988edo: Difference between revisions

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== Theory ==
== Theory ==
988edo provides excellent tuning for the 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, 59th harmonics and a reasonable tuning for 23, 29, 31, 41st harmonics, making a strong higher-limit system. In lower limits, it is [[enfactored]] in the [[17-limit]], with the same tuning as [[494edo]], which is notable for being a zeta edo. If considered in the 19-limit, it provides a good correction for the 19th harmonic over 494edo. The comma basis for 988edo in the 19-limit is {1156/1155, 1275/1274, 1445/1444, 1716/1715, 2080/2079, 2431/2430, 4096/4095}.
988edo is [[enfactoring|enfactored]] in the [[17-limit]], with the same tuning as [[494edo]], which is notable for being a zeta edo. If considered in the 19-limit, it provides a good correction for the 19th harmonic over 494edo. The comma basis for 988edo in the 19-limit is {[[1156/1155]], [[1275/1274]], [[1445/1444]], [[1716/1715]], [[2080/2079]], [[2431/2430]], [[4096/4095]]}. An alternate mapping for 17 would be the 988g val, where it tempers out [[2025/2023]], 13013/13005, 15625/15606, 31213/31212. In addition, in the 988ccd val it is a tuning for [[quadritikleismic]] temperament in the 7-limit.
 
=== Prime harmonics ===
{{Harmonics in equal|988|columns=11}}


An alternate mapping for 17 would be the 988g val, where it tempers out 2025/2023, 13013/13005, 15625/15606, 31213/31212. In addition, in the 988ccd val it is a tuning for [[quadritikleismic]] temperament in the 7-limit.
=== Higher limits ===
=== Higher limits ===
988edo provides excellent approximations for harmonics 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, and 59, and reasonable approximations for harmonics 23, 29, 31, and 41, making it a strong higher-limit system.
In the 2.5.11.13.19.29.31 it supports period-52 temperament called [[french deck]], with the tempering out of [[6656/6655]] inherited from 494edo.
In the 2.5.11.13.19.29.31 it supports period-52 temperament called [[french deck]], with the tempering out of [[6656/6655]] inherited from 494edo.


988edo is similar to [[2016edo]] in the fact that both tune well the 2.5.11.13.19.41.47 subgroup. The result is the 988 & 2016 temperament, which reaches [[13/8]] in four generators and has a comma basis {7943/7942, 322465/322373, 16777475/16777216, 22151168/22150865, 12998046875/12994428928}.
988edo is similar to [[2016edo]] in the fact that both tune well the 2.5.11.13.19.41.47 subgroup. The result is the 988 & 2016 temperament, which reaches [[13/8]] in four generators and has a comma basis {7943/7942, 322465/322373, 16777475/16777216, 22151168/22150865, 12998046875/12994428928}.
=== Subsets and supersets ===
=== Subsets and supersets ===
988edo has subset edos {{EDOs|1, 2, 4, 13, 19, 26, 38, 52, 76, 247, 494}}.
988edo has subset edos {{EDOs| 2, 4, 13, 19, 26, 38, 52, 76, 247, and 494 }}.


One step of 988edo is named ''semisqub'', given the strong relation to 494edo and the fact that 1 step of 494edo is called a squb.
One step of 988edo is named ''semisqub'', given the strong relation to 494edo and the fact that 1 step of 494edo is called a squb.
=== Prime harmonics ===
{{Harmonics in equal|988|columns=11}}


== Regular temperament properties ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: temperaments represented by 494edo are not included.
Note: 17-limit temperaments supported by 494edo are not included.
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
Line 44: Line 46:
| 394.736<br>(2.429)
| 394.736<br>(2.429)
| 134560000/107132311<br>(?)
| 134560000/107132311<br>(?)
|[[French deck]]
| [[French deck]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=c7BW2xnQBb4 Alien ethnic motive in 13edo and 12rdo] by [[Eliora]]
; [[Eliora]]
 
* [https://www.youtube.com/watch?v=c7BW2xnQBb4 ''Alien ethnic motive in 13edo and 12rdo''] (2023)
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 15:50, 19 October 2023

← 987edo 988edo 989edo →
Prime factorization 22 × 13 × 19
Step size 1.21457 ¢ 
Fifth 578\988 (702.024 ¢) (→ 289\494)
Semitones (A1:m2) 94:74 (114.2 ¢ : 89.88 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

Theory

988edo is enfactored in the 17-limit, with the same tuning as 494edo, which is notable for being a zeta edo. If considered in the 19-limit, it provides a good correction for the 19th harmonic over 494edo. The comma basis for 988edo in the 19-limit is {1156/1155, 1275/1274, 1445/1444, 1716/1715, 2080/2079, 2431/2430, 4096/4095}. An alternate mapping for 17 would be the 988g val, where it tempers out 2025/2023, 13013/13005, 15625/15606, 31213/31212. In addition, in the 988ccd val it is a tuning for quadritikleismic temperament in the 7-limit.

Prime harmonics

Approximation of prime harmonics in 988edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.069 -0.079 +0.405 +0.099 -0.042 -0.502 +0.058 -0.339 +0.382 +0.309
Relative (%) +0.0 +5.7 -6.5 +33.3 +8.2 -3.4 -41.3 +4.8 -27.9 +31.5 +25.4
Steps
(reduced) 		988
(0) 		1566
(578) 		2294
(318) 		2774
(798) 		3418
(454) 		3656
(692) 		4038
(86) 		4197
(245) 		4469
(517) 		4800
(848) 		4895
(943)

Higher limits

988edo provides excellent approximations for harmonics 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, and 59, and reasonable approximations for harmonics 23, 29, 31, and 41, making it a strong higher-limit system.

In the 2.5.11.13.19.29.31 it supports period-52 temperament called french deck, with the tempering out of 6656/6655 inherited from 494edo.

988edo is similar to 2016edo in the fact that both tune well the 2.5.11.13.19.41.47 subgroup. The result is the 988 & 2016 temperament, which reaches 13/8 in four generators and has a comma basis {7943/7942, 322465/322373, 16777475/16777216, 22151168/22150865, 12998046875/12994428928}.

Subsets and supersets

988edo has subset edos 2, 4, 13, 19, 26, 38, 52, 76, 247, and 494.

One step of 988edo is named semisqub, given the strong relation to 494edo and the fact that 1 step of 494edo is called a squb.

Regular temperament properties

Rank-2 temperaments

Note: 17-limit temperaments supported by 494edo are not included.

Periods
per 8ve 		Generator* Cents* Associated
Ratio 		Temperaments
4 261\988
(14\988) 		317.004
(17.004) 		6/5
(126/125) 		Quadritikleismic (988ccd)
19 141\988
(37\988) 		171.255
(44.939) 		6545/5928
(?) 		Kalium
52 325\988
(2\988) 		394.736
(2.429) 		134560000/107132311
(?) 		French deck

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Eliora
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