10edo: Difference between revisions
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 2 × 5 | | Prime factorization = 2 × 5 | ||
| Step size = | | Step size = 120.000¢ | ||
| Fifth = 6\10 | | Fifth = 6\10 (720¢) (→[[5edo|3\5]]) | ||
| Major 2nd = 2\10 | | Major 2nd = 2\10 (240¢) (→1\5) | ||
| Semitones = 2 : 0 | | Semitones = 2:0 (240¢ : 0¢) | ||
| Consistency = 7 | | Consistency = 7 | ||
}} | }} | ||
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' (''' | '''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s. | ||
== Theory == | == Theory == | ||
{{Harmonics in equal|10}} | {{Harmonics in equal|10}} | ||
10edo can be thought of as two circles of [[ | 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 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 [[MOS scales|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. | ||
== Intervals == | == Intervals == | ||
| Line 154: | Line 153: | ||
genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1... | genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1... | ||
== | == JI approximation == | ||
=== Selected just intervals by error === | === Selected just intervals by error === | ||
==== Selected 13-limit intervals ==== | ==== Selected 13-limit intervals ==== | ||
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
== | == Regular temperament properties == | ||
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. | The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
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* 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 ETs better in those subgroups are 17, 19, 36 and 31, 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 ETs better in those subgroups are 17, 19, 36 and 31, respectively. | ||
== | === Rank-2 temperaments === | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|- | |- | ||
| Line 233: | Line 232: | ||
| [[Blackwood]]/[[blacksmith]] | | [[Blackwood]]/[[blacksmith]] | ||
|} | |} | ||
== Scales == | |||
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]] | [[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]] | ||
[[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]] | ||
== Pathological | === Pathological scales === | ||
2 1 1 1 2 1 1 1 [[2L 6s]] MOS | * 2 1 1 1 2 1 1 1 [[2L 6s]] MOS | ||
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS | |||
3 1 1 1 1 1 1 1 [[1L 7s]] MOS | * 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS | ||
2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS | |||
== Commas == | == Commas == | ||
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. | |||
{| class="commatable wikitable center-1 center-2 right-4 center-5" | {| class="commatable wikitable center-1 center-2 right-4 center-5" | ||
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== Instruments == | == 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 | 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|20edo]] which approximates the 11th harmonic with relative accuracy, among other features. | 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|20edo]] which approximates the 11th harmonic with relative accuracy, among other features. | ||
Revision as of 12:55, 26 March 2022
| ← 9edo | 10edo | 11edo → |
10 equal divisions of the octave (10edo), or 10-tone equal temperament (10tet, 10et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into ten equal steps of exactly 120 cents.
Theory
| 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) | |
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 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 moment of symmetry scale of the form 1 2 1 2 1 2 1 (3L 4s - mosh). While not an integral or gap edo, it is a 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.
Intervals
| Degree | Cents | Approximate Ratios[1] | Additional Ratios of 3, 5 and 9[2] |
Interval Names | Ups and Downs Notation | ||
|---|---|---|---|---|---|---|---|
| 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 | 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 |
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...
JI approximation
Selected just intervals by error
Selected 13-limit intervals
Regular temperament properties
The following table shows TE temperament measures (RMS normalized) of 10ET.
| 3-limit | 2.3.7 | 2.3.7.13 | 2.3.7.13.17 | 5-limit | 7-limit | 2.3.5.7.13 | 2.3.5.7.13.17 | ||
|---|---|---|---|---|---|---|---|---|---|
| Octave stretch (¢) | -5.69 | -2.77 | -2.05 | -2.37 | -0.09 | +0.72 | +0.60 | -0.11 | |
| Error | absolute (¢) | 5.66 | 6.23 | 5.54 | 5.00 | 9.27 | 8.15 | 7.30 | 6.85 |
| relative (%) | 4.74 | 5.20 | 4.62 | 4.17 | 7.73 | 6.79 | 6.08 | 5.70 | |
- 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs 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 ETs better in those subgroups are 17, 19, 36 and 31, respectively.
Rank-2 temperaments
| Periods per octave |
Generator | Temperament(s) |
|---|---|---|
| 1 | 1\10 | Messed-up negri (or miracle) |
| 1 | 3\10 | Dicot/beatles/neutral thirds scale |
| 2 | 1\10 | Messed-up pajara |
| 2 | 2\10 | Decimal / messed-up lemba |
| 5 | 1\10 | Blackwood/blacksmith |
Scales
Pathological scales
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 | Limma, Pythagorean diatonic semitone |
| 5 | 25/24 | [-3 -1 2⟩ | 70.67 | Yoyo | Classic chromatic semitone, dicot comma |
| 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 | Slendro diesis |
| 7 | 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyo | Tritonic diesis, jubilisma |
| 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 | Septimal kleisma, marvel comma |
| 7 | 16875/16807 | [0 3 4 -5⟩ | 6.99 | Quinru-aquadyo | Mirkwai |
| 7 | (24 digits) | [11 -10 -10 10⟩ | 5.57 | Saquinbizogu | Linus |
| 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 |
| 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
Images
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
- ZIA Space "Who Loves You, Me?", "Champagne", and "Avatar" by Elaine Walker
- Fiat Circadia by Stephen Weigel
- Ten Fingers[dead link] play[dead link] by Bill Sethares (synth guitar)
- Circle of Thirds[dead link] play[dead link] by Bill Sethares (synth ens.)
- 10_fantasy[dead link] play[dead link] by Aaron Krister Johnson (synth monody)
- Prelude in 10ET[dead link] by Aaron Andrew Hunt
- Fugue in 10ET | SoundCloud by Aaron Andrew Hunt, 2015
- Future play and Sol play by ZIA (synths and voice in 10)
- Prelude by Rick McGowan (Rhino synthesizer)
- Ideas on the Waterfall of Expression[dead link] by City of the Asleep (synth)
- For two violas and gongs by Chris Vaisvil Details
- Blues 10 by Carlo Serafini (blog entry)
- Waltz 10 by Carlo Serafini (blog entry)
- Smooth 10 by Carlo Serafini (blog entry)
- 10 PRS by Carlo Serafini (blog entry)
- 10preview.ogg A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
- decexperiment.ogg 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
- 10 Earwigs Invasive by Chris Vaisvil Details
- Comets Over Flatland 9[dead link] by Randy Winchester
- The Dramatic Squirrel Overture by Chris Vaisvil Details
- Shimmerwing by Andrew Heathwaite and Chris Vaisvil
- Shall I Refuse My Dinner by Steve Martin on SoundCloud
- 10-tone demo by Clem Fortuna[dead link]
- Hey, ule! by Dmitry Bazhenov (second part in 10-edo)
- Sad Mike (10EDO) by City of the Asleep
- Bit Crystals by Userminusone
- Струнка (10-РДО) | Strunka (10-EDO) - YouTube by Dmitry Bazhenov
- Microtonal moment musicale 2 by Evan Bennet
- Three weeping tyrants by Stephen Weigel