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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|10}}
{{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 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 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 ===
=== Prime harmonics ===
Line 17: Line 30:
== 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
Line 34: Line 47:
| P1, m2
| P1, m2
| D, Eb
| D, Eb
| unison
| [[File:0-0 unison.mp3|frameless]]
| C
|[[File:0-0 unison.mp3|frameless]]
|-
|-
| 1
| 1
Line 46: Line 57:
| ~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
Line 58: Line 67:
| 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
Line 70: Line 77:
| ~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
Line 82: Line 87:
| 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
Line 94: Line 97:
| ^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
Line 106: Line 107:
| 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
Line 118: Line 117:
| ~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
Line 130: Line 127:
| 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
Line 142: Line 137:
| ~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
Line 154: Line 147:
| M7, P8
| M7, P8
| C#, D
| C#, D
| octave
| [[File:0-1200 octave.mp3|frameless]]
| C
|[[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 169: 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 182: Line 180:


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 ==
Line 190: 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]]
! 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]] (¢)
Line 201: 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 208: 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
|-
| 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.  
Line 217: Line 331:


=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}
{{Uniform map|edo=10}}


=== Commas ===
=== Commas ===
Line 224: Line 338:
{| 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)
|-
|-
Line 236: Line 350:
| 90.22
| 90.22
| Sawa
| Sawa
| Limma, Pythagorean diatonic semitone
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
Line 243: Line 357:
| 70.67
| 70.67
| Yoyo
| Yoyo
| Classic chromatic semitone, dicot comma
| Dicot comma, classic chroma
|-
|-
| 5
| 5
Line 257: Line 371:
| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 278: Line 392:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 285: Line 399:
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
Line 320: Line 434:
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal kleisma, marvel comma
| Marvel comma, septimal kleisma
|-
|-
| 7
| 7
Line 327: Line 441:
| 6.99
| 6.99
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 334: Line 448:
| 5.57
| 5.57
| Saquinbizogu
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
| [[Linus comma]]
|-
|-
| 7
| 7
Line 383: Line 497:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 399: Line 513:
| Island comma, parizeksma
| Island comma, parizeksma
|}
|}
<references/>
<references group="note"/>


=== 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)
Line 414: Line 528:
| 1
| 1
| 3\10
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
|-
| 2
| 2
Line 426: Line 540:
| 5
| 5
| 1\10
| 1\10
| [[Blackwood]] / [[blacksmith]]
| [[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 ==
== 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 ===
 
* [[Pinetone #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
* [[Pinetone #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
* [[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
Line 447: Line 563:


=== 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 570:


== 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 ==
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{{Catrel|10edo tracks}}
{{Catrel|10edo tracks}}


[[Category:Macrotonal]]
== References ==
<references/>
 
[[Category:10-tone scales]]

Latest revision as of 18:04, 12 March 2026

← 9edo 10edo 11edo →
Prime factorization 2 × 5
Step size 120 ¢ 
Fifth 6\10 (720 ¢) (→ 3\5)
Semitones (A1:m2) 2:0 (240 ¢ : 0 ¢)
Consistency limit 7
Distinct consistency limit 3

10 equal divisions of the octave (abbreviated 10edo or 10ed2), also called 10-tone equal temperament (10tet) or 10 equal temperament (10et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 10 equal parts of exactly 120 ¢ each. Each step represents a frequency ratio of 21/10, or the 10th root of 2.

Theory

10edo contains all the intervals of 5edo, but also adds another copy of it separated by 120 cents. The new intervals have sizes of 120 ¢, 360 ¢, 600 ¢, 840 ¢, and 1080 ¢. The 120 ¢ 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 ¢ and 1080 ¢ intervals being close (about 0.6 ¢ off) to 15/14 and 28/15 respectively. The 360 ¢ interval is a large neutral third, being about 0.5 ¢ sharp of 16/13, with its inversion being equally close to 13/8. Finally, the 600 ¢ interval is the tritone that appears in every even-numbered edo, including 12edo.

Taking the the 360 ¢ large neutral third as a generator produces a heptatonic moment of symmetry scale with step sizes 2 1 1 2 1 2 1 (pattern 3L 4s, or "mosh"), which is the most 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 ¢, 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 ¢ flat, and this interval is also equated with 6/5 (which is even more inaccurate, at 44 ¢ 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 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 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 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 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[1].

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 ¢. 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 ¢ as 11/9, we arrive at 11/8 = 600 ¢ (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 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

Approximation of prime harmonics in 10edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +18.0 -26.3 -8.8 +48.7 -0.5 +15.0 -57.5 -28.3 +50.4 +55.0
Relative (%) +0.0 +15.0 -21.9 -7.4 +40.6 -0.4 +12.5 -47.9 -23.6 +42.0 +45.8
Steps
(reduced) 		10
(0) 		16
(6) 		23
(3) 		28
(8) 		35
(5) 		37
(7) 		41
(1) 		42
(2) 		45
(5) 		49
(9) 		50
(0)

