywp-page-play-pause ywp-page-video ywp-link-hover"><span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link">[[http://www.youtube.com/watch?v=WMtp9Wk0tO0|Improvisation in 22-equal temperament]]</span></span>, Mike Battaglia - youtube * Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - [[http://www.angelfire.com/mo/oljare/midicomp.html|MIDI files]] by Mats Öljare ** [[file:xenharmonic/sunday3.pdf|Sagittal score of Sunday Pipes]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3|Phobos Light]]</span> by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo. * //[[http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3|The Capture and Release of the Fairy]]// by [[Chris Vaisvil]] => [[http://chrisvaisvil.com/?p=2494|blog post presentation]] * //[[http://www.youtube.com/watch?v=oNJr1YOOqF8|Yak Butter]]// by The Stern Brocot Band, 1L6s MOS, compressed period/generator * [[http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_03_-_Dirty_Drummer_vbr.mp3|Dirty Drummer]], Sevish * [[http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_12_-_Ganymede_vbr.mp3|Ganymede]], Sevish (doesn't sound that xen, but it's in 22-edo) * [[http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3|Ambrosia]], Sevish * //[[http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3|From the Sky Islands They Came]]// by [[Chris Vaisvil]] => [[http://chrisvaisvil.com/?p=2523|blog post presentation]] * [[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3|Smoke Filled Bar]] by [[Chris Vaisvil]] => [[@http://chrisvaisvil.com/smoke-filled-bar/|blog presentation]] * [[http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3|Recurring Mimosa]] by [[https://soundcloud.com/redrick-sultan/recurring-mimosa|Redrick Sultan]] * The Saharan Pump by Chris Vaisvil [[http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/|blog post]] * [[@http://www.youtube.com/watch?v=qHHv3mwJTlg|Short piece and demonstration]] (video) by [[@http://brendanbyrnes.com/|Brendan Byrnes]] (electric guitar) * [[http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3|22 EDO Guitar Etude]] by [[http://brendanbyrnes.bandcamp.com/|Brendan Byrnes]] * [[http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%20Llurion.mp3|Llurion]] by [[http://brendanbyrnes.bandcamp.com/track/llurion|Brendan Byrnes]] * [[@https://youtu.be/0VLJXecjYK4|Imzadi]] by [[@http://omega9.github.io/|Omega9]] * [[http://micro.soonlabel.com/22-ET/20150910_22edo.mp3|22 edo electric guitar duet]] by [[Chris Vaisvil]] * [[https://soundcloud.com/gareth-hearne/mass-in-22edo-sanctus|Mass in 22edo - Sanctus]] by [[Gareth Hearne]] * [[https://soundcloud.com/gareth-hearne/mass-in-22edo-agnus-dei|Mass in 22edo - Agnus Dei]] by Gareth Hearne [[media type="custom" key="27813215"]]
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<html><head><title>22edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/22edo">Deutsch</a> - <a class="wiki_link" href="/22%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
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<!-- ws:start:WikiTextTocRule:38:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Theory">Theory</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Theory-Intervalic Naming Systems">Intervalic Naming Systems</a></div>
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Theory-Intervals by degree (Superpyth/Porcupine)">Intervals by degree (Superpyth/Porcupine)</a></div>
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Theory-Intervals by degree (ups and downs)">Intervals by degree (ups and downs)</a></div>
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Theory-Intervals by degree (Porcupine)">Intervals by degree (Porcupine)</a></div>
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Theory-Intervals by degree (pentatonic)">Intervals by degree (pentatonic)</a></div>
<!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#toc6"> </a></div>
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Theory-Selected just intervals by error">Selected just intervals by error</a></div>
<!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Theory-Properties of 22 equal temperament">Properties of 22 equal temperament</a></div>
<!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 3em;"><a href="#Theory-Properties of 22 equal temperament-Rank Two Temperaments">Rank Two Temperaments</a></div>
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 3em;"><a href="#Theory-Properties of 22 equal temperament-Commas">Commas</a></div>
<!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 3em;"><a href="#Theory-Properties of 22 equal temperament-How to Notate 22edo in Sagittal">How to Notate 22edo in Sagittal</a></div>
<!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#The Decatonic System">The Decatonic System</a></div>
<!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#The Decatonic System-Decatonic Alphabet">Decatonic Alphabet</a></div>
<!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#toc14"> </a></div>
<!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#The Decatonic System-External links">External links</a></div>
<!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 2em;"><a href="#The Decatonic System-References">References</a></div>
<!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div>
<!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --></div>
<!-- ws:end:WikiTextTocRule:57 --><hr />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:2 -->Theory</h1>
<br />
In music, <em>22 equal temperament</em>, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 22 equally large steps. Each step represents a frequency ratio of the twenty-second root of 2, or 54.55 <a class="wiki_link" href="/cent">cent</a>s.<br />
<br />
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the <a class="wiki_link" href="/Indian">music theory of India</a>, Bosenquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after <a class="wiki_link" href="/19edo">19 equal temperament</a>, and J. Murray Barbour in his classic survey of tuning history, ''Tuning and Temperament''.<br />
<br />
The 22-et system is in fact the third equal division, after 12 and 19, which is capable of approximating the <a class="wiki_link" href="/5-limit">5-limit</a> to within a TE error of 4 cents/oct. While not an integral or gap edo it at least qualifies as a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta peak</a>. Moreover, there is more to it than just the 5-limit; unlike 12 or 19 it is able to approximate the <a class="wiki_link" href="/7-limit">7-</a> and <a class="wiki_link" href="/11-limit">11-limit</a>s to within 3 cents/oct of error. While <a class="wiki_link" href="/31edo">31 equal temperament</a> does much better, 22-et still allows the use of these higher-limit harmonies, and in fact 22 is the smallest equal division to represent the 11-limit<a class="wiki_link" href="/consistent"> consistent</a>ly. Furthermore, 22-et, unlike 12 and <a class="wiki_link" href="/19edo">19</a>, is not a <a class="wiki_link" href="/Regular%20Temperaments#meantone">meantone</a> system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like.<br />
<br />
22-et can also be treated as adding harmonics 3 and 5 to 11-EDO's 2.7.9.11.15.17 subgroup, making it a (rather accurate) 2.3.5.7.11.17 subgroup temperament. Let us also mind it's approximation of the 31st harmonic is within half a cent, which is fairly accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31.<br />
<br />
22-et is very close to an extended "quarter-comma superpyth", a tuning analogous to quarter-comma meantone except that it tempers out the septimal comma 64:63 instead of the syntonic comma 81:80. Because of this it has nearly pure septimal major thirds (9:7).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc1"><a name="Theory-Intervalic Naming Systems"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervalic Naming Systems</h2>
The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents procupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc2"><a name="Theory-Intervals by degree (Superpyth/Porcupine)"></a><!-- ws:end:WikiTextHeadingRule:6 -->Intervals by degree (Superpyth/Porcupine)</h2>
<table class="wiki_table">
<tr>
<td>Degree<br />
</td>
<td>Name and Abbreviation<br />
</td>
<td>Cents<br />
</td>
