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In music, ''22 equal temperament'', called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the [[ | == 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|cents]]. Because it distinguishes 10/9 and 9/8, it's not meantone. | |||
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 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 [[ | 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 EDO lists|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|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|consistently]]. Furthermore, 22-et, unlike 12 and [[19edo|19]], is not a [[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 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. | ||
| Line 18: | Line 20: | ||
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). | 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). | ||
==Properties of 22 equal temperament== | == 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, 19-EDO, 31-EDO, ... do 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 [[ | 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, 19-EDO, 31-EDO, ... do 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 [[ | 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 [[ | 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 a flat minor whole tone of [[10/9]], two of which is a slightly sharp [[6/5]], and three of which is a slightly flat [[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 [[ | 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 [[ | 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|diatonic tritone]] [[45/32|45/32]], which is a major third above a [[major_whole_tone|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. | 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|diatonic tritone]] [[45/32|45/32]], which is a major third above a [[major_whole_tone|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 [[ | 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]], (the [[jubilee comma]]), and [[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|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 [[ | As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo|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. | ||
==Notation== | == Notation == | ||
=== Superpyth/Porcupine Notation === | === Superpyth/Porcupine Notation === | ||
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 porcupine 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. | 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 porcupine 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. | ||
{| class="wikitable" | {| class="wikitable center-1 right-3" | ||
|- | |- | ||
! Degree | ! Degree | ||
! Name and Abbreviation | ! Name and Abbreviation | ||
! Cents | ! Cents | ||
! | ! Approximate Ratios* | ||
|- | |- | ||
| 0 | | 0 | ||
| Natural Unison, 1 | | Natural Unison, 1 | ||
|0.000 | | 0.000 | ||
| | | [[1/1]] | ||
|- | |- | ||
| 1 | | 1 | ||
| s-minor second, sm2 | | s-minor second, sm2 | ||
| 54.545 | | 54.545 | ||
| | | [[36/35]], [[34/33]], [[33/32]], [[32/31]] | ||
|- | |- | ||
| 2 | | 2 | ||
| p-diminished second, pd2 | | p-diminished second, pd2 | ||
| 109. | | 109.091 | ||
| | | [[18/17]], [[17/16]], [[16/15]], [[15/14]] | ||
|- | |- | ||
| 3 | | 3 | ||
| p-minor second, pm2 | | p-minor second, pm2 | ||
| 163. | | 163.636 | ||
| | | [[12/11]], [[11/10]], [[10/9]] | ||
|- | |- | ||
| 4 | | 4 | ||
| (s/p) Major second, M2 | | (s/p) Major second, M2 | ||
| 218. | | 218.182 | ||
| | | [[9/8]], [[17/15]], [[8/7]] | ||
|- | |- | ||
| 5 | | 5 | ||
| s-minor third, sm3 | | s-minor third, sm3 | ||