Intervals

Degree Cents Approximate ratios[note 1] Additional ratios
of 3, 5, and 9[note 2] 		Interval names Ups and downs notation
(EUs: vvA1 and m2) 		Audio
0 0 1/1 256/243, 50/49, 25/24 unison unison, min 2nd P1, m2 D, Eb
1 120 16/15, 15/14, 14/13 10/9, 13/12, 81/80 neutral second mid 2nd ~2 ^D, vE
2 240 8/7, 15/13, 144/125, 224/195 9/8, 7/6 hemifourth, major second, minor third maj 2nd, min 3rd M2, m3 E, F
3 360 16/13 5/4 neutral third mid 3rd ~3 ^F, vG
4 480 64/49, 169/128 4/3, 9/7, 13/10 perfect fourth maj 3rd, perf 4th M3, P4 F#, G
5 600 91/64, 128/91, 169/120, 240/169 7/5, 10/7, 13/9, 18/13 hemioctave, tritone up 4th, down 5th ^4, v5 ^G, vA
6 720 49/32, 256/169 3/2, 14/9, 20/13 perfect fifth perf 5th, min 6th P5, m6 A, Bb
7 840 13/8 8/5 neutral sixth mid 6th ~6 ^A, vB
8 960 7/4, 26/15, 125/72, 195/112 16/9, 12/7 hemitwelfth, major sixth, minor seventh maj 6th, min 7th M6, m7 B, C
9 1080 15/8, 28/15, 13/7 9/5, 24/13, 160/81 neutral seventh mid 7th ~7 ^C, vD
10 1200 2/1 243/128, 49/25, 48/25 octave maj 7th, octave M7, P8 C#, D
  1. ↑ Based on treating 10edo as a 2.15.7.13-subgroup temperament; other approaches are also possible.
  2. ↑ Adding the ratios of 3, 5, and 9 introduces greater error while giving several more harmonic identities to the 10-edo intervals

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.

Pentatonic 5th-generated

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

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

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 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

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

Enharmonic unison: d2

See below: 3L 4s mosh notation

3L 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 ♭.

# 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 20 and 30 and a superset of the notation for 5edo.

Evo and Revo flavors

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

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

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Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 25/24, 256/243 [⟨10 16 23]] -0.089 9.27 7.73
2.3.5.7 25/24, 28/27, 49/48 [⟨10 16 23 28]] +0.718 8.15 6.79
2.3.5.7.13 25/24, 28/27, 40/39, 49/48 [⟨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 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.

Uniform maps

13-limit uniform maps between 9.8 and 10.2
Min. size Max. size Wart notation Map
9.7957 9.8637 10eff ⟨10 16 23 28 34 36]
9.8637 9.9727 10e ⟨10 16 23 28 34 37]
9.9727 10.1209 10 ⟨10 16 23 28 35 37]
10.1209 10.1339 10c ⟨10 16 24 28 35 37]
10.1339 10.1519 10cf ⟨10 16 24 28 35 38]
10.1519 10.2618 10cdf ⟨10 16 24 29 35 38]

Commas

10et tempers out the following commas. This assumes the val ⟨10 16 23 28 35 37].

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name(s)
3 256/243 [8 -5⟩ 90.22 Sawa Blackwood comma, Pythagorean limma
5 25/24 [-3 -1 2⟩ 70.67 Yoyo Dicot comma, classic chroma
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri comma, double augmentation diesis
5 (16 digits) [-25 7 6⟩ 31.57 Lala-tribiyo Ampersand comma
5 2048/2025 [11 -4 -2⟩ 19.55 Sagugu Diaschisma
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 49/48 [-4 -1 0 2⟩ 35.70 Zozo Semaphoresma, slendro diesis
7 50/49 [1 0 2 -2⟩ 34.98 Biruyo Jubilisma, tritonic diesis
7 686/675 [1 -3 -2 3⟩ 27.99 Trizo-agugu Senga
7 64/63 [6 -2 0 -1⟩ 27.26 Ru Septimal comma, Archytas' comma, Leipziger Komma
7 (18 digits) [-10 7 8 -7⟩ 22.41 Lasepru-aquadbiyo Blackjackisma
7 1029/1024 [-10 1 0 3⟩ 8.43 Latrizo Gamelisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma, septimal kleisma
7 16875/16807 [0 3 4 -5⟩ 6.99 Quinru-aquadyo Mirkwai comma
7 (24 digits) [11 -10 -10 10⟩ 5.57 Saquinbizogu Linus comma
7 2401/2400 [-5 -1 -2 4⟩ 0.72 Bizozogu Breedsma
11 243/242 [-1 5 0 0 -2⟩ 7.14 Lulu Rastma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Luzozogu Werckisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Lururuyo Swetisma
11 3025/3024 [-4 -3 2 -1 2⟩ 0.57 Loloruyoyo Lehmerisma
13 91/90 [-1 -2 -1 1 0 1⟩ 19.13 Thozogu Superleap comma, biome comma
13 196/195 [2 -1 -1 2 0 -1⟩ 8.86 Thuzozogu Mynucuma
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Island comma, parizeksma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 temperaments

Periods
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. 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. 28ed7 is a stretched version of 10edo.

Scales

MOS scales

  • Decimal/Lemba[6] 4L 2s (period = 5\10, gen = 2\10): 2 2 1 2 2 1
  • Dicot[7] 3L 4s (gen = 3\10): 1 2 1 2 1 2 1
  • Negri[9] 1L 8s (gen = 1\10): 1 1 1 1 2 1 1 1 1

Other scales

Horagrams

1\10 MOS
1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s
3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s

Diagrams

10edo wheel.png

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 ¢) 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.

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 which approximates the 11th harmonic with relative accuracy, among other features.

Decaphonic_Classic_Guitar.png
A Decaphonic (10edo) Classical Guitar

decaphonic-uke.JPG

Lumatone

See Lumatone mapping for 10edo.

Music

Main article: 10edo/Music
See also: Category:10edo tracks

References

  1. ↑ Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
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