<td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>Natural Unison, 1<br />
</td>
<td>0<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>s-minor second, sm2<br />
</td>
<td>54.55<br />
16°<span style="background-color: #ffffff;">21'49"</span><br />
</td>
<td style="text-align: center;">33/32, 34/33, 32/31<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>p-diminished second, pd2<br />
</td>
<td>109.09<br />
<span style="background-color: #ffffff;">32°43'38"</span><br />
</td>
<td style="text-align: center;">18/17, 17/16, 16/15, 15/14<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>p-minor second, pm2<br />
</td>
<td>163.64<br />
49°5'27"<br />
</td>
<td style="text-align: center;">11/10, 10/9, 32/29<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>(s/p) Major second, M2<br />
</td>
<td>218.18<br />
<span style="background-color: #ffffff;">65°27'16"</span><br />
</td>
<td style="text-align: center;">9/8, 8/7, 17/15<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>s-minor third, sm3<br />
</td>
<td>272.73<br />
81°<span style="background-color: #ffffff;">49'5"</span><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/20_17">20/17</a><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>p-minor third, pm3<br />
</td>
<td>327.27<br />
<span style="background-color: #ffffff;">98°10'55"</span><br />
</td>
<td style="text-align: center;">6/5, 17/14, 11/9, 29/24<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>p-Major third, pM3<br />
</td>
<td>381.82<br />
114°<span style="background-color: #ffffff;">31'44"</span><br />
</td>
<td style="text-align: center;">5/4<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>s-Major third, sM3<br />
</td>
<td>436.36<br />
<span style="background-color: #ffffff;">130°54'33"</span><br />
</td>
<td style="text-align: center;">9/7, 14/11, 22/17<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>Natural Fourth, 4, N4<br />
</td>
<td>490.91<br />
147°<span style="background-color: #ffffff;">16'22"</span><br />
</td>
<td style="text-align: center;">4/3<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>p-Major Fourth, pM4<br />
s-dim fifth<br />
</td>
<td>545.45<br />
<span style="background-color: #ffffff;">163°38'11"</span><br />
</td>
<td style="text-align: center;">11/8, 15/11<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>Augmented Fourth, A4,<br />
Half-Octave, HO<br />
</td>
<td>600<br />
180°<br />
</td>
<td style="text-align: center;">7/5, 10/7, 17/12, 24/17<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>p-minor Fifth, pm5<br />
s-aug fourth<br />
</td>
<td>654.55<br />
<span style="background-color: #ffffff;">196°21'49"</span><br />
</td>
<td style="text-align: center;">16/11, 22/15<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>Natural Fifth, 5, N5<br />
</td>
<td>709.09<br />
212°<span style="background-color: #ffffff;">43'38"</span><br />
</td>
<td style="text-align: center;">3/2<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>s-minor sixth, sm6<br />
</td>
<td>763.64<br />
<span style="background-color: #ffffff;">229°5'27"</span><br />
</td>
<td style="text-align: center;">11/7, 14/9, 17/11<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>p-minor sixth, pm6<br />
</td>
<td>818.18<br />
245°<span style="background-color: #ffffff;">16'22"</span><br />
</td>
<td style="text-align: center;">8/5<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>p-Major sixth, pM6<br />
</td>
<td>872.73<br />
<span style="background-color: #ffffff;">261°49'5"</span><br />
</td>
<td style="text-align: center;">5/3, 18/11, 28/17<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>s-Major sixth, sM6<br />
</td>
<td>927.27<br />
278°<span style="background-color: #ffffff;">10'55"</span><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/12_7">12/7</a>, <a class="wiki_link" href="/17_10">17/10</a><br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>(s/p) minor seventh, m7<br />
</td>
<td>981.82<br />
<span style="background-color: #ffffff;">294°31'44"</span><br />
</td>
<td style="text-align: center;">7/4, 16/9, 30/17<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>p-Major seventh, pM7<br />
</td>
<td>1036.36<br />
310°<span style="background-color: #ffffff;">54'33"</span><br />
</td>
<td style="text-align: center;">20/11, 9/5, 29/16<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>p-Augmented Seventh<br />
</td>
<td>1090.91<br />
<span style="background-color: #ffffff;">327°16'22"</span><br />
</td>
<td style="text-align: center;">15/8, 32/17, 17/9, 28/15<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>s-Major Seventh, sM7<br />
</td>
<td>1145.45<br />
343°<span style="background-color: #ffffff;">38'11"</span><br />
</td>
<td style="text-align: center;">33/17, 64/33, 31/16<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>Octave, 8<br />
</td>
<td>1200<br />
360°<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
</tr>
</table>
<br />
22edo intervals can also be notated using <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>: The keyboard runs C * * * D * * * E F * * * G * * * A * * * B C.<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc3"><a name="Theory-Intervals by degree (ups and downs)"></a><!-- ws:end:WikiTextHeadingRule:8 -->Intervals by degree (ups and downs)</h2>
<table class="wiki_table">
<tr>
<td style="text-align: center;">Degree<br />
</td>
<td style="text-align: center;">Interval Name<br />
</td>
<td style="text-align: center;">Abbreviation<br />
</td>
<td style="text-align: center;">Cents<br />
</td>
<td style="text-align: center;">Example<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">perfect unison<br />
</td>
<td style="text-align: center;">P1<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">C<br />
</td>
</tr>
<tr>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">minor second<br />
</td>
<td style="text-align: center;">m2<br />
</td>
<td style="text-align: center;">55<br />
</td>
<td style="text-align: center;">Db<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">upminor 2nd<br />
</td>
<td style="text-align: center;">^m2<br />
</td>
<td style="text-align: center;">109<br />
</td>
<td style="text-align: center;">Db^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3<br />
</td>
<td style="text-align: center;">downmajor 2nd<br />
</td>
<td style="text-align: center;">vM2<br />
</td>
<td style="text-align: center;">164<br />
</td>
<td style="text-align: center;">Dv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">major second<br />
</td>
<td style="text-align: center;">M2<br />
</td>
<td style="text-align: center;">218<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5<br />
</td>
<td style="text-align: center;">minor third<br />
</td>
<td style="text-align: center;">m3<br />
</td>
<td style="text-align: center;">273<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6<br />
</td>
<td style="text-align: center;">upminor third<br />
</td>
<td style="text-align: center;">^m3<br />
</td>
<td style="text-align: center;">327<br />
</td>
<td style="text-align: center;">Eb^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7<br />
</td>
<td style="text-align: center;">downmajor third<br />
</td>
<td style="text-align: center;">vM3<br />
</td>
<td style="text-align: center;">382<br />
</td>
<td style="text-align: center;">Ev<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8<br />
</td>
<td style="text-align: center;">major third<br />
</td>
<td style="text-align: center;">M3<br />
</td>
<td style="text-align: center;">436<br />
</td>
<td style="text-align: center;">E<br />
</td>
</tr>
<tr>
<td style="text-align: center;">9<br />
</td>
<td style="text-align: center;">perfect fourth<br />
</td>
<td style="text-align: center;">P4<br />
</td>
<td style="text-align: center;">491<br />
</td>
<td style="text-align: center;">F<br />
</td>
</tr>
<tr>
<td style="text-align: center;">10<br />
</td>
<td style="text-align: center;">up-fourth or dim fifth<br />
</td>
<td style="text-align: center;">^4 or d5<br />
</td>
<td style="text-align: center;">545<br />
</td>
<td style="text-align: center;">F^ or Gb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">11<br />
</td>
<td style="text-align: center;">downaug fourth or updim fifth<br />
</td>
