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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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{| class="wikitable"
{| class="wikitable"
|-
! Degree
! | Degree
! Cents
! | Cents
! Approximate Ratios <ref>based on treating 10-EDO as a 2.7.13.15 subgroup temperament</ref>
! | Approximate Ratios*
! 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*
! Interval Names
 
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
|-
| 0
| 0
| style="text-align:right;" | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| 256/243, 50/49, 25/24
| | unison
| unison
| | unison, min 2nd
| unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
| style="text-align:center;" | D, Eb
|-
|-
| | 1
| 1
| style="text-align:right;" | 120
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| 10/9, 13/12, 81/80
| | small neutral second, large minor second
| small neutral second, large minor second
| | mid 2nd
| mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
| style="text-align:center;" | ^D, vE
|-
|-
| | 2
| 2
| style="text-align:right;" | 240
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| 9/8, 7/6
| | second/third
| second/third
| | maj 2nd, min 3rd
| maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
| style="text-align:center;" | E, F
|-
|-
| | 3
| 3
| style="text-align:right;" | 360
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| style="text-align:center;" | 16/13
| | 5/4
| 5/4
| | large neutral third
| large neutral third
| | mid 3rd
| mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
| style="text-align:center;" | ^F, vG
|-
|-
| | 4
| 4
| style="text-align:right;" | 480
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| 4/3, 9/7, 13/10
| | smaller fourth
| smaller fourth
| | maj 3rd, perf 4th
| maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
| style="text-align:center;" | F#, G
|-
|-
| | 5
| 5
| style="text-align:right;" | 600
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| style="text-align:center;" | 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
| tritone
| | up 4th, down 5th
| up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
| style="text-align:center;" | ^G, vA
|-
|-
| | 6
| 6
| style="text-align:right;" | 720
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| 3/2, 14/9, 20/13
| | bigger fifth
| bigger fifth
| | perf 5th, min 6th
| perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
| style="text-align:center;" | A, Bb
|-
|-
| | 7
| 7
| style="text-align:right;" | 840
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| style="text-align:center;" | 13/8
| | 8/5
| 8/5
| | neutral sixth
| neutral sixth
| | mid 6th
| mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
| style="text-align:center;" | ^A, vB
|-
|-
| | 8
| 8
| style="text-align:right;" | 960
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| 16/9, 12/7
| | sixth/seventh
| sixth/seventh
| | maj 6th, min 7th
| maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
| style="text-align:center;" | B, C
|-
|-
| | 9
| 9
| style="text-align:right;" | 1080
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| 9/5, 24/13, 160/81
| | small major 7th
| small major 7th
| | mid 7th
| mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
| style="text-align:center;" | ^C, vD
|-
|-
| | 10
| 10
| style="text-align:right;" | 1200
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| 243/128, 49/25, 48/25
| | octave
| octave
| | maj 7th, octave
| maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
| style="text-align:center;" | C#, D
|}
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.


*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.
<references/>


This is a heptatonic notation generated by 5ths (5th meaning 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.
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{| class="wikitable"
{| class="wikitable"
|-
|-
! | Periods
! Periods <br> per octave
 
! Generator
per octave
! Temperament(s)
! | Generator
! | Temperament(s)
|-
|-
| | 1
| 1
| | 1\10
| 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
|-
| | 1
| 1
| | 3\10
| 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
|-
| | 2
| 2
| | 1\10
| 1\10
| | Messed-up [[pajara|pajara]]
| Messed-up [[pajara|pajara]]
|-
|-
| | 2
| 2
| | 2\10
| 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
|-
| | 5
| 5
| | 1\10
| 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
| [[Blackwood|Blackwood]]/[[blacksmith|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 11.13.09 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]]
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{| class="wikitable"
{| class="wikitable"
|-
|-
! | [[Ratios|Ratio]]
! [[Ratios|Ratio]]
! | [[Monzo]]
! [[Monzo]]
! | [[Cent|Cents]]
! [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! [[Color notation/Temperament Names|Color Name]]
! | Name 1
! Name 1
! | Name 2
! Name 2
!Name 3
! Name 3
|-
|-
| style="text-align:center;" | 256/243
| style="text-align:center;" | 256/243
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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"


[https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]


[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] <span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''</span></span> by [[Bill_Sethares|Bill Sethares]] (synth guitar)
* [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] <span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''</span></span> by Bill Sethares (synth ens.)
* [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] <span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''</span></span> by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)
* [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)


<span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''</span></span> by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]
* [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] <span style=""><span style="">''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''</span></span> and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] <span style=""><span style="">''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''</span></span> by ZIA (synths and voice in 10)
* [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)


<span style=""><span style="">''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''</span></span> by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)
* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)


<span style=""><span style="">''[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]''</span></span> by Igliashon Jones (synth)
* ''[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)


<span style=""><span style="">''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''</span></span> by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]


''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|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|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|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|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|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: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)
* ''[[: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 [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]


''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]


''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]


''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]
* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]


''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud
* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud


''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]
* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]


[[Category:10-tone]]
[[Category:10-tone]]
Retrieved from "https://en.xen.wiki/w/10edo"