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{{interwiki
{{interwiki
| de = 10edo
| de = 10-EDO
| en = 10edo  
| en = 10edo  
| es =  
| es =  
| ja = 10平均律
| ja = 10平均律
}}
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]].
{{Infobox ET}}
{{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]].


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 [[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.
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}}


== 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 10-EDO 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
|-
|-
| 0
| 0
Line 29: Line 47:
| P1, m2
| P1, m2
| D, Eb
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
|-
| 1
| 1
| 120
| 120
| 16/15, 15/14, 13/14
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| neutral second
| mid 2nd
| mid 2nd
| ~2
| ~2
| ^D, vE
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
|-
| 2
| 2
| 240
| 240
| 8/7, 15/13, 144/125
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| 9/8, 7/6
| second/third
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| maj 2nd, min 3rd
| M2, m3
| M2, m3
| E, F
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
|-
| 3
| 3
| 360
| 360
| 16/13
| [[16/13]]
| 5/4
| 5/4
| large neutral third
| neutral third
| mid 3rd
| mid 3rd
| ~3
| ~3
| ^F, vG
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
|-
| 4
| 4
Line 61: Line 83:
| 64/49, 169/128
| 64/49, 169/128
| 4/3, 9/7, 13/10
| 4/3, 9/7, 13/10
| smaller fourth
| perfect fourth
| maj 3rd, perf 4th
| maj 3rd, perf 4th
| M3, P4
| M3, P4
| F#, G
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
|-
| 5
| 5
Line 70: 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
| tritone
| 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]]
|-
|-
| 6
| 6
Line 79: Line 103:
| 49/32, 256/169
| 49/32, 256/169
| 3/2, 14/9, 20/13
| 3/2, 14/9, 20/13
| bigger fifth
| perfect fifth
| perf 5th, min 6th
| perf 5th, min 6th
| P5, m6
| P5, m6
| A, Bb
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
|-
| 7
| 7
| 840
| 840
| 13/8
| [[13/8]]
| 8/5
| 8/5
| neutral sixth
| neutral sixth
Line 92: Line 117:
| ~6
| ~6
| ^A, vB
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
|-
| 8
| 8
| 960
| 960
| 7/4, 26/15, 125/72
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| 16/9, 12/7
| sixth/seventh
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| maj 6th, min 7th
| M6, m7
| M6, m7
| B, C
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
|-
| 9
| 9
Line 106: Line 133:
| 15/8, 28/15, 13/7
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| 9/5, 24/13, 160/81
| small major 7th
| neutral seventh
| mid 7th
| mid 7th
| ~7
| ~7
| ^C, vD
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
|-
| 10
| 10
Line 119: Line 147:
| M7, P8
| M7, P8
| C#, D
| C#, D
| [[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 a heptatonic notation generated by 5ths (5th meaning 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 131: 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 circles 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, see [[5edo#Alternative%20notations|5edo notation]])


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)
[[Enharmonic unison]]: vvs3


'''<u>Heptatonic 3rd-generated</u>: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)
==== 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 145: 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...


== Just approximation ==
[[Enharmonic unison]]: d2
=== Selected just intervals by error ===
 
{| class="wikitable center-all"
See below: 3L&nbsp;4s mosh notation
! colspan="2" |
 
! prime 2
=== 3L&nbsp;4s (mosh) notation ===
! prime 3
See above: Heptatonic 3rd-generated notation.
! prime 5
 
! prime 7
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 ♭.
! prime 11
 
! prime 13
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! prime 17
! #
! 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
|-
|-
! rowspan="2" |Error
| 9
! absolute (¢)
| 1080
| 0.0
| B
| +18.0
| Maj 7th
| -26.3
| 11/6
| -8.8
| +48.7
| -0.5
| +12.5
|-
|-
! [[Relative error|relative]] (%)
| 10
| 0.0
| 1200
| +15.0
| C
| -21.9
| Perf 8ve
| -7.3
| 2/1
| +40.6
| -0.4
| +15.0
|}
|}


=== 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 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.]]


=== Temperament measures ===
== Regular temperament properties ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et.
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-all"
|-
! colspan="2" |
! rowspan="2" | [[Subgroup]]
! 3-limit
! rowspan="2" | [[Comma list]]
! 2.3.7
! rowspan="2" | [[Mapping]]
! 2.3.7.13
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 2.3.7.13.17
! colspan="2" | Tuning error
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
|-
! colspan="2" |Octave stretch (¢)
! [[TE error|Absolute]] (¢)
| -5.69
! [[TE simple badness|Relative]] (%)
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
|-
! rowspan="2" |Error
| 2.3.5
![[TE error|absolute]] (¢)
| 25/24, 256/243
| 5.66
| {{Mapping| 10 16 23 }}
| 6.23
| -0.089
| 5.54
| 5.00
| 9.27
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 6.08
| 5.70
|}
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively.  
* 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 ET that does better in these subgroups is 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 equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.


== Linear temperaments (with images for MOS horagrams) ==
=== Uniform maps ===
{{Uniform map|edo=10}}


{| class="wikitable center-1 center-2"
=== Commas ===
|-
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.  
! Periods <br> 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]]
|}
[[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]]
 
== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &lt; 10 16 23 28 35 37 |.)