<td style="text-align: center;">vA4 or ^d5<br />
</td>
<td style="text-align: center;">600<br />
</td>
<td style="text-align: center;">F#v or Gb^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">12<br />
</td>
<td style="text-align: center;">aug fourth or down-fifth<br />
</td>
<td style="text-align: center;">A4 or v5<br />
</td>
<td style="text-align: center;">655<br />
</td>
<td style="text-align: center;">F# or Gv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">13<br />
</td>
<td style="text-align: center;">perfect fifth<br />
</td>
<td style="text-align: center;">P5<br />
</td>
<td style="text-align: center;">709<br />
</td>
<td style="text-align: center;">G<br />
</td>
</tr>
<tr>
<td style="text-align: center;">14<br />
</td>
<td style="text-align: center;">minor sixth<br />
</td>
<td style="text-align: center;">m6<br />
</td>
<td style="text-align: center;">764<br />
</td>
<td style="text-align: center;">Ab<br />
</td>
</tr>
<tr>
<td style="text-align: center;">15<br />
</td>
<td style="text-align: center;">upminor sixth<br />
</td>
<td style="text-align: center;">^m6<br />
</td>
<td style="text-align: center;">818<br />
</td>
<td style="text-align: center;">Ab^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">16<br />
</td>
<td style="text-align: center;">downmajor sixth<br />
</td>
<td style="text-align: center;">vM6<br />
</td>
<td style="text-align: center;">873<br />
</td>
<td style="text-align: center;">Av<br />
</td>
</tr>
<tr>
<td style="text-align: center;">17<br />
</td>
<td style="text-align: center;">major sixth<br />
</td>
<td style="text-align: center;">M6<br />
</td>
<td style="text-align: center;">927<br />
</td>
<td style="text-align: center;">A<br />
</td>
</tr>
<tr>
<td style="text-align: center;">18<br />
</td>
<td style="text-align: center;">minor seventh<br />
</td>
<td style="text-align: center;">m7<br />
</td>
<td style="text-align: center;">982<br />
</td>
<td style="text-align: center;">Bb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">19<br />
</td>
<td style="text-align: center;">upminor seventh<br />
</td>
<td style="text-align: center;">^m7<br />
</td>
<td style="text-align: center;">1036<br />
</td>
<td style="text-align: center;">Bb^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">20<br />
</td>
<td style="text-align: center;">downmajor seventh<br />
</td>
<td style="text-align: center;">vM7<br />
</td>
<td style="text-align: center;">1091<br />
</td>
<td style="text-align: center;">Bv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">21<br />
</td>
<td style="text-align: center;">major seventh<br />
</td>
<td style="text-align: center;">M7<br />
</td>
<td style="text-align: center;">1145<br />
</td>
<td style="text-align: center;">B<br />
</td>
</tr>
<tr>
<td style="text-align: center;">22<br />
</td>
<td style="text-align: center;">octave<br />
</td>
<td style="text-align: center;">P8<br />
</td>
<td style="text-align: center;">1200<br />
</td>
<td style="text-align: center;">C<br />
</td>
</tr>
</table>
<br />
Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D.<br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc4"><a name="Theory-Intervals by degree (Porcupine)"></a><!-- ws:end:WikiTextHeadingRule:10 -->Intervals by degree (Porcupine)</h2>
<table class="wiki_table">
<tr>
<td style="text-align: center;">Degree<br />
</td>
<td style="text-align: center;">Interval Name<br />
</td>
<td style="text-align: center;">Abbreviation<br />
</td>
<td style="text-align: center;">Cents<br />
</td>
<td style="text-align: center;">Example<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">perfect unison<br />
</td>
<td style="text-align: center;">P1<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">aug unison<br />
</td>
<td style="text-align: center;">A1<br />
</td>
<td style="text-align: center;">55<br />
</td>
<td style="text-align: center;">D#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">dim second<br />
</td>
<td style="text-align: center;">d2<br />
</td>
<td style="text-align: center;">109<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3<br />
</td>
<td style="text-align: center;">perfect second<br />
</td>
<td style="text-align: center;">P2<br />
</td>
<td style="text-align: center;">164<br />
</td>
<td style="text-align: center;">E<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">aug second<br />
</td>
<td style="text-align: center;">A2<br />
</td>
<td style="text-align: center;">218<br />
</td>
<td style="text-align: center;">E#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5<br />
</td>
<td style="text-align: center;">dim third<br />
</td>
<td style="text-align: center;">d3<br />
</td>
<td style="text-align: center;">273<br />
</td>
<td style="text-align: center;">Fb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6<br />
</td>
<td style="text-align: center;">minor third<br />
</td>
<td style="text-align: center;">m3<br />
</td>
<td style="text-align: center;">327<br />
</td>
<td style="text-align: center;">F<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7<br />
</td>
<td style="text-align: center;">major third<br />
</td>
<td style="text-align: center;">M3<br />
</td>
<td style="text-align: center;">382<br />
</td>
<td style="text-align: center;">F#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8<br />
</td>
<td style="text-align: center;">aug third or dim fourth<br />
</td>
<td style="text-align: center;">A3 or d4<br />
</td>
<td style="text-align: center;">436<br />
</td>
<td style="text-align: center;">Gb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">9<br />
</td>
<td style="text-align: center;">minor fourth<br />
</td>
<td style="text-align: center;">m4<br />
</td>
<td style="text-align: center;">491<br />
</td>
<td style="text-align: center;">G<br />
</td>
</tr>
<tr>
<td style="text-align: center;">10<br />
</td>
<td style="text-align: center;">major fourth<br />
</td>
<td style="text-align: center;">M4<br />
</td>
<td style="text-align: center;">545<br />
</td>
<td style="text-align: center;">G#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">11<br />
</td>
<td style="text-align: center;">aug fourth or dim fifth<br />
</td>
<td style="text-align: center;">A4 or d5<br />
</td>
<td style="text-align: center;">600<br />
</td>
<td style="text-align: center;">Gx or Abb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">12<br />
</td>
<td style="text-align: center;">minor fifth<br />
</td>
<td style="text-align: center;">m5<br />
</td>
<td style="text-align: center;">655<br />
</td>
<td style="text-align: center;">Ab<br />
</td>
</tr>
<tr>
<td style="text-align: center;">13<br />
</td>
<td style="text-align: center;">major fifth<br />
</td>
<td style="text-align: center;">M5<br />
</td>
<td style="text-align: center;">709<br />
</td>
<td style="text-align: center;">A<br />
</td>
</tr>
<tr>
<td style="text-align: center;">14<br />
</td>
<td style="text-align: center;">aug fifth or dim sixth<br />
</td>
<td style="text-align: center;">A5 or d6<br />
</td>
<td style="text-align: center;">764<br />
</td>
<td style="text-align: center;">A#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">15<br />
</td>
<td style="text-align: center;">minor sixth<br />
</td>
<td style="text-align: center;">m6<br />
</td>
<td style="text-align: center;">818<br />
</td>
<td style="text-align: center;">Bb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">16<br />
</td>
<td style="text-align: center;">major sixth<br />
</td>
<td style="text-align: center;">M6<br />
</td>
<td style="text-align: center;">873<br />
</td>
<td style="text-align: center;">B<br />
</td>
</tr>
<tr>
<td style="text-align: center;">17<br />
</td>
<td style="text-align: center;">aug sixth<br />
</td>
<td style="text-align: center;">A6<br />
</td>
<td style="text-align: center;">927<br />
</td>
<td style="text-align: center;">B#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">18<br />
</td>
<td style="text-align: center;">dim seventh<br />
</td>
<td style="text-align: center;">d7<br />
</td>
<td style="text-align: center;">982<br />
</td>
<td style="text-align: center;">Cb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">19<br />
</td>
<td style="text-align: center;">perfect seventh<br />
</td>
<td style="text-align: center;">P7<br />
</td>
