1200edo: Difference between revisions

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~~The '''1200 edo''' divides the octave in 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]].

~~{{primes in edo|1200|columns=15}}~~

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.

== 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 ~~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 which offers detuning by cents~~.

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.

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

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 ===

=== 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" | [[Comma list]]

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5.7

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

| {{mapping| 1200 1902 2786 3369 }}

| +0.0112

| 0.0748

| 7.48

|-

| 2.3.5.7.11

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

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

| +0.0273

| 0.0743

| 7.43

|}

== 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: @@
-The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.
+{{Infobox ET}}
+{{ED intro}}
-==Theory==
+edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].
-{{primes in edo|1200|columns=15}}
-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.
+== 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 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 which offers detuning by cents.
+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.
-[[Category:Equal divisions of the octave]]
+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}}
+=== 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" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3.5.7
+| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
+| {{mapping| 1200 1902 2786 3369 }}
+| +0.0112
+| 0.0748
+| 7.48
+|-
+| 2.3.5.7.11
+| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
+| {{mapping| 1200 1902 2786 3369 4151 }}
+| +0.0273
+| 0.0743
+| 7.43
+|}
+== 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]]