{| class="wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
!Prime<br>Limit
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! [[Color name]]
! Name(s)
! Name(s)
|-
|-
| 3
| 3
| 256/243
| [[256/243]]
| <nowiki> | 8 -5 </nowiki>&gt;
| {{Monzo| 8 -5 }}
| 90.22
| 90.22
| Sawa
| Sawa
| Limma, Pythagorean minor 2nd
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
| 25/24
| [[25/24]]
| <nowiki> | -3 -1 2 </nowiki>&gt;
| {{Monzo| -3 -1 2 }}
| 70.67
| 70.67
| Yoyo
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
| Dicot comma, classic chroma
|-
|-
| "
| 5
| 16875/16384
| [[16875/16384]]
| <nowiki> | -14 3 4 </nowiki>&gt;
| {{Monzo| -14 3 4 }}
| 51.12
| 51.12
| Laquadyo
| Laquadyo
| Negri comma, double augmentation diesis
| Negri comma, double augmentation diesis
|-
|-
| "
| 5
| 9931568/9752117
| [[34171875/33554432|(16 digits)]]
| <nowiki> | -25 7 6 </nowiki>&gt;
| {{Monzo| -25 7 6 }}
| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| Ampersand's comma
| [[Ampersand comma]]
|-
|-
| "
| 5
| 2048/2025
| [[2048/2025]]
| <nowiki> | 11 -4 -2 </nowiki>&gt;
| {{Monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Sagugu
Line 306: Line 381:
|-
|-
| 7
| 7
| 525/512
| [[525/512]]
| <nowiki> | -9 1 2 1 </nowiki>&gt;
| {{Monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
| Avicennma, Avicenna's enharmonic diesis
|-
|-
| "
| 7
| 49/48
| [[49/48]]
| <nowiki> | -4 -1 0 2 </nowiki>&gt;
| {{Monzo| -4 -1 0 2 }}
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| "
| 7
| 50/49
| [[50/49]]
| <nowiki> | 1 0 2 -2 </nowiki>&gt;
| {{Monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, jubilisma
| Jubilisma, tritonic diesis
|-
|-
| "
| 7
| 686/675
| [[686/675]]
| <nowiki> | 1 -3 -2 3 </nowiki>&gt;
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| 27.99
| Trizo-agugu
| Trizo-agugu
| Senga
| Senga
|-
|-
| "
| 7
| 64/63
| [[64/63]]
| <nowiki> | 6 -2 0 -1 </nowiki>&gt;
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|-
| "
| 7
| 9859966/9733137
| <abbr title="854296875/843308032">(18 digits)</abbr>
| <nowiki> | -10 7 8 -7 </nowiki>&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| 22.41
| Lasepru-aquadbiyo
| Lasepru-aquadbiyo
| Blackjackisma
| [[Blackjackisma]]
|-
|-
| "
| 7
| 1029/1024
| [[1029/1024]]
| <nowiki> | -10 1 0 3 </nowiki>&gt;
| {{Monzo| -10 1 0 3 }}
| 8.43
| 8.43
| Latrizo
| Latrizo
| Gamelisma
| Gamelisma
|-
|-
| "
| 7
| 225/224
| [[225/224]]
| <nowiki> | -5 2 2 -1 </nowiki>&gt;
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal kleisma, marvel comma
| Marvel comma, septimal kleisma
|-
|-
| "
| 7
| 16875/16807
| [[16875/16807]]
| <nowiki> | 0 3 4 -5 </nowiki>&gt;
| {{Monzo| 0 3 4 -5 }}
| 6.99
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| "
| 7
| 6772805/6751042
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| <nowiki> | 11 -10 -10 10 </nowiki>&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| 5.57
| Saquinbizogu
| Saquinbizogu
| Linus
| [[Linus comma]]
|-
|-
| "
| 7
| 2401/2400
| [[2401/2400]]
| <nowiki> | -5 -1 -2 4 </nowiki>&gt;
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| 0.72
| Bizozogu
| Bizozogu
Line 383: Line 458:
|-
|-
| 11
| 11
| 243/242
| [[243/242]]
| <nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| 7.14
| Lulu
| Lulu
| Rastma
| Rastma
|-
|-
| "
| 11
| 385/384
| [[385/384]]
| <nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
|-
| "
| 11
| 441/440
| [[441/440]]
| <nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozogu
| Werckisma
| Werckisma
|-
|-
| "
| 11
| 540/539
| [[540/539]]
| <nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| 3.21
| Lururuyo
| Lururuyo
| Swetisma
| Swetisma
|-
|-
| "
| 11
| 3025/3024
| [[3025/3024]]
| <nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| 0.57
| Loloruyoyo
| Loloruyoyo
Line 418: Line 493:
|-
|-
| 13
| 13
| 91/90
| [[91/90]]
| <nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| "
| 13
| 676/675
| [[196/195]]
| <nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Bithogu
| Bithogu
| Parizeksma
| Island comma, parizeksma
|}
|}
<references group="note"/>


== Images ==
=== 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 ==
[[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.
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.
Line 444: Line 578:
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
|-
| A Decaphonic (10-EDO) Classical Guitar
| A Decaphonic (10edo) Classical Guitar
|}
|}
[[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 ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}


* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
== References ==
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
<references/>
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)


[[Category:Edo]]
[[Category:10-tone scales]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]
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