<td style="text-align: center;">1036<br />
</td>
<td style="text-align: center;">C<br />
</td>
</tr>
<tr>
<td style="text-align: center;">20<br />
</td>
<td style="text-align: center;">aug seventh<br />
</td>
<td style="text-align: center;">A7<br />
</td>
<td style="text-align: center;">1091<br />
</td>
<td style="text-align: center;">C#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">21<br />
</td>
<td style="text-align: center;">dim octave<br />
</td>
<td style="text-align: center;">d8<br />
</td>
<td style="text-align: center;">1145<br />
</td>
<td style="text-align: center;">Db<br />
</td>
</tr>
<tr>
<td style="text-align: center;">22<br />
</td>
<td style="text-align: center;">octave<br />
</td>
<td style="text-align: center;">P8<br />
</td>
<td style="text-align: center;">1200<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
</table>
Yet another notation is pentatonic. This is the only way to use a chain-of-fifths notation without additional accidentals. The pentatonic degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The keyboard runs D * * * E * * * * G * * * A * * * * C * * * D.<br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc5"><a name="Theory-Intervals by degree (pentatonic)"></a><!-- ws:end:WikiTextHeadingRule:12 -->Intervals by degree (pentatonic)</h2>
<table class="wiki_table">
<tr>
<td style="text-align: center;">Degree<br />
</td>
<td style="text-align: center;">Interval Name<br />
</td>
<td style="text-align: center;">Abbreviation<br />
</td>
<td style="text-align: center;">Cents<br />
</td>
<td style="text-align: center;">Example<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">perfect unison<br />
</td>
<td style="text-align: center;">P1<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">aug unison<br />
</td>
<td style="text-align: center;">A1<br />
</td>
<td style="text-align: center;">55<br />
</td>
<td style="text-align: center;">D#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">double-aug unison or <br />
double-dim sub3rd<br />
</td>
<td style="text-align: center;">AA1 or dds3<br />
</td>
<td style="text-align: center;">109<br />
</td>
<td style="text-align: center;">Dx or Ebb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3<br />
</td>
<td style="text-align: center;">dim sub3rd<br />
</td>
<td style="text-align: center;">ds3<br />
</td>
<td style="text-align: center;">164<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">minor sub3rd<br />
</td>
<td style="text-align: center;">ms3<br />
</td>
<td style="text-align: center;">218<br />
</td>
<td style="text-align: center;">E<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5<br />
</td>
<td style="text-align: center;">major sub3rd<br />
</td>
<td style="text-align: center;">Ms3<br />
</td>
<td style="text-align: center;">273<br />
</td>
<td style="text-align: center;">E#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6<br />
</td>
<td style="text-align: center;">aug sub3rd<br />
</td>
<td style="text-align: center;">As3<br />
</td>
<td style="text-align: center;">327<br />
</td>
<td style="text-align: center;">Ex<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7<br />
</td>
<td style="text-align: center;">double-aug sub3rd or <br />
double-dim 4thoid<br />
</td>
<td style="text-align: center;">AAs3 or dd4d<br />
</td>
<td style="text-align: center;">382<br />
</td>
<td style="text-align: center;">Gbb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8<br />
</td>
<td style="text-align: center;">dim 4thoid<br />
</td>
<td style="text-align: center;">d4d<br />
</td>
<td style="text-align: center;">436<br />
</td>
<td style="text-align: center;">Gb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">9<br />
</td>
<td style="text-align: center;">perfect 4thoid<br />
</td>
<td style="text-align: center;">P4d<br />
</td>
<td style="text-align: center;">491<br />
</td>
<td style="text-align: center;">G<br />
</td>
</tr>
<tr>
<td style="text-align: center;">10<br />
</td>
<td style="text-align: center;">aug 4thoid<br />
</td>
<td style="text-align: center;">A4d<br />
</td>
<td style="text-align: center;">545<br />
</td>
<td style="text-align: center;">G#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">11<br />
</td>
<td style="text-align: center;">double-aug 4thoid or <br />
double-dim 5thoid<br />
</td>
<td style="text-align: center;">AA4d or dd5d<br />
</td>
<td style="text-align: center;">600<br />
</td>
<td style="text-align: center;">Gx or Abb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">12<br />
</td>
<td style="text-align: center;">dim 5thoid<br />
</td>
<td style="text-align: center;">d5d<br />
</td>
<td style="text-align: center;">655<br />
</td>
<td style="text-align: center;">Ab<br />
</td>
</tr>
<tr>
<td style="text-align: center;">13<br />
</td>
<td style="text-align: center;">perfect 5thoid<br />
</td>
<td style="text-align: center;">P5d<br />
</td>
<td style="text-align: center;">709<br />
</td>
<td style="text-align: center;">A<br />
</td>
</tr>
<tr>
<td style="text-align: center;">14<br />
</td>
<td style="text-align: center;">aug 5thoid<br />
</td>
<td style="text-align: center;">A5d<br />
</td>
<td style="text-align: center;">764<br />
</td>
<td style="text-align: center;">A#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">15<br />
</td>
<td style="text-align: center;">double-aug 5thoid or <br />
double-dim sub7th<br />
</td>
<td style="text-align: center;">AA5d or dds7<br />
</td>
<td style="text-align: center;">818<br />
</td>
<td style="text-align: center;">Ax<br />
</td>
</tr>
<tr>
<td style="text-align: center;">16<br />
</td>
<td style="text-align: center;">dim sub7th<br />
</td>
<td style="text-align: center;">ds7<br />
</td>
<td style="text-align: center;">873<br />
</td>
<td style="text-align: center;">Cbb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">17<br />
</td>
<td style="text-align: center;">minor sub7th<br />
</td>
<td style="text-align: center;">ms7<br />
</td>
<td style="text-align: center;">927<br />
</td>
<td style="text-align: center;">Cb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">18<br />
</td>
<td style="text-align: center;">major sub7th<br />
</td>
<td style="text-align: center;">Ms7<br />
</td>
<td style="text-align: center;">982<br />
</td>
<td style="text-align: center;">C<br />
</td>
</tr>
<tr>
<td style="text-align: center;">19<br />
</td>
<td style="text-align: center;">aug sub7th<br />
</td>
<td style="text-align: center;">As7<br />
</td>
<td style="text-align: center;">1036<br />
</td>
<td style="text-align: center;">C#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">20<br />
</td>
<td style="text-align: center;">double-aug sub7th or <br />
double-dim octoid<br />
</td>
<td style="text-align: center;">AAs7 or dd8d<br />
</td>
<td style="text-align: center;">1091<br />
</td>
<td style="text-align: center;">Cx or Dbb<br />
</td>
</tr>
<tr>
<td style="text-align: center;">21<br />
</td>
<td style="text-align: center;">dim octoid<br />
</td>
<td style="text-align: center;">d8d<br />
</td>
<td style="text-align: center;">1145<br />
</td>
<td style="text-align: center;">Db<br />
</td>
</tr>
<tr>
<td style="text-align: center;">22<br />
</td>
<td style="text-align: center;">octave<br />
</td>
<td style="text-align: center;">P8<br />
</td>
<td style="text-align: center;">1200<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc6"><!-- ws:end:WikiTextHeadingRule:14 --> </h2>
<!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc7"><a name="Theory-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:16 -->Selected just intervals by error</h2>
The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 22edo (ordered by absolute error).<br />
<table class="wiki_table">
<tr>
<td><strong>Interval, complement</strong><br />
</td>
<td><strong>Error (abs., in <a class="wiki_link" href="/cent">cents</a>)</strong><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br />
</td>
<td style="text-align: center;">1.280<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_10">11/10</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
<td style="text-align: center;">1.368<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_8">15/8</a><br />
</td>
<td style="text-align: center;">2.640<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br />
</td>
