10edo
← 9edo | 10edo | 11edo → |
10 equal divisions of the octave (abbreviated 10edo or 10ed2), also called 10-tone equal temperament (10tet) or 10 equal temperament (10et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 10 equal parts of exactly 120 ¢ each. Each step represents a frequency ratio of 21/10, or the 10th root of 2.
Theory
10edo can be thought of as two circles of 5edo separated by 120 cents (or 5 circles of 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 and its inversion 16/13; and the familiar 600-cent tritone that appears in every even-numbered edo.
Taking the the 360 cent large neutral third as a generator produces a heptatonic moment of symmetry scale of the form 1 2 1 2 1 2 1 (3L 4s, or "mosh"), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale.
While not an integral or gap edo, 10edo is a zeta peak edo. 10edo is also the smallest edo that maintains 25% or lower relative error on all of the first eight harmonics of the harmonic series.
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, 43904/43875, 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.
Thanks to its sevenths, 10edo is an ideal tuning for its size for metallic harmony.
Prime harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | +18.0 | -26.3 | -8.8 | +48.7 | -0.5 | +15.0 | -57.5 | -28.3 | +50.4 | +55.0 |
Relative (%) | +0.0 | +15.0 | -21.9 | -7.4 | +40.6 | -0.4 | +12.5 | -47.9 | -23.6 | +42.0 | +45.8 | |
Steps (reduced) |
10 (0) |
16 (6) |
23 (3) |
28 (8) |
35 (5) |
37 (7) |
41 (1) |
42 (2) |
45 (5) |
49 (9) |
50 (0) |
Intervals
Degree | Cents | Approximate ratios[1] | Additional ratios of 3, 5 and 9[2] |
Interval names | Ups and downs notation | Audio | ||
---|---|---|---|---|---|---|---|---|
0 | 0 | 1/1 | 256/243, 50/49, 25/24 | unison | unison, min 2nd | P1, m2 | D, Eb | |
1 | 120 | 16/15, 15/14, 14/13 | 10/9, 13/12, 81/80 | neutral second | mid 2nd | ~2 | ^D, vE | |
2 | 240 | 8/7, 15/13, 144/125, 224/195 | 9/8, 7/6 | hemifourth, major second, minor third | maj 2nd, min 3rd | M2, m3 | E, F | |
3 | 360 | 16/13 | 5/4 | neutral third | mid 3rd | ~3 | ^F, vG | |
4 | 480 | 64/49, 169/128 | 4/3, 9/7, 13/10 | perfect fourth | maj 3rd, perf 4th | M3, P4 | F#, G | |
5 | 600 | 91/64, 128/91, 169/120, 240/169 | 7/5, 10/7, 13/9, 18/13 | hemioctave, tritone | up 4th, down 5th | ^4, v5 | ^G, vA | |
6 | 720 | 49/32, 256/169 | 3/2, 14/9, 20/13 | perfect fifth | perf 5th, min 6th | P5, m6 | A, Bb | |
7 | 840 | 13/8 | 8/5 | neutral sixth | mid 6th | ~6 | ^A, vB | |
8 | 960 | 7/4, 26/15, 125/72, 195/112 | 16/9, 12/7 | hemitwelfth, major sixth, minor seventh | maj 6th, min 7th | M6, m7 | B, C | |
9 | 1080 | 15/8, 28/15, 13/7 | 9/5, 24/13, 160/81 | neutral seventh | mid 7th | ~7 | ^C, vD | |
10 | 1200 | 2/1 | 243/128, 49/25, 48/25 | octave | maj 7th, octave | M7, P8 | C#, D |
Notation
This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.
Pentatonic 5th-generated: D * E * G * A * C * D (generator = 3/2 = 6\10 = perfect 5thoid)
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)
pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)
pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)
Heptatonic 3rd-generated: D E * F G * A B * C D (generator = 3\10 = perfect 3rd)
D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D
P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8
genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")
genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
3L 4s (mosh) notation
The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭.
Degree | Cents | Note | Name | Associated ratio |
---|---|---|---|---|
0 | 0 | C | Perf 1sn | 1/1 |
1 | 120 | Db | Min 2nd | 12/11 |
2 | 240 | D, Eb | Maj 2nd, dim 3rd | 9/8, 32/27 |
3 | 360 | E | Perf 3rd | 11/9, 27/22 |
4 | 480 | Fb | Min 4th | 4/3 |
5 | 600 | F, Gb | Maj 4th, min 5th | 11/8, 16/11 |
6 | 720 | G | Maj 5th | 3/2 |
7 | 840 | A | Perf 6th | 18/11, 44/27 |
8 | 960 | A#, Bb | Aug 6th, min 7th | 16/9, 27/16 |
9 | 1080 | B | Maj 7th | 11/6 |
10 | 1200 | C | Perf 8ve | 2/1 |
Approximation to JI
Selected just intervals by error
Selected 13-limit intervals
Regular temperament properties
Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
---|---|---|---|---|---|
Absolute (¢) | Relative (%) | ||||
2.3.5 | 25/24, 256/243 | [⟨10 16 23]] | -0.089 | 9.27 | 7.73 |
2.3.5.7 | 25/24, 28/27, 49/48 | [⟨10 16 23 28]] | +0.718 | 8.15 | 6.79 |
2.3.5.7.13 | 25/24, 28/27, 40/39, 49/48 | [⟨10 16 23 28 37]] | +0.603 | 7.30 | 6.08 |
- 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively.
