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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 {{nowrap|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, 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.
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 {{nowrap|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, 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.


Rather surprisingly, 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.
Rather surprisingly, 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.


=== Prime harmonics ===
=== Prime harmonics ===
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! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref>based on treating 10edo as a 2.7.13.15 subgroup temperament</ref>
! Approximate ratios<ref>based on treating 10edo as a 2.15.7.13-subgroup temperament</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>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]]
! colspan="2" | [[3L 4s]] notation
! Audio
!Audio
|-
|-
| 0
| 0
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| P1, m2
| P1, m2
| D, Eb
| D, Eb
| unison
| [[File:0-0 unison.mp3|frameless]]
| C
|[[File:0-0 unison.mp3|frameless]]
|-
|-
| 1
| 1
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| ~2
| ~2
| ^D, vE
| ^D, vE
| minor second
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
|-
| 2
| 2
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| M2, m3
| M2, m3
| E, F
| E, F
| major second, diminished third
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
|-
| 3
| 3
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| ~3
| ~3
| ^F, vG
| ^F, vG
| perfect/neutral third
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
|-
| 4
| 4
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| M3, P4
| M3, P4
| F#, G
| F#, G
| minor fourth
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
|-
| 5
| 5
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| ^4, v5
| ^4, v5
| ^G, vA
| ^G, vA
| major fourth, minor fifth
| [[File:0-600 (12-EDO).mp3|frameless]]
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
|-
| 6
| 6
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| P5, m6
| P5, m6
| A, Bb
| A, Bb
| major fifth
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
|-
| 7
| 7
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| ~6
| ~6
| ^A, vB
| ^A, vB
| perfect/neutral sixth
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
|-
| 8
| 8
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| M6, m7
| M6, m7
| B, C
| B, C
| augmented sixth, minor seventh
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
|-
| 9
| 9
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| ~7
| ~7
| ^C, vD
| ^C, vD
| major seventh
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
|-
| 10
| 10
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| M7, P8
| M7, P8
| C#, D
| C#, D
| octave
| [[File:0-1200 octave.mp3|frameless]]
| C
|[[File:0-1200 octave.mp3|frameless]]
|}
|}


<references />
<references />


== Notation ==
This is 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.


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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...
=== 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 ♭.
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! 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 ==
== Approximation to JI ==
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