<td style="text-align: center;">4.496<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br />
</td>
<td style="text-align: center;">5.856<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br />
</td>
<td style="text-align: center;">5.863<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br />
</td>
<td style="text-align: center;">7.136<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/22_15">22/15</a><br />
</td>
<td style="text-align: center;">8.504<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/28_15">28/15</a><br />
</td>
<td style="text-align: center;">10.352<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br />
</td>
<td style="text-align: center;">11.631<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
<td style="text-align: center;">12.992<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a>, <a class="wiki_link" href="/11_6">11/6</a><br />
</td>
<td style="text-align: center;">12.999<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br />
</td>
<td style="text-align: center;">14.272<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_11">13/11</a>, <a class="wiki_link" href="/22_13">22/13</a><br />
</td>
<td style="text-align: center;">16.482<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br />
</td>
<td style="text-align: center;">17.488<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_10">13/10</a>, <a class="wiki_link" href="/20_13">20/13</a><br />
</td>
<td style="text-align: center;">17.850<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/18_13">18/13</a>, <a class="wiki_link" href="/13_9">13/9</a><br />
</td>
<td style="text-align: center;">17.928<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br />
</td>
<td style="text-align: center;">18.767<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/11_7">11/7</a><br />
</td>
<td style="text-align: center;">18.856<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/13_7">13/7</a><br />
</td>
<td style="text-align: center;">19.207<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br />
</td>
<td style="text-align: center;">20.135<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_13">16/13</a>, <a class="wiki_link" href="/13_8">13/8</a><br />
</td>
<td style="text-align: center;">22.346<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_13">15/13</a>, <a class="wiki_link" href="/26_15">26/15</a><br />
</td>
<td style="text-align: center;">24.986<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_12">13/12</a>, <a class="wiki_link" href="/24_13">24/13</a><br />
</td>
<td style="text-align: center;">25.064<br />
</td>
</tr>
</table>
<br />
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See also: <a class="wiki_link" href="/22edo%20Solfege">22edo Solfege</a>, <a class="wiki_link" href="/22edo%20Tetrachords">22edo Tetrachords</a>, <a class="wiki_link" href="/22%20EDO%20Chords">22 EDO Chords</a>, <a class="wiki_link" href="/22edo%20Modes">22edo Modes</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc8"><a name="Theory-Properties of 22 equal temperament"></a><!-- ws:end:WikiTextHeadingRule:18 -->Properties of 22 equal temperament</h2>
<br />
Possibly the most striking characteristic of 22-et to those not used to it is that it does <strong>not</strong> "temper out" the syntonic comma of 81/80, and therefore is not a system of <a class="wiki_link" href="/Regular%20Temperaments#meantone">meantone</a> temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12-EDO does not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/41edo">41edo</a> and <a class="wiki_link" href="/53edo">53edo</a>.<br />
<br />
The diatonic scale it produces is instead derived from <a class="wiki_link" href="/superpyth">superpyth</a> temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, <a class="wiki_link" href="/5L%202s">5L 2s</a>), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22-EDO. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12-equal and meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.<br />
<br />
It additionally tempers out the porcupine comma or maximal diesis of 250/243, which means that 22-EDO supports <a class="wiki_link" href="/porcupine">porcupine</a> temperament. The generator for porcupine is is a flat minor whole tone of <a class="wiki_link" href="/10_9">10/9</a>, two of which is a slightly sharp <a class="wiki_link" href="/6_5">6/5</a>, and three of which is a slightly flat <a class="wiki_link" href="/4_3">4/3</a>, implying the existence of an equal-step tetrachord, which is characteristic of Porcupine. Porcupine is notable for being the 5-limit temperament lowest in <a class="wiki_link" href="/badness">badness</a> which is <em>not</em> approximated by the familiar 12-tone equal temperament, and as such represents one excellent point of departure for examining the harmonic properties of 22-EDO. It forms <a class="wiki_link" href="/MOSScales">MOS</a>'s of 7 and 8, which in 22-EDO are tuned respectively as 4 3 3 3 3 3 3 and 3 1 3 3 3 3 3 3 (and their respective modes).<br />
<br />
The 164¢ "flat minor whole tone" is a key interval in 22-EDO, in part because it functions as no less than three different consonant ratios in the <a class="wiki_link" href="/11-limit">11-limit</a>: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22-EDO can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.<br />
<br />
22-EDO also supports Orwell temperament, which uses the septimal subminor third as a generator (5 degrees) and forms MOS scales with step patterns 3 2 3 2 3 2 3 2 2 and 1 2 2 1 2 2 1 2 2 1 2 2 2. Harmonically, Orwell can be tuned more accurately in other temperaments, such as <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/53edo">53edo</a> and <a class="wiki_link" href="/84edo">84edo</a>. But 22-equal Orwell has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish in 22.<br />
<br />
Other 5-limit commas 22-EDO tempers out include the diaschisma, 2048/2025 and the magic comma or small diesis, 3125/3072. In a diaschismic system, such as 12-et or 22-et, the <a class="wiki_link" href="/diatonic%20tritone">diatonic tritone</a> <a class="wiki_link" href="/45_32">45/32</a>, which is a major third above a <a class="wiki_link" href="/major%20whole%20tone">major whole tone</a> representing <a class="wiki_link" href="/9_8">9/8</a>, is equated to its inverted form, <a class="wiki_link" href="/64_45">64/45</a>. That the magic comma is tempered out means that 22-et is a <a class="wiki_link" href="/Regular%20Temperaments#magic">magic</a> system, where five major thirds make up a perfect fifth.<br />
<br />
In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both <a class="wiki_link" href="/50_49">50/49</a>, (the <a class="wiki_link" href="/jubilee%20comma">jubilee comma</a>), and <a class="wiki_link" href="/64_63">64/63</a>, (the <a class="wiki_link" href="/septimal%20comma">septimal comma</a>), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the <a class="wiki_link" href="/septimal%20kleisma">septimal kleisma</a>, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the <a class="wiki_link" href="/orwell%20comma">orwell comma</a>; and the <a class="wiki_link" href="/orwell%20tetrad">orwell tetrad</a> is also a chord of 22-et.<br />
<br />
As 22 is divisible by 11, a 22edo instrument can play any music in <a class="wiki_link" href="/11edo">11edo</a>, in the same way that 12edo can play 6edo (the whole tone scale). 11-equal is interesting for sounding melodically very similar to 12-equal (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to 24edo as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In <a class="wiki_link" href="/Sagittal">Sagittal</a>, 11 can be notated as every other note of 22.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc9"><a name="Theory-Properties of 22 equal temperament-Rank Two Temperaments"></a><!-- ws:end:WikiTextHeadingRule:20 -->Rank Two Temperaments</h3>
<a class="wiki_link" href="/List%20of%2022et%20rank%20two%20temperaments%20by%20badness">List of 22et rank two temperaments by badness</a><br />
<a class="wiki_link" href="/List%20of%2022et%20rank%20two%20temperaments%20by%20complexity">List of 22et rank two temperaments by complexity</a><br />
<a class="wiki_link" href="/List%20of%20edo-distinct%2022et%20rank%20two%20temperaments">List of edo-distinct 22et rank two temperaments</a><br />
