12edo: Difference between revisions

Line 23:

=== Octave stretch ===

Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense.

Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

=== Subsets and supersets ===

12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

[[24edo]], which doubles it, ~~provides a great correction for the approximate harmonics~~ 11 and 13. [[36edo]], which triples it, ~~provides a great correction for~~ the ~~approximate harmonic 7~~. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===

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== Intervals ==

{| class="wikitable center-all"

|+ Intervals of 12edo

|+ style="font-size: 105%;" | Intervals of 12edo

|-

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

! rowspan="2" | [[Cent]]s

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| Minor second

|

| [[25/24]] (+29.328) [[16/15]] (~~-11~~.731)

| [[25/24]] (+29.328) [[16/15]] (−11.731)

| [[28/27]] (+37.039) [[21/20]] (+15.533) [[15/14]] (~~-19~~.443)

| [[28/27]] (+37.039) [[21/20]] (+15.533) [[15/14]] (−19.443)

| [[18/17]] (+1.045) [[17/16]] (-4.955)

| [[18/17]] (+1.045) [[17/16]] (−4.955)

| [[File:piano_1_12edo.mp3]]

|-

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| 200

| Major second

| [[9/8]] (-3.910)

| [[9/8]] (−3.910)

| [[10/9]] (+17.596)

| [[28/25]] (+3.802) [[8/7]] (~~-31~~.174)

| [[28/25]] (+3.802) [[8/7]] (−31.174)

| [[19/17]] (+7.442) [[55/49]] (+0.020) [[64/57]] (-0.532) [[17/15]] (~~-16~~.687)

| [[19/17]] (+7.442) [[55/49]] (+0.020) [[64/57]] (−0.532) [[17/15]] (−16.687)

| [[File:piano_1_6edo.mp3]]

|-

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| Minor third

| [[32/27]] (+5.865)

| [[6/5]] (~~-15~~.641)

| [[6/5]] (−15.641)

| [[7/6]] (+33.129) [[25/21]] (-1.847)

| [[7/6]] (+33.129) [[25/21]] (−1.847)

| [[19/16]] (+2.487) [[44/37]] (+0.026)

| [[File:piano_1_4edo.mp3]]

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| 400

| Major third

| [[81/64]] (-7.820)

| [[81/64]] (−7.820)

| [[5/4]] (+13.686)

| [[63/50]] (-0.108) [[9/7]] (~~-35~~.084)

| [[63/50]] (−0.108) [[9/7]] (−35.084)

| [[34/27]] (+0.910) [[24/19]] (-4.442)

| [[34/27]] (+0.910) [[24/19]] (−4.442)

| [[File:piano_1_3edo.mp3]]

|-

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|

| [[7/5]] (+17.488) [[10/7]] (~~-17~~.488)

| [[7/5]] (+17.488) [[10/7]] (−17.488)

| [[24/17]] (+3.000) [[99/70]] (-0.088) [[17/12]] (-3.000)

| [[24/17]] (+3.000) [[99/70]] (−0.088) [[17/12]] (−3.000)

| [[File:piano_1_2edo.mp3]]

|-

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| 700

| Fifth

| [[3/2]] (-1.955)

| [[3/2]] (−1.955)

|

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| Minor sixth

| [[128/81]] (+7.820)

| [[8/5]] (~~-13~~.686)

| [[8/5]] (−13.686)

| [[14/9]] (+35.084) [[100/63]] (+0.108)

| [[19/12]] (+4.442) [[27/17]] (-0.910)

| [[19/12]] (+4.442) [[27/17]] (−0.910)

| [[File:piano_2_3edo.mp3]]

|-

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| 900

| Major sixth

| [[27/16]] (-5.865)

| [[27/16]] (−5.865)

| [[5/3]] (+15.641)

| [[42/25]] (+1.847) [[12/7]] (~~-33~~.129)

| [[42/25]] (+1.847) [[12/7]] (−33.129)

| [[37/22]] (-0.026) [[32/19]] (-2.487)

