36edo: Difference between revisions

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That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.

~~For those interested in approximations to just intonation~~, 36edo offers no improvement over 12edo ~~in the 5-limit~~, since its nearest approximation to 5:4 is the overly-familiar 400-cent ~~sharp~~ third. ~~However~~, it excels ~~at approximations involving 3 and 7~~. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent mmajor third. In the 7-limit, however, it excels. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

36edo also offers a good approximation to the frequency ratio phi, as 25\36.

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Another 5-limit alternative val {{monzo|36 57 83}} (36c-edo), which is similar to the patent val but maps 5/4 to the 367-cent submajor third rather than the major third, supports very sharp [[porcupine]] temperament using 5\36 as a generator.

Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17

Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.

=== Prime harmonics ===

=== Divisors ===

36edo is the 7th [[highly composite EDO]], with subset edos {{EDOs|1, 2, 3, 4, 6, 9, 12, 18}}.

== Intervals ==

{| class="wikitable center-all right-2"

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== Relation to 12edo ==

For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.

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The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still pleasing). In contrast, the smallest interval in 24edo, which is 50 cents, sounds very bad to most ears.

People with perfect (absolute) pitch often have a ~~harder~~ time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding ~~bad~~). This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes too.

People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding out of tune). This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes too.

=== "Quark" ===

In particle physics, [https://en.wikipedia.org/wiki/Baryon baryons] , which are the main building blocks of atomic nuclei, are always comprised of three quarks. One could draw an analogy between baryons and semitones (the main building block of 12edo); the baryon is comprised of three quarks and the semitone of three sixth-tones. The number of quarks in a [https://en.wikipedia.org/wiki/Color_charge colorless] particle is always a multiple of three; similarly, the width of "colorless" intervals (the 12-edo intervals, which are neither red nor blue), expressed in terms of sixth-tones, is always a multiple of three. Because of this amusing coincidence, Mason Green proposes referring to the 33.333-cent sixth-tone interval as a "quark".

== JI Approximations ==

=== 3-limit (Pythagorean) approximations (same as 12edo): ===

3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.

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=== 7-limit approximations: ===

==== 7 only: ====

7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.

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63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.

== ~~Rank~~-2 ~~temperaments~~ ==

== Regular temperament properties ==

=== Commas ===

This is a partial list of the [[commas]] that 36edo [[tempers out]] with its patent [[val]], {{val| 36 57 84 101 125 133 }}.

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

|-

! [[Harmonic limit|Prime<br>Limit]]

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

! [[Monzo]]

! [[Cent]]s

! [[Color name]]

! Name(s)

|-

| 3

| [[531441/524288|(12 digits)]]

| {{monzo| -19 12 }}

| 23.46

| Lalawa

| [[Pythagorean comma]]

|-

| 5

| [[648/625]]

| {{monzo| 3 4 -4 }}

| 62.57

| Quadgu

| Greater diesis, diminished domma

|-

| 5

| [[128/125]]

| {{monzo| 7 0 -3 }}

| 41.06

| Trigu

| Lesser diesis, augmented comma

|-

| 5

| [[81/80]]

| {{monzo| -4 4 -1 }}

| 21.51

| Gu

| Syntonic comma, Didymus comma, meantone comma

|-

| 5

| [[2048/2025]]

| {{monzo| 11 -4 -2 }}

| 19.55

| Sagugu

| Diaschisma

|-

| 5

| [[67108864/66430125|(16 digits)]]

| {{monzo| 26 -12 -3 }}

| 17.60

| Sasa-trigu

| [[Misty comma]]

|-

| 5

| [[32805/32768]]

| {{monzo| -15 8 1 }}

| 1.95

| Layo

| Schisma

|-

| 5

| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>

| {{monzo| 161 -84 -12 }}

| 0.02

| Sepbisa-quadbigu

| [[Kirnberger's atom]]

|-

| 7

| [[1029/1000]]

| {{monzo| -3 1 -3 3 }}

| 49.49

| Trizogu

| Keega

|-

| 7

| [[686/675]]

| {{monzo| 1 -3 -2 3 }}

| 27.99

| Trizo-agugu

| Senga

|-

| 7

| [[1728/1715]]

| {{monzo| 6 3 -1 -3 }}

| 13.07

| Triru-agu

| Orwellisma

|-

| 7

| [[1029/1024]]

| {{monzo| -10 1 0 3 }}

| 8.43

| Latrizo

| Gamelisma

|-

| 7

| 118098/117649

| {{monzo| 1 10 0 -6 }}

| 6.59

|

|-

| 7

| [[10976/10935]]

| {{monzo| 5 -7 -1 3 }}

| 6.48

| Satrizo-agu

| Hemimage comma

|-

| 7

| <abbr title="40353607/40310784">(16 digits)</abbr>

| {{monzo| -11 -9 0 9 }}

| 1.84

| Tritrizo

| [[Septimal ennealimma]]