| 272. | | 272.737 | ||
| [[20/17]], [[7/6]] | |||
|- | |- | ||
| 6 | | 6 | ||
| p-minor third, pm3 | | p-minor third, pm3 | ||
| 327. | | 327.273 | ||
| | | [[6/5]], [[17/14]], [[11/9]] | ||
|- | |- | ||
| 7 | | 7 | ||
| p-Major third, pM3 | | p-Major third, pM3 | ||
| 381. | | 381.818 | ||
| | | [[5/4]] | ||
|- | |- | ||
| 8 | | 8 | ||
| s-Major third, sM3 | | s-Major third, sM3 | ||
| 436. | | 436.364 | ||
| | | [[14/11]], [[9/7]], [[22/17]] | ||
|- | |- | ||
| 9 | | 9 | ||
| Natural Fourth, 4, N4 | | Natural Fourth, 4, N4 | ||
| 490. | | 490.909 | ||
| | | [[4/3]] | ||
|- | |- | ||
| 10 | | 10 | ||
| p-Major Fourth, pM4, s-dim fifth | | p-Major Fourth, pM4, s-dim fifth | ||
|545.455 | | 545.455 | ||
| | | [[15/11]], [[11/8]] | ||
|- | |- | ||
| 11 | | 11 | ||
| Augmented Fourth, A4, Half-Octave, HO | | Augmented Fourth, A4, Half-Octave, HO | ||
| 600 | | 600.000 | ||
| | | [[7/5]], [[24/17]], [[17/12]], [[10/7]] | ||
|- | |- | ||
| 12 | | 12 | ||
| p-minor Fifth, pm5, s-aug fourth | | p-minor Fifth, pm5, s-aug fourth | ||
|654.545 | | 654.545 | ||
| | | [[16/11]], [[22/15]] | ||
|- | |- | ||
| 13 | | 13 | ||
| Natural Fifth, 5, N5 | | Natural Fifth, 5, N5 | ||
| 709. | | 709.091 | ||
| | | [[3/2]] | ||
|- | |- | ||
| 14 | | 14 | ||
| s-minor sixth, sm6 | | s-minor sixth, sm6 | ||
| 763. | | 763.637 | ||
| | | [[17/11]], [[14/9]], [[11/7]] | ||
|- | |- | ||
| 15 | | 15 | ||
| p-minor sixth, pm6 | | p-minor sixth, pm6 | ||
| 818. | | 818.182 | ||
| | | [[8/5]] | ||
|- | |- | ||
| 16 | | 16 | ||
| p-Major sixth, pM6 | | p-Major sixth, pM6 | ||
| 872. | | 872.727 | ||
| | | [[18/11]], [[28/17]], [[5/3]] | ||
|- | |- | ||
| 17 | | 17 | ||
| s-Major sixth, sM6 | | s-Major sixth, sM6 | ||
| 927. | | 927.273 | ||
| [[17/10]], [[12/7]] | |||
|- | |- | ||
| 18 | | 18 | ||
| (s/p) minor seventh, m7 | | (s/p) minor seventh, m7 | ||
| 981. | | 981.818 | ||
| [[7/4]], [[30/17]], [[16/9]] | |||
|- | |- | ||
| 19 | | 19 | ||
| p-Major seventh, pM7 | | p-Major seventh, pM7 | ||
| 1036. | | 1036.364 | ||
| | | [[9/5]], [[11/6]], [[20/11]] | ||
|- | |- | ||
| 20 | | 20 | ||
| p-Augmented Seventh | | p-Augmented Seventh | ||
| 1090. | | 1090.909 | ||
| | | [[28/15]], [[15/8]], [[32/17]], [[17/9]] | ||
|- | |- | ||
| 21 | | 21 | ||
| s-Major Seventh, sM7 | | s-Major Seventh, sM7 | ||
| 1145.455 | | 1145.455 | ||
| | | [[31/16]], [[64/33]], [[33/17]], [[35/18]] | ||
|- | |- | ||
| 22 | | 22 | ||
| Octave, 8 | | Octave, 8 | ||
| 1200 | | 1200.000 | ||
| | | [[2/1]] | ||
|} | |} | ||
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible. | <nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible. | ||
===Ups and Downs, Porcupine and Pentatonic Notations=== | === Ups and Downs, Porcupine and Pentatonic Notations === | ||
22edo intervals can also be notated using [[Ups_and_Downs_Notation|ups and downs]]. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D. The natural notes represent the conventional chain of 5ths FCGDAEB. | 22edo intervals can also be notated using [[Ups_and_Downs_Notation|ups and downs]]. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D. The natural notes represent the conventional chain of 5ths FCGDAEB. | ||
| Line 173: | Line 175: | ||
Yet another notation is pentatonic. The degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. This is the only way to use a chain-of-fifths notation without additional accidentals. The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D. The natural notes represent a chain of 5ths FCGDA. | Yet another notation is pentatonic. The degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. This is the only way to use a chain-of-fifths notation without additional accidentals. The keyboard runs D * * * * F * * * G * * * A * * * * C * * * D. The natural notes represent a chain of 5ths FCGDA. | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