| [[37/22]] (−0.026) [[32/19]] (−2.487)

| [[File:piano_3_4edo.mp3]]

|-

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| Minor seventh

| [[16/9]] (+3.910)

| [[9/5]] (~~-17~~.596)

| [[9/5]] (−17.596)

| [[7/4]] (+31.174) [[25/14]] (-3.802)

| [[7/4]] (+31.174) [[25/14]] (−3.802)

| [[30/17]] (+16.687) [[57/32]] (+0.532) [[98/55]] (-0.020) [[34/19]] (-7.442)

| [[30/17]] (+16.687) [[57/32]] (+0.532) [[98/55]] (−0.020) [[34/19]] (−7.442)

| [[File:piano_5_6edo.mp3]]

|-

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| Major seventh

|

| [[15/8]] (+11.731) [[48/25]] (~~-29~~.328)

| [[15/8]] (+11.731) [[48/25]] (−29.328)

| [[28/15]] (+19.443) [[40/21]] (~~-15~~.533) [[27/14]] (~~-37~~.039)

| [[28/15]] (+19.443) [[40/21]] (−15.533) [[27/14]] (−37.039)

| [[32/17]] (+4.955) [[17/9]] (-1.045)

| [[32/17]] (+4.955) [[17/9]] (−1.045)

| [[File:piano_11_12edo.mp3]]

|-

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{| class="wikitable center-all"

|+~~Notation of 12edo~~

|+ style="font-size: 105%;" | Notation of 12edo

~~! rowspan~~="2" |~~[[Degree]]~~

~~! rowspan="2" |[[Cent]]s~~

~~! colspan="2" |[[Chain-~~of~~-fifths notation|Standard notation]]~~

|-

! Diatonic ([[5L 2s]]) interval names

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

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Chain-of-fifths notation|Standard notation]]

|-

! Diatonic ([[5L 2s]]) interval names

! Note names (on D)

|-

| 0

|'''Perfect unison (P1)'''

| '''Perfect unison (P1)'''

|'''D'''

| '''D'''

|-

| 1

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| 2

| 200

|'''Major second (M2)''' Diminished third (d3)

| '''Major second (M2)''' Diminished third (d3)

|'''E''' Fb

| '''E''' Fb

|-

| 3

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Line 212:

| 5

| 500

|'''Perfect fourth (P4)'''

| '''Perfect fourth (P4)'''

|'''G'''

| '''G'''

|-

| 6

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| 7

| 700

|'''Perfect fifth (P5)'''

| '''Perfect fifth (P5)'''

| A

|-

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| 9

| 900

|'''Major sixth (M6)''' Diminished seventh (d7)

| '''Major sixth (M6)''' Diminished seventh (d7)

|'''B''' Cb

| '''B''' Cb

|-

| 10

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| 12

| 1200

|'''Perfect octave (P8)'''

| '''Perfect octave (P8)'''

|'''D'''

| '''D'''

|}

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* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===

=== Sagittal notation ===

~~====Evo flavor====~~

==== Evo flavor ====

File:12-EDO_Evo_Sagittal.svg

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Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

~~====Revo flavor====~~

==== Revo flavor ====

File:12-EDO_Revo_Sagittal.svg

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== Solfege ==

{| class="wikitable center-all"

|+ Solfege of 12edo

|+ style="font-size: 105%;" | Solfege of 12edo

|-

! [[Degree]]

! [[Cents]]

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* [[Lumatone mapping for 12edo]]

