1200edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 24 × 3 × 52/sup>~~

~~| Step size = 1¢by definition~~

~~| Fifth = 702\1200 = 702¢ (→[[200edo|117\200]])~~

~~| Major 2nd = 204\1200 = 204¢~~

}}

The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==

1200edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].

== Theory ==

1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]].

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 & 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 & 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200.

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 & 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system.

=== Prime harmonics ===

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

~~1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals~~ of ~~the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]]~~ {{~~val~~| ~~1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[~~5~~-limit]]~~, ~~having the same mapping as 600edo. In the [[7-limit]]~~, ~~it tempers out 2460375/2458624 and 95703125/95551488~~, ~~leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200~~, ~~also supported by [[171edo]]. In the 11-limit~~, ~~it tempers out 9801/9800~~, ~~234375/234256 and 825000/823543~~, ~~leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200~~, ~~also supported by [[494edo]]. In the 7-limit~~, ~~it provides a val~~, ~~1200ccd~~, ~~which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368~~ }}. ~~It also provides~~ the ~~optimal patent val for the 224&752 temperament tempering out 2200/2197, 4096/4095~~, ~~9801/9800~~ and ~~35750/35721~~.

=== Subsets and supersets ===

The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==

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

! rowspan="2" | Subgroup

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

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

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

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

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

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

Line 26:

Line 32:

| 2.3.5.7

| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}

| [{{~~val~~| 1200 1902 2786 3369 }}]

| {{mapping| 1200 1902 2786 3369 }}

| +0.0112

| 0.0748

Line 33:

Line 39:

| 2.3.5.7.11

| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472

| [{{~~val~~| 1200 1902 2786 3369 4151 }}]

| {{mapping| 1200 1902 2786 3369 4151 }}

| +0.0273

| 0.0743

| 7.43

|-

~~| 2.3.7.17.31.41.47~~

~~| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416~~

~~| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]~~

~~| -0.0244~~

~~| 0.0351~~

~~| 3.51~~

|}

[[Category:~~Equal divisions of the octave~~]]

== Music ==

; [[Hideya]]

* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)

; [[Sevish]]

* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>4</sup> × 3 × 5<sup>2/sup>
+{{ED intro}}
-| Step size = 1¢<sup>by definition</sup>
-| Fifth = 702\1200 = 702¢ (→[[200edo|117\200]])
-| Major 2nd = 204\1200 = 204¢
-}}
-The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.
-==Theory==
+edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].
+== Theory ==
+edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]].
+The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 & 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 & 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200.
+It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.
+Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system.
+=== Prime harmonics ===
 {{Harmonics in equal|1200}}
-Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.
-edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.
+=== Subsets and supersets ===
+The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! 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
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 26: / Line 32: @@
 | 2.3.5.7
 | 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
-| [{{val| 1200 1902 2786 3369 }}]
+| {{mapping| 1200 1902 2786 3369 }}
 | +0.0112
 | 0.0748
@@ Line 33: / Line 39: @@
 | 2.3.5.7.11
 | 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
-| [{{val| 1200 1902 2786 3369 4151 }}]
+| {{mapping| 1200 1902 2786 3369 4151 }}
 | +0.0273
 | 0.0743
 | 7.43
-|-
-| 2.3.7.17.31.41.47
-| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
-| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
-| -0.0244
-| 0.0351
-| 3.51
 |}
-[[Category:Equal divisions of the octave]]
+== Music ==
+; [[Hideya]]
+* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)
+; [[Sevish]]
+* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)
+[[Category:Listen]]