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|-

@@ Line 8: / Line 8: @@
 That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.
-For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
+In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400-cent mmajor third. In the 7-limit, however, it excels. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
 edo also offers a good approximation to the frequency ratio phi, as 25\36.
@@ Line 18: / Line 18: @@
 Another 5-limit alternative val {{monzo|36 57 83}} (36c-edo), which is similar to the patent val but maps 5/4 to the 367-cent submajor third rather than the major third, supports very sharp [[porcupine]] temperament using 5\36 as a generator.
-Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17
+Heinz Bohlen proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.
 === Prime harmonics ===
 {{harmonics in equal|36}}
 === Divisors ===
 edo is the 7th [[highly composite EDO]], with subset edos {{EDOs|1, 2, 3, 4, 6, 9, 12, 18}}.
 == Intervals  ==
 {| class="wikitable center-all right-2"
@@ Line 332: / Line 335: @@
 == Relation to 12edo ==
 For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.
@@ Line 345: / Line 347: @@
 The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still pleasing). In contrast, the smallest interval in 24edo, which is 50 cents, sounds very bad to most ears.
-People with perfect (absolute) pitch often have a harder time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding bad). This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes too.
+People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding out of tune). This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes too.
 === "Quark" ===
 In particle physics, [https://en.wikipedia.org/wiki/Baryon baryons] , which are the main building blocks of atomic nuclei, are always comprised of three quarks. One could draw an analogy between baryons and semitones (the main building block of 12edo); the baryon is comprised of three quarks and the semitone of three sixth-tones. The number of quarks in a [https://en.wikipedia.org/wiki/Color_charge colorless] particle is always a multiple of three; similarly, the width of "colorless" intervals (the 12-edo intervals, which are neither red nor blue), expressed in terms of sixth-tones, is always a multiple of three. Because of this amusing coincidence, Mason Green proposes referring to the 33.333-cent sixth-tone interval as a "quark".
 == JI Approximations ==
 === 3-limit (Pythagorean) approximations (same as 12edo): ===
 /2 = 701.955... cents; 21 degrees of 36edo = 700 cents.
@@ Line 372: / Line 371: @@
 === 7-limit approximations: ===
 ==== 7 only: ====
 /4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.
@@ Line 411: / Line 409: @@
 /32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.
-== Rank-2 temperaments ==
+== Regular temperament properties ==
+=== Commas ===
+This is a partial list of the [[commas]] that 36edo [[tempers out]] with its patent [[val]], {{val| 36 57 84 101 125 133 }}.
+{| class="commatable wikitable center-1 center-2 right-4 center-5"
+|-
+! [[Harmonic limit|Prime<br>Limit]]
+! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
+! [[Monzo]]
+! [[Cent]]s
+! [[Color name]]
+! Name(s)
+|-
+| 3
+| [[531441/524288|(12 digits)]]
+| {{monzo| -19 12 }}
+| 23.46
+| Lalawa
+| [[Pythagorean comma]]
+|-
+| 5
+| [[648/625]]
+| {{monzo| 3 4 -4 }}
+| 62.57
+| Quadgu
+| Greater diesis, diminished domma
+|-
+| 5
+| [[128/125]]
+| {{monzo| 7 0 -3 }}
+| 41.06
+| Trigu
+| Lesser diesis, augmented comma
+|-
+| 5
+| [[81/80]]
+| {{monzo| -4 4 -1 }}
+| 21.51
+| Gu
+| Syntonic comma, Didymus comma, meantone comma
+|-
+| 5
+| [[2048/2025]]
+| {{monzo| 11 -4 -2 }}
+| 19.55
+| Sagugu
+| Diaschisma
+|-
+| 5
+| [[67108864/66430125|(16 digits)]]
+| {{monzo| 26 -12 -3 }}
+| 17.60
+| Sasa-trigu
+| [[Misty comma]]
+|-
+| 5
+| [[32805/32768]]
+| {{monzo| -15 8 1 }}
+| 1.95
+| Layo
+| Schisma
+|-
+| 5
+| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
+| {{monzo| 161 -84 -12 }}
+| 0.02
+| Sepbisa-quadbigu
+| [[Kirnberger's atom]]
+|-
+| 7
+| [[1029/1000]]
+| {{monzo| -3 1 -3 3 }}
+| 49.49
+| Trizogu
+| Keega
+|-
+| 7
+| [[686/675]]
+| {{monzo| 1 -3 -2 3 }}
+| 27.99
+| Trizo-agugu
+| Senga
+|-
+| 7
+| [[1728/1715]]
+| {{monzo| 6 3 -1 -3 }}
+| 13.07
+| Triru-agu
+| Orwellisma
+|-
+| 7
+| [[1029/1024]]
+| {{monzo| -10 1 0 3 }}
+| 8.43
+| Latrizo
+| Gamelisma
+|-
+| 7
+| 118098/117649
+| {{monzo| 1 10 0 -6 }}
+| 6.59
+|
+|
+|-
+| 7
+| [[10976/10935]]
+| {{monzo| 5 -7 -1 3 }}
+| 6.48
+| Satrizo-agu
+| Hemimage comma
+|-
+| 7
+| <abbr title="40353607/40310784">(16 digits)</abbr>
+| {{monzo| -11 -9 0 9 }}
+| 1.84
+| Tritrizo
+| [[Septimal ennealimma]]
+|}
+=== Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
 |-