- 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.
Uniform maps
Min. size | Max. size | Wart notation | Map |
---|---|---|---|
9.5000 | 9.5935 | 10bccddeeeffff | ⟨10 15 22 27 33 35] |
9.5935 | 9.6837 | 10bccddeeeff | ⟨10 15 22 27 33 36] |
9.6837 | 9.6902 | 10bccddeff | ⟨10 15 22 27 34 36] |
9.6902 | 9.7794 | 10bddeff | ⟨10 15 23 27 34 36] |
9.7794 | 9.7957 | 10ddeff | ⟨10 16 23 27 34 36] |
9.7957 | 9.8637 | 10eff | ⟨10 16 23 28 34 36] |
9.8637 | 9.9727 | 10e | ⟨10 16 23 28 34 37] |
9.9727 | 10.1209 | 10 | ⟨10 16 23 28 35 37] |
10.1209 | 10.1339 | 10c | ⟨10 16 24 28 35 37] |
10.1339 | 10.1519 | 10cf | ⟨10 16 24 28 35 38] |
10.1519 | 10.2618 | 10cdf | ⟨10 16 24 29 35 38] |
10.2618 | 10.4042 | 10cdeef | ⟨10 16 24 29 36 38] |
10.4042 | 10.4103 | 10cdeefff | ⟨10 16 24 29 36 39] |
10.4103 | 10.5000 | 10bbcdeefff | ⟨10 17 24 29 36 39] |
Commas
10et tempers out the following commas. This assumes the val ⟨10 16 23 28 35 37].
Prime limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
---|---|---|---|---|---|
3 | 256/243 | [8 -5⟩ | 90.22 | Sawa | Blackwood comma, Pythagorean limma |
5 | 25/24 | [-3 -1 2⟩ | 70.67 | Yoyo | Dicot comma, classic chroma |
5 | 16875/16384 | [-14 3 4⟩ | 51.12 | Laquadyo | Negri comma, double augmentation diesis |
5 | (16 digits) | [-25 7 6⟩ | 31.57 | Lala-tribiyo | Ampersand, Ampersand's comma |
5 | 2048/2025 | [11 -4 -2⟩ | 19.55 | Sagugu | Diaschisma |
7 | 525/512 | [-9 1 2 1⟩ | 43.41 | Lazoyoyo | Avicennma, Avicenna's enharmonic diesis |
7 | 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Semaphoresma, slendro diesis |
7 | 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyo | Jubilisma, tritonic diesis |
7 | 686/675 | [1 -3 -2 3⟩ | 27.99 | Trizo-agugu | Senga |
7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal comma, Archytas' comma, Leipziger Komma |
7 | (18 digits) | [-10 7 8 -7⟩ | 22.41 | Lasepru-aquadbiyo | Blackjackisma |
7 | 1029/1024 | [-10 1 0 3⟩ | 8.43 | Latrizo | Gamelisma |
7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyo | Marvel comma, septimal kleisma |
7 | 16875/16807 | [0 3 4 -5⟩ | 6.99 | Quinru-aquadyo | Mirkwai comma |
7 | (24 digits) | [11 -10 -10 10⟩ | 5.57 | Saquinbizogu | Linus comma |
7 | 2401/2400 | [-5 -1 -2 4⟩ | 0.72 | Bizozogu | Breedsma |
11 | 243/242 | [-1 5 0 0 -2⟩ | 7.14 | Lulu | Rastma |
11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Luzozogu | Werckisma |
11 | 540/539 | [2 3 1 -2 -1⟩ | 3.21 | Lururuyo | Swetisma |
11 | 3025/3024 | [-4 -3 2 -1 2⟩ | 0.57 | Loloruyoyo | Lehmerisma |
13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap comma, biome comma |
13 | 196/195 | [2 -1 -1 2 0 -1⟩ | 8.86 | Thuzozogu | Mynucuma |
13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.56 | Bithogu | Island comma, parizeksma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Rank-2 temperaments
Periods per 8ve |
Generator | Temperament(s) |
---|---|---|
1 | 1\10 | Negri, miracle (out-of-tune) |
1 | 3\10 | Dicot / beatles / neutral |
2 | 1\10 | Pajara (out-of-tune) |
2 | 2\10 | Decimal, lemba (out-of-tune) |
5 | 1\10 | Blackwood / blacksmith |
Scales
MOS scales
- Decimal/Lemba[6] 4L 2s (period = 5\10, gen = 2\10): 2 2 1 2 2 1
- Dicot[7] 3L 4s (gen = 3\10): 1 2 1 2 1 2 1
- Negri[9] 1L 8s (gen = 1\10): 1 1 1 1 2 1 1 1 1
Other scales
- Pinetone major pentatonic (subset of Dicot[7]): 2 1 3 1 3
- Pinetone minor pentatonic (subset of Dicot[7]): 3 1 2 3 1
- Marvel augmented hexatonic (subset of Dicot[7]): 2 1 3 1 2 1
- Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
- Decimal/Lemba[6] 4M: 2 1 2 2 2 1
- Dicot[7] 4M: 2 1 1 2 2 1 1
Horagrams
Diagrams
Instruments
10edo 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 12edo, 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.
Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from 20edo which approximates the 11th harmonic with relative accuracy, among other features.
A Decaphonic (10edo) Classical Guitar |
Music
- See also: Category:10edo tracks