10edo: Difference between revisions

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{{Infobox ET

| Prime factorization = 2 × 5

| Step size = ~~120¢~~

| Step size = 120.000¢

| Fifth = 6\10 = 720¢ (→[[5edo|3\5]])

| Fifth = 6\10 (720¢) (→[[5edo|3\5]])

| Major 2nd = 2\10 = 240¢ (→1\5)

| Major 2nd = 2\10 (240¢) (→1\5)

| Semitones = 2 : 0

| Semitones = 2:0 (240¢ : 0¢)

| Consistency = 7

~~| Monotonicity = 9~~

}}

'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''~~10-TET~~''', '''~~10ET~~''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==

10edo can be thought of as two circles of [[~~5edo|~~5edo]] separated by 120 cents (or 5 circles of [[~~2edo|~~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 happy 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]]). 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, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 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]]). 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, 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.

== Intervals ==

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

== ~~Just~~ approximation ==

== JI approximation ==

=== Selected just intervals by error ===

==== Selected 13-limit intervals ====

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

==~~= Temperament measures =~~==

== Regular temperament properties ==

The following table shows [[TE temperament measures]] (RMS normalized) of 10ET.

{| class="wikitable center-all"

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* 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 ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== ~~Linear~~ temperaments ~~(with images for MOS horagrams)~~ ==

=== Rank-2 temperaments ===

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

|-

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| [[Blackwood]]/[[blacksmith]]

|}

== Scales ==

[[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]]

== Pathological ~~Modes~~ ==

=== Pathological scales ===

2 1 1 1 2 1 1 1 [[2L 6s]] MOS

* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS

* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS

* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==

~~10edo~~ tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.

10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.

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

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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 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.

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 7: / Line 7: @@
 {{Infobox ET
 | Prime factorization = 2 × 5
-| Step size = 120¢
+| Step size = 120.000¢
-| Fifth = 6\10 = 720¢ (&rarr;[[5edo|3\5]])
+| Fifth = 6\10 (720¢) (&rarr;[[5edo|3\5]])
-| Major 2nd = 2\10 = 240¢ (&rarr;1\5)
+| Major 2nd = 2\10 (240¢) (&rarr;1\5)
-| Semitones = 2 : 0
+| Semitones = 2:0 (240¢ : 0¢)
 | Consistency = 7
-| Monotonicity = 9
 }}
-'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.
+'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.
 == Theory ==
 {{Harmonics in equal|10}}
-edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|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 happy 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]]). 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, 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.
+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 happy 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]]). 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, 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.
 == Intervals ==
@@ Line 154: / Line 153: @@
 genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...
-== Just approximation ==
+== JI approximation ==
 === Selected just intervals by error ===
 ==== Selected 13-limit intervals ====
 [[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]
-=== Temperament measures ===
+== Regular temperament properties ==
 The following table shows [[TE temperament measures]] (RMS normalized) of 10ET.
 {| class="wikitable center-all"
@@ Line 206: / Line 205: @@
 * 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 ETs better in those subgroups are 17, 19, 36 and 31, respectively.
-== Linear temperaments (with images for MOS horagrams) ==
+=== Rank-2 temperaments ===
 {| class="wikitable center-1 center-2"
 |-
@@ Line 233: / Line 232: @@
 | [[Blackwood]]/[[blacksmith]]
 |}
+== Scales ==
 [[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|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 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]
-== Pathological Modes ==
+=== Pathological scales ===
-1 1 1 2 1 1 1 [[2L 6s]] MOS
+* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
+* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS
-1 1 1 1 1 1 1 [[1L 7s]] MOS
+* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS
-1 1 1 1 1 1 1 1 [[1L 8s]] MOS
 == Commas ==
-edo tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.
+et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}.
 {| class="commatable wikitable center-1 center-2 right-4 center-5"
@@ Line 429: / Line 428: @@
 == 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 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.
 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.