10edo: Difference between revisions

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== 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~~ {{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~~.

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.

~~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.

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 ===

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== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"

|-

! Degree

! Cents

! Approximate ratios<ref>~~based on treating 10edo as a~~ 2.15.7.13-subgroup ~~temperament~~</ref>

! Approximate ratios<ref group="note">{{sg|limit=2.15.7.13-subgroup}}</ref>

! Additional ratios 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 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

! colspan="3" | [[Ups and downs notation]]

! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)

! Audio

|-

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| [[File:0-1200 octave.mp3|frameless]]

|}

== Notation ==

~~This is~~ the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

=== 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)

==== 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

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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

~~'''~~Heptatonic 3rd-generated~~:~~ 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

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

=== 3L 4s (mosh) notation ===

[[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 ♭.

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"

! ~~Degree~~

! #

! Cents

! Note

! Name

! Associated ~~ratio~~

! Associated ratios

|-

| 0

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| 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 ====

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 ====

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 ==

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

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| 2.3.5

| 25/24, 256/243

| {{~~mapping~~| 10 16 23 }}

| {{Mapping| 10 16 23 }}

| -0.089

| 9.27

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| 2.3.5.7

| 25/24, 28/27, 49/48

| {{~~mapping~~| 10 16 23 28 }}

| {{Mapping| 10 16 23 28 }}

| +0.718

| 8.15

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| 2.3.5.7.13

| 25/24, 28/27, 40/39, 49/48

| {{~~mapping~~| 10 16 23 28 37 }}

| {{Mapping| 10 16 23 28 37 }}

| +0.603

| 7.30

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=== Uniform maps ===

=== Commas ===

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|-

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref>~~Ratios longer than 10 digits are presented by placeholders with informative hints~~</ref>

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cent]]s

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| 31.57

| Lala-tribiyo

| [[Ampersand]]~~, Ampersand's comma~~

| [[Ampersand comma]]

|-

| 5

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| Island comma, parizeksma

|}

=== Rank-2 temperaments ===

{| class="wikitable center-1 center-2"

|-

! Periods per 8ve

! Periods per 8ve

! Generator

! Temperament(s)

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| 1

| 3\10

| [[Dicot]] / [[beatles]] / [[neutral]]

| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)

|-

| 2

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| 5

| 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 ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 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 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 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 ===

* [[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 double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1

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=== Horagrams ===

[[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]]

== Diagrams ==

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== 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.

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|}

[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===

''See [[Lumatone mapping for 10edo]]''.

== Music ==

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[[Category:~~Macrotonal~~]]

== References ==

~~{{todo|add lumatone mapping}}~~

[[Category:10-tone scales]]

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|10}}
+{{ED intro}}
 == Theory ==
+edo 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]].
-edo 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&nbsp;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.
+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&nbsp;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.
-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.
+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.
+edo 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 ===
@@ Line 19: / Line 30: @@
 == Intervals ==
 {| class="wikitable right-1 right-2 center-7 center-8"
+|-
 ! Degree
 ! Cents
-! Approximate ratios<ref>based on treating 10edo as a 2.15.7.13-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
-! 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
 |-
@@ Line 137: / Line 149: @@
 | [[File:0-1200 octave.mp3|frameless]]
 |}
+<references group="note" />
-<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.
+=== 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.
-'''<u>Pentatonic 5th-generated</u>: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)
+==== 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
@@ Line 149: / Line 162: @@
 - ^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
@@ Line 163: / Line 181: @@
 genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
-=== 3L 4s (mosh) notation ===
+[[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"
-! Degree
+! #
 ! Cents
 ! Note
 ! Name
-! Associated ratio
+! Associated ratios
 |-
 | 0
@@ Line 239: / Line 263: @@
 | 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 ==
@@ Line 247: / Line 296: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
@@ Line 258: / Line 308: @@
 | 2.3.5
 | 25/24, 256/243
-| {{mapping| 10 16 23 }}
+| {{Mapping| 10 16 23 }}
 | -0.089
 | 9.27
@@ Line 265: / Line 315: @@
 | 2.3.5.7
 | 25/24, 28/27, 49/48
-| {{mapping| 10 16 23 28 }}
+| {{Mapping| 10 16 23 28 }}
 | +0.718
 | 8.15
@@ Line 272: / Line 322: @@
 | 2.3.5.7.13
 | 25/24, 28/27, 40/39, 49/48
-| {{mapping| 10 16 23 28 37 }}
+| {{Mapping| 10 16 23 28 37 }}
 | +0.603
 | 7.30
@@ Line 281: / Line 331: @@
 === Uniform maps ===
-{{Uniform map|13|9.5|10.5}}
+{{Uniform map|edo=10}}
 === Commas ===
@@ Line 289: / Line 339: @@
 |-
 ! [[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]]
 ! [[Cent]]s
@@ Line 321: / Line 371: @@
 | 31.57
 | Lala-tribiyo
-| [[Ampersand]], Ampersand's comma
+| [[Ampersand comma]]
 |-
 | 5
@@ Line 463: / Line 513: @@
 | Island comma, parizeksma
 |}
-<references/>
+<references group="note"/>
 === Rank-2 temperaments ===
 {| class="wikitable center-1 center-2"
 |-
-! Periods <br> per 8ve
+! Periods<br>per 8ve
 ! Generator
 ! Temperament(s)
@@ Line 478: / Line 528: @@
 | 1
 | 3\10
-| [[Dicot]] / [[beatles]] / [[neutral]]
+| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
 |-
 | 2
@@ Line 490: / Line 540: @@
 | 5
 | 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 ==
 === 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 ===
+* [[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 double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
@@ Line 511: / Line 563: @@
 === 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 ==
@@ Line 518: / Line 570: @@
 == Instruments ==
-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 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.
+edo 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.
@@ Line 529: / Line 581: @@
 |}
 [[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
+=== Lumatone ===
+''See [[Lumatone mapping for 10edo]]''.
 == Music ==
@@ Line 534: / Line 589: @@
 {{Catrel|10edo tracks}}
-[[Category:Macrotonal]]
+== References ==
-{{todo|add lumatone mapping}}
+<references/>
+[[Category:10-tone scales]]