|- | |- | ||
! | [[ | ! | [[Degree]] | ||
! | [[cent|Cents]] | ! | [[cent|Cents]] | ||
! colspan="3" | [[Ups and Downs Notation | ! colspan="3" | [[Ups and Downs Notation]] | ||
! colspan="3" | Porcupine | ! colspan="3" | Porcupine | ||
! colspan="3" | Pentatonic | ! colspan="3" | Pentatonic | ||
|- | |- | ||
| 0 | |||
| 0.000 | |||
| perfect unison | |||
| P1 | |||
| D | |||
| perfect unison | |||
| P1 | |||
| D | |||
| perfect unison | |||
| P1 | |||
| D | |||
|- | |- | ||
| 1 | |||
| 54.5 | |||
| minor 2nd | |||
| m2 | |||
| Eb | |||
| aug unison | |||
| A1 | |||
| D# | |||
| aug unison | |||
| A1 | |||
| D# | |||
|- | |- | ||
| 2 | |||
| 109 | |||
| upminor 2nd | |||
| ^m2 | |||
| ^Eb | |||
| dim 2nd | |||
| d2 | |||
| Eb | |||
| double-aug unison, <br>double-dim sub3rd | |||
| AA1, <br>dds3 | |||
double-dim sub3rd | | Dx, <br>Fb<span style="vertical-align: super;">3 </span> | ||
dds3 | |||
Fb<span style="vertical-align: super;">3 </span> | |||
|- | |- | ||
| 3 | |||
| 164 | |||
| downmajor 2nd | |||
| vM2 | |||
| vE | |||
| perfect 2nd | |||
| P2 | |||
| E | |||
| dim sub3rd | |||
| ds3 | |||
| Fbb | |||
|- | |- | ||
| 4 | |||
| 218 | |||
| major 2nd | |||
| M2 | |||
| E | |||
| aug 2nd | |||
| A2 | |||
| E# | |||
| minor sub3rd | |||
| ms3 | |||
| Fb | |||
|- | |- | ||
| 5 | |||
| 273 | |||
| minor 3rd | |||
| m3 | |||
| F | |||
| dim 3rd | |||
| d3 | |||
| Fb | |||
| major sub3rd | |||
| Ms3 | |||
| F | |||
|- | |- | ||
| 6 | |||
| 327 | |||
| upminor 3rd | |||
| ^m3 | |||
| ^F | |||
| minor 3rd | |||
| m3 | |||
| F | |||
| aug sub3rd | |||
| As3 | |||
| F# | |||
|- | |- | ||
| 7 | |||
| 382 | |||
| downmajor 3rd | |||
| vM3 | |||
| vF# | |||
| major 3rd | |||
| M3 | |||
| F# | |||
| double-aug sub3rd, <br>double-dim 4thoid | |||
| AAs3, <br>dd4d | |||
double-dim 4thoid | | Fx, <br>Gbb | ||
dd4d | |||
Gbb | |||
|- | |- | ||
| 8 | |||
| 436 | |||
| major 3rd | |||
| M3 | |||
| F# | |||
| aug 3rd, dim 4th | |||
| A3, d4 | |||
| Fx, Gb | |||
| dim 4thoid | |||
| d4d | |||
| Gb | |||
|- | |- | ||
| 9 | |||
| 491 | |||
| perfect fourth | |||
| P4 | |||
| G | |||
| minor 4th | |||
| m4 | |||
| G | |||
| perfect 4thoid | |||
| P4d | |||
| G | |||
|- | |- | ||
| 10 | |||
| 545.5 | |||
| up-4th, dim 5th | |||
| ^4, d5 | |||
| ^G, Ab | |||
| major 4th | |||
| M4 | |||
| G# | |||
| aug 4thoid | |||
| A4d | |||
| G# | |||
|- | |- | ||
| 11 | |||
| 600 | |||
| downaug 4th, <br>updim 5th | |||
| vA4, ^d5 | |||
updim 5th | | vG#, <br>^Ab | ||
| aug 4th, <br>dim 5th | |||
| A4, d5 | |||
| Gx, <br>Abb | |||
^Ab | | double-aug 4thoid, <br>double-dim 5thoid | ||
| AA4d, <br>dd5d | |||
| Gx, <br>Abb | |||
dim 5th | |||
Abb | |||
double-dim 5thoid | |||
dd5d | |||
Abb | |||
|- | |- | ||
| 12 | |||
| 654.5 | |||
| aug 4th, down-5th | |||
| A4, v5 | |||
| G#, vA | |||
| minor 5th | |||
| m5 | |||
| Ab | |||
| dim 5thoid | |||
| d5d | |||
| Ab | |||
|- | |- | ||
| 13 | |||
| 709 | |||
| perfect 5th | |||
| P5 | |||
| A | |||
| major 5th | |||
| M5 | |||
| A | |||
| perfect 5thoid | |||
| P5d | |||
| A | |||
|- | |- | ||
| 14 | |||
| 764 | |||
| minor 6th | |||
| m6 | |||
| Bb | |||
| aug 5th, dim 6th | |||
| A5, d6 | |||
| A#, Bbb | |||
| aug 5thoid | |||
| A5d | |||
| A# | |||
|- | |- | ||
| 15 | |||
| 818 | |||
| upminor 6th | |||
| ^m6 | |||
| ^Bb | |||
| minor 6th | |||
| m6 | |||
| Bb | |||
| double-aug 5thoid, <br>double-dim sub7th | |||
| AA5d, <br>dds7 | |||
double-dim sub7th | | Ax, <br>Cb<span style="vertical-align: super;">3</span> | ||
