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{{interwiki
{{interwiki
| de = 10edo
| de = 10-EDO
| en = 10edo  
| en = 10edo  
| es =  
| es =  
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}}
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{{Infobox ET}}
{{Infobox ET}}
'''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.
{{ED intro}}


== Theory ==
== Theory ==
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interval is the tritone that appears in every even-numbered edo, including [[12edo]].
Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L 4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale.
It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.
Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic<ref>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]</ref>.
Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15. 
By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]] and [[243/242]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system. 
10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].
Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].
=== Prime harmonics ===
{{Harmonics in equal|10}}
{{Harmonics in equal|10}}
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 [[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]], [[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.


== Intervals ==
== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref>based on treating 10edo as a 2.7.13.15 subgroup temperament</ref>
! Approximate ratios<ref group="note">{{sg|limit=2.15.7.13-subgroup}}</ref>
! Additional Ratios <br> of 3, 5 and 9<ref>adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals</ref>
! Additional ratios<br />of 3, 5, and 9<ref group="note">Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals</ref>
! Interval Names
! Interval names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]<br />([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
!Audio
! Audio
|-
|-
| 0
| 0
Line 31: Line 47:
| P1, m2
| P1, m2
| D, Eb
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
| [[File:0-0 unison.mp3|frameless]]
|-
|-
| 1
| 1
Line 41: Line 57:
| ~2
| ~2
| ^D, vE
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
|-
| 2
| 2
Line 51: Line 67:
| M2, m3
| M2, m3
| E, F
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
|-
| 3
| 3
Line 61: Line 77:
| ~3
| ~3
| ^F, vG
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
|-
| 4
| 4
Line 71: Line 87:
| M3, P4
| M3, P4
| F#, G
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
|-
| 5
| 5
Line 77: Line 93:
| 91/64, 128/91, 169/120, 240/169
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| hemioctave, tritone
| up 4th, down 5th
| up 4th, down 5th
| ^4, v5
| ^4, v5
| ^G, vA
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
|-
| 6
| 6
Line 91: Line 107:
| P5, m6
| P5, m6
| A, Bb
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
|-
| 7
| 7
Line 101: Line 117:
| ~6
| ~6
| ^A, vB
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
|-
| 8
| 8
Line 111: Line 127:
| M6, m7
| M6, m7
| B, C
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
|-
| 9
| 9
Line 121: Line 137:
| ~7
| ~7
| ^C, vD
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
|-
| 10
| 10
Line 131: Line 147:
| M7, P8
| M7, P8
| C#, D
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
| [[File:0-1200 octave.mp3|frameless]]
|}
|}
<references group="note" />


<references />
== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.


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)
'''<u>Pentatonic 5th-generated</u>: 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
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
Line 144: Line 162:
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)
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 circles 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)
pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)


'''<u>Heptatonic 3rd-generated</u>: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)
(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])
 
[[Enharmonic unison]]: vvs3
 
==== 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
D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D
Line 158: Line 181:
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 ==
[[Enharmonic unison]]: d2
 
See below: 3L&nbsp;4s mosh notation
 
=== 3L&nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated 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 ♭.
 
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 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
|}
 
=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].
 
==== Evo and Revo flavors ====
<imagemap>
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.
 
== Approximation to JI ==
=== Selected just intervals by error ===
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
==== Selected 13-limit intervals ====
Line 165: Line 296:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 176: Line 308:
| 2.3.5
| 2.3.5
| 25/24, 256/243
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| {{Mapping| 10 16 23 }}
| -0.09
| -0.089
| 9.27
| 9.27
| 7.73
| 7.73
Line 183: Line 315:
| 2.3.5.7
| 2.3.5.7
| 25/24, 28/27, 49/48
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| {{Mapping| 10 16 23 28 }}
| +0.72
| +0.718
| 8.15
| 8.15
| 6.79
| 6.79
|}
* 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.
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods <br> per octave
! Generator
! Temperament(s)
|-
|-
| 1
| 2.3.5.7.13
| 1\10
| 25/24, 28/27, 40/39, 49/48
| [[Negri]], [[miracle]] (out-of-tune)
| {{Mapping| 10 16 23 28 37 }}
|-
| +0.603
| 1
| 7.30
| 3\10
| 6.08
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}
|}
* 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 [[12edo|12]] and [[19edo|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 [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.


== Scales ==
=== Uniform maps ===
[[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]]
{{Uniform map|edo=10}}
[[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 scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] 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 }}.  
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"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 245: Line 350:
| 90.22
| 90.22
| Sawa
| Sawa
| Limma, Pythagorean diatonic semitone
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
Line 252: Line 357:
| 70.67
| 70.67
| Yoyo
| Yoyo
| Classic chromatic semitone, dicot comma
| Dicot comma, classic chroma
|-
|-
| 5
| 5
Line 266: Line 371:
| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 287: Line 392:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 294: Line 399:
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
Line 329: Line 434:
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal kleisma, marvel comma
| Marvel comma, septimal kleisma
|-
|-
| 7
| 7
Line 336: Line 441:
| 6.99
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 343: Line 448:
| 5.57
| 5.57
| Saquinbizogu
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
| [[Linus comma]]
|-
|-
| 7
| 7
Line 392: Line 497:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 408: Line 513:
| Island comma, parizeksma
| Island comma, parizeksma
|}
|}
<references/>
<references group="note"/>
 
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}
 
== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.
 
If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.
 
== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1
 
=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|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 ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&nbsp;1s, 1L&nbsp;2s, 1L&nbsp;3s, 1L&nbsp;4s, 1L&nbsp;5s, 1L&nbsp;6s, 1L&nbsp;7s, and 1L&nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&nbsp;1s, 1L&nbsp;2s, 3L&nbsp;1s, 3L&nbsp;4s]]


== Diagrams ==
== Diagrams ==
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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 × 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.
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|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 {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|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.
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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]]
=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.


== Music ==
== Music ==
{{See also|:Category:10edo tracks}}
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}


'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
== References ==
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
<references/>
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)
 
'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&index=15 ''Microtonal Moment Musicale 2''] (2018)
 
'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)
 
'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)
 
'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)
 
'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)
 
'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])
 
'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)
 
'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)
 
'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)
 
'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])
 
'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])
 
'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)
 
'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)
 
'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)
 
'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])
 
'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&index=4 ''Three Weeping Tyrants''] (2020)
 
'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)
 
'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)
 
'''[https://jonlervold.com/ Syzygy]'''
 
* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)


[[Category:10edo| ]] <!-- main article -->
[[Category:10-tone scales]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
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