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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|10}}
{{ED intro}}


== Theory ==
== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. 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.


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]]), 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.
Taking the the 360{{c}} 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 scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate 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), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all. 
 
This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15. 
 
By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup. 
 
10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning 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 ===
=== Prime harmonics ===
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== 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)
! 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 />
== 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 169: Line 159:
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
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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...
[[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 ==
== Approximation to JI ==
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== 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]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 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
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| 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
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
|}
* 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 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.  
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=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}
{{Uniform map|edo=10}}


=== Commas ===
=== Commas ===
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{| 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
! [[Color name|Color Name]]
! [[Color name]]
! Name(s)
! Name(s)
|-
|-
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| 90.22
| 90.22
| Sawa
| Sawa
| Limma, Pythagorean diatonic semitone
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
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| 70.67
| 70.67
| Yoyo
| Yoyo
| Classic chromatic semitone, dicot comma
| Dicot comma, classic chroma
|-
|-
| 5
| 5
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| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
| [[Ampersand comma]]
|-
|-
| 5
| 5
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| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
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| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
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| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal kleisma, marvel comma
| Marvel comma, septimal kleisma
|-
|-
| 7
| 7
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| 6.99
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
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| 5.57
| 5.57
| Saquinbizogu
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
| [[Linus comma]]
|-
|-
| 7
| 7
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| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
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| Island comma, parizeksma
| Island comma, parizeksma
|}
|}
<references/>


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Periods <br> per 8ve
! Periods<br />per 8ve
! Generator
! Generator
! Temperament(s)
! Temperament(s)
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== Scales ==
== Scales ==
=== MOS scales ===
=== MOS scales ===
 
* Decimal/Lemba[6] [[4L&nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Decimal/Lemba[6] [[4L 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
* Dicot[7] [[3L 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
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1


=== Other scales ===
=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 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
* 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
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1


=== Horagrams ===
=== 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 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|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 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&nbsp;1s, 1L&nbsp;2s, 3L&nbsp;1s, 3L&nbsp;4s]]


== Diagrams ==
== Diagrams ==
Line 454: Line 561:


== 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.
Line 465: Line 572:
|}
|}
[[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 ==
Line 470: Line 580:
{{Catrel|10edo tracks}}
{{Catrel|10edo tracks}}


[[Category:Macrotonal]]
== Notes ==
<references group="note" />
 
[[Category:10-tone scales]]
Retrieved from "https://en.xen.wiki/w/10edo"