* [[:purdal:12-EDD]]{{dead link}}

* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==

@@ Line 23: / Line 23: @@
 === Octave stretch ===
-Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense.
+Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.
 === Subsets and supersets ===
-edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.
+edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.
-[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].
+[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].
 === Miscellany ===
@@ Line 35: / Line 35: @@
 == Intervals ==
 {| class="wikitable center-all"
-|+ Intervals of 12edo
+|+ style="font-size: 105%;" | Intervals of 12edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 60: / Line 61: @@
 | Minor second
 |
-| [[25/24]] (+29.328)<br>[[16/15]] (-11.731)
+| [[25/24]] (+29.328)<br>[[16/15]] (−11.731)
-| [[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (-19.443)
+| [[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (−19.443)
-| [[18/17]] (+1.045)<br>[[17/16]] (-4.955)
+| [[18/17]] (+1.045)<br>[[17/16]] (−4.955)
 | [[File:piano_1_12edo.mp3]]
 |-
@@ Line 68: / Line 69: @@
 | 200
 | Major second
-| [[9/8]] (-3.910)
+| [[9/8]] (−3.910)
 | [[10/9]] (+17.596)
-| [[28/25]] (+3.802)<br>[[8/7]] (-31.174)
+| [[28/25]] (+3.802)<br>[[8/7]] (−31.174)
-| [[19/17]] (+7.442)<br>[[55/49]] (+0.020)<br>[[64/57]] (-0.532)<br>[[17/15]] (-16.687)
+| [[19/17]] (+7.442)<br>[[55/49]] (+0.020)<br>[[64/57]] (−0.532)<br>[[17/15]] (−16.687)
 | [[File:piano_1_6edo.mp3]]
 |-
@@ Line 78: / Line 79: @@
 | Minor third
 | [[32/27]] (+5.865)
-| [[6/5]] (-15.641)
+| [[6/5]] (−15.641)
-| [[7/6]] (+33.129)<br>[[25/21]] (-1.847)
+| [[7/6]] (+33.129)<br>[[25/21]] (−1.847)
 | [[19/16]] (+2.487)<br>[[44/37]] (+0.026)
 | [[File:piano_1_4edo.mp3]]
@@ Line 86: / Line 87: @@
 | 400
 | Major third
-| [[81/64]] (-7.820)
+| [[81/64]] (−7.820)
 | [[5/4]] (+13.686)
-| [[63/50]] (-0.108)<br>[[9/7]] (-35.084)
+| [[63/50]] (−0.108)<br>[[9/7]] (−35.084)
-| [[34/27]] (+0.910)<br>[[24/19]] (-4.442)
+| [[34/27]] (+0.910)<br>[[24/19]] (−4.442)
 | [[File:piano_1_3edo.mp3]]
 |-
@@ Line 106: / Line 107: @@
 |
 |
-| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)
+| [[7/5]] (+17.488)<br>[[10/7]] (−17.488)
-| [[24/17]] (+3.000)<br>[[99/70]] (-0.088)<br>[[17/12]] (-3.000)
+| [[24/17]] (+3.000)<br>[[99/70]] (−0.088)<br>[[17/12]] (−3.000)
 | [[File:piano_1_2edo.mp3]]
 |-
@@ Line 113: / Line 114: @@
 | 700
 | Fifth
-| [[3/2]] (-1.955)
+| [[3/2]] (−1.955)
 |
 |
@@ Line 123: / Line 124: @@
 | Minor sixth
 | [[128/81]] (+7.820)
-| [[8/5]] (-13.686)
+| [[8/5]] (−13.686)
 | [[14/9]] (+35.084)<br>[[100/63]] (+0.108)
-| [[19/12]] (+4.442)<br>[[27/17]] (-0.910)
+| [[19/12]] (+4.442)<br>[[27/17]] (−0.910)
 | [[File:piano_2_3edo.mp3]]
 |-
@@ Line 131: / Line 132: @@
 | 900
 | Major sixth
-| [[27/16]] (-5.865)
+| [[27/16]] (−5.865)
