10edo: Difference between revisions

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}}

== 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 {{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 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.

~~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), 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 ===

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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]]

|}

~~<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.

~~'''~~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 ~~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...)

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

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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]]

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

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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=== Uniform maps ===

=== Commas ===

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{| class="commatable wikitable center-1 center-2 right-4 center-5"

|-

! [[Harmonic limit|Prime limit]]

! [[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

|}

~~<references/>~~

=== Rank-2 temperaments ===

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

|-

! Periods per 8ve

! Periods per 8ve

! Generator

! Temperament(s)

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== 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 ===

* [[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

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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~~]]

== Notes ==

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

[[Category:10-tone scales]]

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|10}}
+{{ED intro}}
 == Theory ==
+edo 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.
-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 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 scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate 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), 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.
+edo 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 ===
@@ Line 19: / Line 28: @@
 == 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 147: @@
 | [[File:0-1200 octave.mp3|frameless]]
 |}
-<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 159: @@
 - ^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
@@ Line 163: / Line 178: @@
 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 260: @@
 | 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 293: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 281: / Line 328: @@
 === Uniform maps ===
-{{Uniform map|13|9.5|10.5}}
+{{Uniform map|edo=10}}
 === Commas ===
@@ Line 288: / Line 335: @@
 {| 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]]
 ! [[Cent]]s
@@ Line 321: / Line 368: @@
 | 31.57
 | Lala-tribiyo
-| [[Ampersand]], Ampersand's comma
+| [[Ampersand comma]]
 |-
 | 5
@@ Line 463: / Line 510: @@
 | Island comma, parizeksma
 |}
-<references/>
 === Rank-2 temperaments ===
 {| class="wikitable center-1 center-2"
 |-
-! Periods <br> per 8ve
+! Periods<br />per 8ve
 ! Generator
 ! Temperament(s)
@@ Line 494: / Line 540: @@
 == 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 ===
 * [[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
@@ Line 511: / Line 554: @@
 === 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 561: @@
 == 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 572: @@
 |}
 [[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
+=== Lumatone ===
+''See [[Lumatone mapping for 10edo]]''.
 == Music ==
@@ Line 534: / Line 580: @@
 {{Catrel|10edo tracks}}
-[[Category:Macrotonal]]
+== Notes ==
-{{todo|add lumatone mapping}}
+<references group="note" />
+[[Category:10-tone scales]]