1200edo: Difference between revisions

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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.
{{Infobox ET}}
{{ED intro}}


==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 ===
{{Harmonics in equal|1200}}
{{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.


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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5.7.11
| 2.3.5.7
|9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
|[{{val|1200 1902 2786 3369 4151 }}]
| {{mapping| 1200 1902 2786 3369 }}
|
| +0.0112
|
| 0.0748
|
| 7.48
|-
|-
|2.3.7.17.31.41.47
| 2.3.5.7.11
|2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
|[{{val|1200 1902 3369 4905 5945 6429 6666 }}]
| {{mapping| 1200 1902 2786 3369 4151 }}
| -0.0244
| +0.0273
|0.0351
| 0.0743
|3.51
| 7.43
|}
|}
[[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]]

Latest revision as of 11:20, 11 April 2025

← 1199edo 1200edo 1201edo →
Prime factorization 24 × 3 × 52
Step size 1 ¢ 
Fifth 702\1200 (702 ¢) (→ 117\200)
Semitones (A1:m2) 114:90 (114 ¢ : 90 ¢)
Consistency limit 11
Distinct consistency limit 11

1200 equal divisions of the octave (abbreviated 1200edo or 1200ed2), also called 1200-tone equal temperament (1200tet) or 1200 equal temperament (1200et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1200 equal parts of exactly 1 ¢ each. Each step represents a frequency ratio of 21/1200, or the 1200th root of 2.

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

Theory

1200edo is 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 ⟨1200 1902 2786 3369 4151]. It is enfactored in the 5-limit, having the same tuning as 600edo.

The equal temperament 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: ⟨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

Approximation of prime harmonics in 1200edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.045 -0.314 +0.174 -0.318 +0.472 +0.045 +0.487 -0.274 +0.423 -0.036
Relative (%) +0.0 +4.5 -31.4 +17.4 -31.8 +47.2 +4.5 +48.7 -27.4 +42.3 -3.6
Steps
(reduced) 		1200
(0) 		1902
(702) 		2786
(386) 		3369
(969) 		4151
(551) 		4441
(841) 		4905
(105) 		5098
(298) 		5428
(628) 		5830
(1030) 		5945
(1145)

Subsets and supersets

The nontrivial divisors of 1200 are 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

Subgroup Comma list Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5.7 2460375/2458624, 95703125/95551488, [36 -5 0 -10⟩ [⟨1200 1902 2786 3369]] +0.0112 0.0748 7.48
2.3.5.7.11 9801/9800, 234375/234256, 825000/823543, 1771561/1769472 [⟨1200 1902 2786 3369 4151]] +0.0273 0.0743 7.43

Music

Hideya
Sevish
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