<table class="wiki_table">
<tr>
<th>Periods<br />
per octave<br />
</th>
<th>Generator<br />
</th>
<th>Temperaments<br />
</th>
</tr>
<tr>
<td>1<br />
</td>
<td>1\22<br />
</td>
<td><a class="wiki_link" href="/Sensamagic%20clan#Sensa">Sensa</a>/chromo/ceratitid<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>3\22<br />
</td>
<td><a class="wiki_link" href="/Porcupine">Porcupine</a><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>5\22<br />
</td>
<td><a class="wiki_link" href="/Orson">Orson</a>/<a class="wiki_link" href="/orwell">orwell</a>/blair<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>7\22<br />
</td>
<td><a class="wiki_link" href="/Magic">Magic</a>/telepathy<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>9\22<br />
</td>
<td><a class="wiki_link" href="/Superpyth">Superpyth</a>/<a class="wiki_link" href="/suprapyth">suprapyth</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>1\22<br />
</td>
<td><a class="wiki_link" href="/Shrutar">Shrutar</a>/hemipaj/comic<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>2\22<br />
</td>
<td><a class="wiki_link" href="/Srutal">Srutal</a>/<a class="wiki_link" href="/pajara">pajara</a>/pajarous<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>3\22<br />
</td>
<td><a class="wiki_link" href="/Porcupine%20family#Hedgehog">Hedgehog</a>/<a class="wiki_link" href="/echidna">echidna</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>4\22<br />
</td>
<td><a class="wiki_link" href="/Astrology">Astrology</a>/<a class="wiki_link" href="/wizard">wizard</a>/<a class="wiki_link" href="/antikythera">antikythera</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>5\22<br />
</td>
<td><a class="wiki_link" href="/Doublewide">Doublewide</a>/fleetwood<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>1\22<br />
</td>
<td><a class="wiki_link" href="/Hendecatonic">Hendecatonic</a>/undeka<br />
</td>
</tr>
</table>
<!-- ws:start:WikiTextHeadingRule:22:<h3> --><h3 id="toc10"><a name="Theory-Properties of 22 equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule:22 -->Commas</h3>
22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.)<br />
<table class="wiki_table">
<tr>
<th>Comma<br />
</th>
<th>Monzo<br />
</th>
<th>Value (Cents)<br />
</th>
<th>Name 1<br />
</th>
<th>Name 2<br />
</th>
<th>Name 3<br />
</th>
</tr>
<tr>
<td style="text-align: center;">250/243<br />
</td>
<td style="text-align: left;">| 1 -5 3 ><br />
</td>
<td style="text-align: right;">49.17<br />
</td>
<td style="text-align: center;">Maximal Diesis<br />
</td>
<td style="text-align: center;">Porcupine Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3125/3072<br />
</td>
<td style="text-align: left;">| -10 -1 5 ><br />
</td>
<td style="text-align: right;">29.61<br />
</td>
<td style="text-align: center;">Small Diesis<br />
</td>
<td style="text-align: center;">Magic Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2048/2025<br />
</td>
<td style="text-align: left;">| 11 -4 -2 ><br />
</td>
<td style="text-align: right;">19.55<br />
</td>
<td style="text-align: center;">Diaschisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2109375/2097152<br />
</td>
<td style="text-align: left;">| -21 3 7 ><br />
</td>
<td style="text-align: right;">10.06<br />
</td>
<td style="text-align: center;">Semicomma<br />
</td>
<td style="text-align: center;">Fokker Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9193891/9143623<br />
</td>
<td style="text-align: left;">| 32 -7 -9 ><br />
</td>
<td style="text-align: right;">9.49<br />
</td>
<td style="text-align: center;">Escapade Comma<br />
</td>
<td><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4758837/4757272<br />
</td>
<td style="text-align: left;">| -53 10 16 ><br />
</td>
<td style="text-align: right;">0.57<br />
</td>
<td style="text-align: center;">Kwazy<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">50/49<br />
</td>
<td style="text-align: left;">| 1 0 2 -2 ><br />
</td>
<td style="text-align: right;">34.98<br />
</td>
<td style="text-align: center;">Tritonic Diesis<br />
</td>
<td style="text-align: center;">Jubilisma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">64/63<br />
</td>
<td style="text-align: left;">| 6 -2 0 -1 ><br />
</td>
<td style="text-align: right;">27.26<br />
</td>
<td style="text-align: center;">Septimal Comma<br />
</td>
<td style="text-align: center;">Archytas' Comma<br />
</td>
<td style="text-align: center;">Leipziger Komma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">875/864<br />
</td>
<td style="text-align: left;">| -5 -3 3 1 ><br />
</td>
<td style="text-align: right;">21.90<br />
</td>
<td style="text-align: center;">Keema<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2430/2401<br />
</td>
<td style="text-align: left;">| 1 5 1 -4 ><br />
</td>
<td style="text-align: right;">20.79<br />
</td>
<td style="text-align: center;">Nuwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">245/243<br />
</td>
<td style="text-align: left;">| 0 -5 1 2 ><br />
</td>
<td style="text-align: right;">14.19<br />
</td>
<td style="text-align: center;">Sensamagic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1728/1715<br />
</td>
<td style="text-align: left;">| 6 3 -1 -3 ><br />
</td>
<td style="text-align: right;">13.07<br />
</td>
<td style="text-align: center;">Orwellisma<br />
</td>
<td style="text-align: center;">Orwell Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">225/224<br />
</td>
<td style="text-align: left;">| -5 2 2 -1 ><br />
</td>
<td style="text-align: right;">7.71<br />
</td>
<td style="text-align: center;">Septimal Kleisma<br />
</td>
<td style="text-align: center;">Marvel Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">10976/10935<br />
</td>
<td style="text-align: left;">| 5 -7 -1 3 ><br />
</td>
<td style="text-align: right;">6.48<br />
</td>
<td style="text-align: center;">Hemimage<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">6144/6125<br />
</td>
<td style="text-align: left;">| 11 1 -3 -2 ><br />
</td>
<td style="text-align: right;">5.36<br />
</td>
<td style="text-align: center;">Porwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">65625/65536<br />
</td>
<td style="text-align: left;">| -16 1 5 1 ><br />
</td>
<td style="text-align: right;">2.35<br />
</td>
<td style="text-align: center;">Horwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">420175/419904<br />
</td>
<td style="text-align: left;">| -6 -8 2 5 ><br />
</td>
<td style="text-align: right;">1.12<br />
</td>
<td style="text-align: center;">Wizma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">99/98<br />
</td>
<td style="text-align: left;">| -1 2 0 -2 1 ><br />
</td>
<td style="text-align: right;">17.58<br />
</td>
<td style="text-align: center;">Mothwellsma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">100/99<br />
</td>
<td style="text-align: left;">| 2 -2 2 0 -1 ><br />
</td>
<td style="text-align: right;">17.40<br />
</td>
<td style="text-align: center;">Ptolemisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">121/120<br />
</td>
<td style="text-align: left;">| -3 -1 -1 0 2 ><br />
</td>
<td style="text-align: right;">14.37<br />
</td>
<td style="text-align: center;">Biyatisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td>125/124<br />
</td>
<td>|-4 0 3 0 ... -1><br />
</td>
<td>13.91<br />
</td>
<td style="text-align: center;">Twizzler<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">176/175<br />
</td>
<td style="text-align: left;">| 4 0 -2 -1 1 ><br />
</td>
<td style="text-align: right;">9.86<br />
</td>
<td style="text-align: center;">Valinorsma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">896/891<br />
</td>
<td style="text-align: left;">| 7 -4 0 1 -1 ><br />
</td>
<td style="text-align: right;">9.69<br />
</td>
<td style="text-align: center;">Pentacircle<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">65536/65219<br />
</td>
<td style="text-align: left;">| 16 0 0 -2 -3 ><br />
</td>
<td style="text-align: right;">8.39<br />
</td>
<td style="text-align: center;">Orgonisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">385/384<br />
</td>
<td style="text-align: left;">| -7 -1 1 1 1 ><br />
</td>
<td style="text-align: right;">4.50<br />
</td>