dds7 | |||
Cb<span style="vertical-align: super;">3</span> | |||
|- | |- | ||
| 16 | |||
| 873 | |||
| downmajor 6th | |||
| vM6 | |||
| vB | |||
| major 6th | |||
| M6 | |||
| B | |||
| dim sub7th | |||
| ds7 | |||
| Cbb | |||
|- | |- | ||
| 17 | |||
| 927 | |||
| major 6th | |||
| M6 | |||
| B | |||
| aug 6th | |||
| A6 | |||
| B# | |||
| minor sub7th | |||
| ms7 | |||
| Cb | |||
|- | |- | ||
| 18 | |||
| 982 | |||
| minor 7th | |||
| m7 | |||
| C | |||
| dim 7th | |||
| d7 | |||
| Cb | |||
| major sub7th | |||
| Ms7 | |||
| C | |||
|- | |- | ||
| 19 | |||
| 1036 | |||
| upminor 7th | |||
| ^m7 | |||
| ^C | |||
| perfect 7th | |||
| P7 | |||
| C | |||
| aug sub7th | |||
| As7 | |||
| C# | |||
|- | |- | ||
| 20 | |||
| 1091 | |||
| downmajor 7th | |||
| vM7 | |||
| vC# | |||
| aug 7th | |||
| A7 | |||
| C# | |||
| double-aug sub7th, <br>double-dim octave | |||
| AAs7, <br>dd8 | |||
double-dim octave | | Cx, <br>Dbb | ||
dd8 | |||
Dbb | |||
|- | |- | ||
| 21 | |||
| 1145.5 | |||
| major 7th | |||
| M7 | |||
| C# | |||
| dim 8ve | |||
| d8 | |||
| Db | |||
| dim octave | |||
| d8 | |||
| Db | |||
|- | |- | ||
| 22 | |||
| 1200 | |||
| perfect octave | |||
| P8 | |||
| D | |||
| perfect octave | |||
| P8 | |||
| D | |||
| perfect octave | |||
| P8 | |||
| D | |||
|} | |} | ||
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: | Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
|- | |- | ||
! | quality | ! | quality | ||
| Line 505: | Line 469: | ||
! | examples | ! | examples | ||
|- | |- | ||
| minor | |||
| zo | |||
| {a, b, 0, 1} | |||
| 7/6, 7/4 | |||
|- | |- | ||
| " | |||
| fourthward wa | |||
| {a, b}, b < -1 | |||
| 32/27, 16/9 | |||
|- | |- | ||
| upminor | |||
| gu | |||
| {a, b, -1} | |||
| 6/5, 9/5 | |||
|- | |- | ||
| downmajor | |||
| yo | |||
| {a, b, 1} | |||
| 5/4, 5/3 | |||
|- | |- | ||
| major | |||
| fifthward wa | |||
| {a, b}, b > 1 | |||
| 9/8, 27/16 | |||
|- | |- | ||
| " | |||
| ru | |||
| {a, b, 0, -1} | |||
| 9/7, 12/7 | |||
|} | |} | ||
===Decatonic Notation=== | === Decatonic Notation === | ||
The decatonic notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales. | The decatonic notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales. | ||
| Line 549: | Line 513: | ||
==Chord Names== | ==Chord Names== | ||
See also [[ | See also [[22 EDO Chords]], [[Chords of orwell]]. | ||
All 22edo chords can be named using ups and downs notation. Here are the zo, gu, yo and ru triads: | All 22edo chords can be named using ups and downs notation. Here are the zo, gu, yo and ru triads: | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
|- | |- | ||
! | [[Kite's color notation|color of the 3rd]] | ! | [[Kite's color notation|color of the 3rd]] | ||
| Line 562: | Line 526: | ||
! | spoken name | ! | spoken name | ||
|- | |- | ||
| zo | |||
| 6:7:9 | |||
| 0-5-13 | |||
| C Eb G | |||
| Cm | |||
| C minor | |||
|- | |- | ||
| gu | |||
| 10:12:15 | |||
| 0-6-13 | |||
| C ^Eb G | |||
| C^m | |||
| C upminor | |||
|- | |- | ||
| yo | |||
| 4:5:6 | |||
| 0-7-13 | |||
| C vE G | |||
| Cv | |||
| C downmajor or C down | |||
|- | |- | ||
| ru | |||
| 14:18:21 | |||
| 0-8-13 | |||
| C E G | |||
| C | |||
| C major or C | |||
|} | |} | ||
Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). | Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). | ||
| Line 644: | Line 609: | ||
0-6-12-15 = C ^Eb vG vvA = Cm6(^3,v5,vv6), or C ^Eb ^^Gb Bbb = Cdim7(^3,^^5) | 0-6-12-15 = C ^Eb vG vvA = Cm6(^3,v5,vv6), or C ^Eb ^^Gb Bbb = Cdim7(^3,^^5) | ||
For a more complete list, see [[Ups and Downs Notation# | For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | ||
== Scales == | == Scales == | ||