10edo: Difference between revisions
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= Theory = | == Theory == | ||
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup. | 10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup. | ||
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=Intervals= | == Intervals == | ||
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[[:File:10ed2-001.svg|10ed2-001.svg]] | [[:File:10ed2-001.svg|10ed2-001.svg]] | ||
=Linear temperaments (with images for MOS horagrams)= | == Linear temperaments (with images for MOS horagrams) == | ||
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[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]] | [[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]] | ||
=Commas= | == Commas == | ||
10 EDO tempers out the following commas. (Note: This assumes the val < 10 16 23 28 35 37 |.) | 10 EDO tempers out the following commas. (Note: This assumes the val < 10 16 23 28 35 37 |.) | ||
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=Images= | == Images == | ||
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]] | [[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]] | ||
=Instruments= | == Instruments == | ||
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple. | 10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple. | ||
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[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]] | [[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]] | ||
=Music= | == Music == | ||
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar" | [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar" | ||