<td style="text-align: center;">Keenanisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">540/539<br />
</td>
<td style="text-align: left;">| 2 3 1 -2 -1 ><br />
</td>
<td style="text-align: right;">3.21<br />
</td>
<td style="text-align: center;">Swetisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3993<br />
</td>
<td style="text-align: left;">| 5 -1 3 0 -3 ><br />
</td>
<td style="text-align: right;">3.03<br />
</td>
<td style="text-align: center;">Wizardharry<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9801/9800<br />
</td>
<td style="text-align: left;">| -3 4 -2 -2 2 ><br />
</td>
<td style="text-align: right;">0.18<br />
</td>
<td style="text-align: center;">Kalisma<br />
</td>
<td style="text-align: center;">Gauss' Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">91/90<br />
</td>
<td style="text-align: left;">| -1 -2 -1 1 0 1 ><br />
</td>
<td style="text-align: right;">19.13<br />
</td>
<td style="text-align: center;">Superleap<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:24:<h3> --><h3 id="toc11"><a name="Theory-Properties of 22 equal temperament-How to Notate 22edo in Sagittal"></a><!-- ws:end:WikiTextHeadingRule:24 -->How to Notate 22edo in Sagittal</h3>
<br />
When 22edo is treated as generated by a cycle of its fifths, the naturals F C G D A E B represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:<br />
<!-- ws:start:WikiTextLocalImageRule:2023:<img src="/file/view/22edo.png/269078624/22edo.png" alt="" title="" /> --><img src="/file/view/22edo.png/269078624/22edo.png" alt="22edo.png" title="22edo.png" /><!-- ws:end:WikiTextLocalImageRule:2023 --><br />
This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.<br />
<br />
The division of the apotome into three syntonic commas also indicates 22's tempering out of the <a class="wiki_link" href="/250_243">porcupine comma</a> (which is equivalent to three syntonic commas minus a Pythagorean apotome).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc12"><a name="The Decatonic System"></a><!-- ws:end:WikiTextHeadingRule:26 -->The Decatonic System</h1>
<span style="background-color: #ffffff; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">The decatonic system is an approach of notation based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system bases music into a 10 tone scale rather than 7. </span><br />
<span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">This approach requires an entire re-learning of chords, intervals, and notation but the advantage is that it allows 22 EDO to be notated using only one pair of accidentals, as well as</span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">gives the opportunity to escape a heptatonic thinking pattern.</span><br />
<br />
<span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"><br />
<!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc13"><a name="The Decatonic System-Decatonic Alphabet"></a><!-- ws:end:WikiTextHeadingRule:28 --><!-- ws:start:WikiTextAnchorRule:58:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-Decatonic-Alphabet" title="Anchor: TOC-Decatonic-Alphabet"/> --><a name="TOC-Decatonic-Alphabet"></a><!-- ws:end:WikiTextAnchorRule:58 -->Decatonic Alphabet</h2>
</span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">The system is based on two chains of fifths. One represented by latin letters, the other greek. The two chains can be looked at as two juxtaposed pentatonic scales.</span><br />
<br />
<span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">Chain 1: C G D A E</span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">Chain 2: <span style="background-color: transparent; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">γ δ α ε β </span></span><br />
<br />
<span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;">The alphabet is ascending order: C<span style="font-family: helvetica,arial,sans-serif; font-size: 12px;"> δ D ε E γ G α A β C </span></span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"><br />
</span><span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"><span style="font-family: helvetica,arial,sans-serif; font-size: 12px;">In this alphabet, a chain of fifths is preserved because equivalent greek letters also represent fifths if they are the same as their latin counter parts. For example G D is a fifth as well as </span><span style="color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">γ δ. </span></span><br />
<!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc14"><!-- ws:end:WikiTextHeadingRule:30 --> </h1>
<table class="wiki_table">
<tr>
<td><strong>Mode Name</strong><br />
</td>
<td>Notes<br />
</td>
<td>Step Structure<br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Keter</span><br />
</td>
<td><span style="background-color: #ffffff; font-family: helvetica,arial,sans-serif; font-size: 12px;">α A β C δ D ε E</span><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">˥ </span><span style="background-color: #ffffff; font-family: helvetica,arial,sans-serif; font-size: 12px;">γ G α</span><br />
</td>
<td>2 2 3 2 2 2 3 2 2 2<br />
</td>
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<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Chokhmah</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
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<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Binah</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
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<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Chesed</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Gevurah</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Tiferet</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Netzach</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Hod</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Yesod</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="background-color: #ffffff; color: #141823; font-family: helvetica,arial,sans-serif; font-size: 14px;">Malkuth</span><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
<span style="background-color: #ffffff; display: block; font-family: Arial,Verdana,sans-serif; font-size: 13.3333px;"><br />
</span><br />
<!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc15"><a name="The Decatonic System-External links"></a><!-- ws:end:WikiTextHeadingRule:32 -->External links</h2>
<br />
<a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow">Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''</a><br />
<br />
<a class="wiki_link_ext" href="http://porcupinemusic.weebly.com/" rel="nofollow">"Porcupine Music" - Website Focused on the Development of 22 EDO music </a><br />
<!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc16"><a name="The Decatonic System-References"></a><!-- ws:end:WikiTextHeadingRule:34 -->References</h2>
<br />
Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br />
Bosanquet, R.H.M. <a class="wiki_link_ext" href="http://www.webcitation.org/5kjJcrhEx" rel="nofollow">''On the Hindoo division of the octave, with additions to the theory of higher orders''</a>, Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965<br />
<br />
<hr />
<!-- ws:start:WikiTextHeadingRule:36:<h1> --><h1 id="toc17"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:36 -->Music</h1>
<br />