 | [[5/3]] (+15.641)
-| [[42/25]] (+1.847)<br>[[12/7]] (-33.129)
+| [[42/25]] (+1.847)<br>[[12/7]] (−33.129)
-| [[37/22]] (-0.026)<br>[[32/19]] (-2.487)
+| [[37/22]] (−0.026)<br>[[32/19]] (−2.487)
 | [[File:piano_3_4edo.mp3]]
 |-
@@ Line 141: / Line 142: @@
 | Minor seventh
 | [[16/9]] (+3.910)
-| [[9/5]] (-17.596)
+| [[9/5]] (−17.596)
-| [[7/4]] (+31.174)<br>[[25/14]] (-3.802)
+| [[7/4]] (+31.174)<br>[[25/14]] (−3.802)
-| [[30/17]] (+16.687)<br>[[57/32]] (+0.532)<br>[[98/55]] (-0.020)<br>[[34/19]] (-7.442)
+| [[30/17]] (+16.687)<br>[[57/32]] (+0.532)<br>[[98/55]] (−0.020)<br>[[34/19]] (−7.442)
 | [[File:piano_5_6edo.mp3]]
 |-
@@ Line 150: / Line 151: @@
 | Major seventh
 |
-| [[15/8]] (+11.731)<br>[[48/25]] (-29.328)
+| [[15/8]] (+11.731)<br>[[48/25]] (−29.328)
-| [[28/15]] (+19.443)<br>[[40/21]] (-15.533)<br>[[27/14]] (-37.039)
+| [[28/15]] (+19.443)<br>[[40/21]] (−15.533)<br>[[27/14]] (−37.039)
-| [[32/17]] (+4.955)<br>[[17/9]] (-1.045)
+| [[32/17]] (+4.955)<br>[[17/9]] (−1.045)
 | [[File:piano_11_12edo.mp3]]
 |-
@@ Line 175: / Line 176: @@
 {| class="wikitable center-all"
-|+Notation of 12edo
+|+ style="font-size: 105%;" | Notation of 12edo
-! rowspan="2" |[[Degree]]
-! rowspan="2" |[[Cent]]s
-! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
 |-
-! Diatonic ([[5L 2s]]) interval names
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
+|-
+! Diatonic ([[5L&nbsp;2s]]) interval names
 ! Note names (on D)
 |-
 | 0
 | 0
-|'''Perfect unison (P1)'''
+| '''Perfect unison (P1)'''
-|'''D'''
+| '''D'''
 |-
 | 1
@@ Line 195: / Line 197: @@
 | 2
 | 200
-|'''Major second (M2)'''<br>Diminished third (d3)
+| '''Major second (M2)'''<br>Diminished third (d3)
-|'''E'''<br>Fb
+| '''E'''<br>Fb
 |-
 | 3
@@ Line 210: / Line 212: @@
 | 5
 | 500
-|'''Perfect fourth (P4)'''
+| '''Perfect fourth (P4)'''
-|'''G'''
+| '''G'''
 |-
 | 6
@@ Line 220: / Line 222: @@
 | 7
 | 700
-|'''Perfect fifth (P5)'''
+| '''Perfect fifth (P5)'''
 | A
 |-
@@ Line 230: / Line 232: @@
 | 9
 | 900
-|'''Major sixth (M6)'''<br>Diminished seventh (d7)
+| '''Major sixth (M6)'''<br>Diminished seventh (d7)
-|'''B'''<br>Cb
+| '''B'''<br>Cb
 |-
 | 10
@@ Line 245: / Line 247: @@
 | 12
 | 1200
-|'''Perfect octave (P8)'''
+| '''Perfect octave (P8)'''
-|'''D'''
+| '''D'''
 |}
@@ Line 253: / Line 255: @@
 * Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:12-EDO_Evo_Sagittal.svg
@@ Line 266: / Line 268: @@
 Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:12-EDO_Revo_Sagittal.svg
@@ Line 278: / Line 280: @@
 == Solfege ==
 {| class="wikitable center-all"
-|+ Solfege of 12edo
+|+ style="font-size: 105%;" | Solfege of 12edo
+|-
 ! [[Degree]]
 ! [[Cents]]
@@ Line 911: / Line 914: @@
 * [[Lumatone mapping for 12edo]]
 * [[:purdal:12-EDD]]{{dead link}}
-* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step
+* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
 == Notes ==