<ul><li><a class="wiki_link_ext" href="https://soundcloud.com/overtoneshock/dose-of-familiarityode-to-microtonality-22-edo-studio-version" rel="nofollow" target="_blank">Stephen Weigel · Dose Of Familiarity/Ode to Microtonality</a></li><li><a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1145" rel="nofollow" target="_blank">Canon 2 in 1 upon a ground (22edo)</a> by Claudi Meneghin</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3" rel="nofollow">Tibia</a></span></span> by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a><ul><li>Sagittal score of Tibia, <a href="http://xenharmonic.wikispaces.com/file/view/TIBIA.pdf/313029038/TIBIA.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/TIBIA.pdf/313029038/TIBIA.pdf');">in F||\</a> or <a href="http://xenharmonic.wikispaces.com/file/view/tibia%20in%20g.pdf/313029040/tibia%20in%20g.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/tibia%20in%20g.pdf/313029040/tibia%20in%20g.pdf');">in G</a></li></ul></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095" rel="nofollow">Glassic</a></span></span> by Paul Erlich and <a class="wiki_link" href="/Ara%20Sarkissian">Ara Sarkissian</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/decatonic-swing.mp3" rel="nofollow">Decatonic Swing</a></span></span> by Paul Erlich and Ara Sarkissian (jazz)</li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3" rel="nofollow">12-22hexachordal Dirge</a> by <a class="wiki_link" href="/Joel%20Grant%20Taylor">Joel Grant Taylor</a></li><li><a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs" rel="nofollow" target="_blank">Chord sequence in Paul Erlich's 22 EDO decatonic major</a> by <a class="wiki_link" href="/Jake%20Freivald">Jake Freivald</a></li><li><a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/porcupine-comma-pump" rel="nofollow">Porcupine Comma Pump</a> by <a class="wiki_link" href="/Jake%20Freivald">Jake Freivald</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Dragged%20By%20a%20Storm%20Across%20the%20Desert%20Years.mp3" rel="nofollow">Dragged by a Storm Across the Desert Years</a></span></span> by * <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a> (synth with electric guitar)</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Numerology.mp3" rel="nofollow">Numerology</a></span></span> by Iglashion Jones (progressive metal)</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2022-Revenge%20of%20the%20Inorganic%20Compounds.mp3" rel="nofollow">Revenge of the inorganic compounds</a></span></span> by Iglashion Jones (progressive metal)</li><li><a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=267" rel="nofollow">My Crazy Aunt Sophie</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/22edo-piano-my-crazy-aunt-sophie.mp3" rel="nofollow">play</a></span></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a>. Blatantly xenharmonic piano.</li><li><a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839058" rel="nofollow">where words are said to mean</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+wherewordsaresaidtomean.mp3" rel="nofollow">play</a></span></span> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a>, a setting of a text by Herbert Brün to a 22-tone row, thrice repeated. This & the following pieces by Andrew are for 22-tone guitar & voice.</li><li><a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow">I've come with a bucket of roses</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+ivecomewithabucketofroses.mp3" rel="nofollow">play</a></span></span> by Andrew Heathwaite (orwell-9: 3 2 3 2 3 2 3 2 2).</li><li><a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow">one drop of rain</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+onedropofrain.mp3" rel="nofollow">play</a></span></span> by Andrew Heathwaite (orwell-9).</li><li><a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839060" rel="nofollow">being a</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+beinga.mp3" rel="nofollow">play</a></span></span> by Andrew Heathwaite (porcupine-8: 3 1 3 3 3 3 3).</li><li><a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow">my own house</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+myownhouse.mp3" rel="nofollow">play</a></span></span> by Andrew Heathwaite (a pelog-flavored subset of orwell-9: 3 2 7 3 7).</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/17%20-%2017.%2022%20octave.mp3" rel="nofollow">Comets Over Flatland 17</a></span></span> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3" rel="nofollow">Night on Porcupine Mountain</a></span></span> Mussorgsky-Smith</li><li><span class="ywp-page-play-pause ywp-page-video ywp-link-hover"><span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=lO5xSjIHyMg" rel="nofollow">Paul Erlich 22-Equal Guitar Improvisation Shredfest Insanity</a></span></span> - youtube</li><li><span class="ywp-page-play-pause ywp-page-video ywp-link-hover"><span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=WMtp9Wk0tO0" rel="nofollow">Improvisation in 22-equal temperament</a></span></span>, Mike Battaglia - youtube</li><li>Boxwood Forest, Dream Tone, The Eternal Sleep, Sunday Pipes, Twisted Clowns - <a class="wiki_link_ext" href="http://www.angelfire.com/mo/oljare/midicomp.html" rel="nofollow">MIDI files</a> by Mats Öljare<ul><li><a href="http://xenharmonic.wikispaces.com/file/view/sunday3.pdf/269076436/sunday3.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/sunday3.pdf/269076436/sunday3.pdf');">Sagittal score of Sunday Pipes</a></li></ul></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3" rel="nofollow">Phobos Light</a></span> by Chris Vaisvil in Hedgehog[14] <a class="wiki_link" href="/hedgehog14">tuned</a> to 22edo.</li><li><em><a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120716_theorbo_22edo.mp3" rel="nofollow">The Capture and Release of the Fairy</a></em> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a> => <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2494" rel="nofollow">blog post presentation</a></li><li><em><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=oNJr1YOOqF8" rel="nofollow">Yak Butter</a></em> by The Stern Brocot Band, 1L6s MOS, compressed period/generator</li><li><a class="wiki_link_ext" href="http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_03_-_Dirty_Drummer_vbr.mp3" rel="nofollow">Dirty Drummer</a>, Sevish</li><li><a class="wiki_link_ext" href="http://www.archive.org/download/Sevish_-_Golden_Hour/Sevish_-_12_-_Ganymede_vbr.mp3" rel="nofollow">Ganymede</a>, Sevish (doesn't sound that xen, but it's in 22-edo)</li><li><a class="wiki_link_ext" href="http://www.archive.org/download/HumanAstronomy/03Sevish-Ambrosia.mp3" rel="nofollow">Ambrosia</a>, Sevish</li><li><em><a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120726-from-the-sky-islands-they-came.mp3" rel="nofollow">From the Sky Islands They Came</a></em> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a> => <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2523" rel="nofollow">blog post presentation</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3" rel="nofollow">Smoke Filled Bar</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a> => <a class="wiki_link_ext" href="http://chrisvaisvil.com/smoke-filled-bar/" rel="nofollow" target="_blank">blog presentation</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Sultan/__Recurring_Mimosa_by_Redrick_Sultan.mp3" rel="nofollow">Recurring Mimosa</a> by <a class="wiki_link_ext" href="https://soundcloud.com/redrick-sultan/recurring-mimosa" rel="nofollow">Redrick Sultan</a></li><li>The Saharan Pump by Chris Vaisvil <a class="wiki_link_ext" href="http://chrisvaisvil.com/the-saharan-pump-22-edo-rock/" rel="nofollow">blog post</a></li><li><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=qHHv3mwJTlg" rel="nofollow" target="_blank">Short piece and demonstration</a> (video) by <a class="wiki_link_ext" href="http://brendanbyrnes.com/" rel="nofollow" target="_blank">Brendan Byrnes</a> (electric guitar)</li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%2022%20EDO%20Guitar%20Etude.mp3" rel="nofollow">22 EDO Guitar Etude</a> by <a class="wiki_link_ext" href="http://brendanbyrnes.bandcamp.com/" rel="nofollow">Brendan Byrnes</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Byrnes/Brendan%20Byrnes%20-%20Llurion.mp3" rel="nofollow">Llurion</a> by <a class="wiki_link_ext" href="http://brendanbyrnes.bandcamp.com/track/llurion" rel="nofollow">Brendan Byrnes</a></li><li><a class="wiki_link_ext" href="https://youtu.be/0VLJXecjYK4" rel="nofollow" target="_blank">Imzadi</a> by <a class="wiki_link_ext" href="http://omega9.github.io/" rel="nofollow" target="_blank">Omega9</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20150910_22edo.mp3" rel="nofollow">22 edo electric guitar duet</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></li><li><a class="wiki_link_ext" href="https://soundcloud.com/gareth-hearne/mass-in-22edo-sanctus" rel="nofollow">Mass in 22edo - Sanctus</a> by <a class="wiki_link" href="/Gareth%20Hearne">Gareth Hearne</a></li><li><a class="wiki_link_ext" href="https://soundcloud.com/gareth-hearne/mass-in-22edo-agnus-dei" rel="nofollow">Mass in 22edo - Agnus Dei</a> by Gareth Hearne</li></ul><br />
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