988edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|988}}
{{ED intro}}


== Theory ==
== Theory ==
988edo provides excellent tuning for the 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, and 59th harmonics, making a strong higher-limit system. It is double the famous [[494edo]], and with the same mapping for the 17-limit. If considered in the 19-limit, it is basically a spicy 494edo with the 19th harmonic. The comma basis for such regular temperament is 1445/1444, 1716/1715, 2601/2600, 3025/3024, 4225/4224, 10830/10829, 297440/297381.  
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. A [[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.  


An alternate mapping for 17 would be the 988g val. In it, it tempers out 2025/2023, 13013/13005, 15625/15606, 31213/31212.  
In addition, in the 988ccd val provides a tuning that is extremely close to the [[POTE tuning]] for [[quadritikleismic]] temperament in the 7-limit.  


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}}


In the 2.5.11.13.19.41.47 it supports a 988 & [[2016edo|2016]] temperament.<!-- why is it notable? -->
=== 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 2.5.11.13.29.31 it supports period-52 temperament called [[french deck]].
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.


=== Prime harmonics ===
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}.
{{Harmonics in equal|988}}
 
=== Subsets and supersets ===
Since 988 factors into {{factorization|988}}, 988edo has subset edos {{EDOs| 2, 4, 13, 19, 26, 38, 52, 76, 247, and 494 }}.


[[Category:Equal divisions of the octave|###]]
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 ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 17-limit temperaments supported by 494edo are not included.
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
per octave
!Generator
(reduced)
!Cents
(reduced)
!Associated
ratio
!Temperaments
|-
|-
|52
! Periods<br />per 8ve
|325\988
! Generator*
(2\988)
! Cents*
|394.736
! Associated<br />ratio*
(2.429)
! Temperaments
|134560000/107132311
|-
| 4
| 261\988<br />(14\988)
| 317.004<br />(17.004)
| 6/5<br />(126/125)
| [[Quadritikleismic]] (988ccd)
|-
| 19
| 141\988<br />(37\988)
| 171.255<br />(44.939)
| 6545/5928<br />(?)
| [[Kalium]]
|-
| 52
| 325\988<br />(2\988)
| 394.736<br />(2.429)
| 134560000/107132311<br />(?)
| [[French deck]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Music ==
; [[birdshite stalactite]]
* "clagworks" from ''clagworks / it's probably gout'' (2024) &ndash; [https://open.spotify.com/track/1Abk4KcVUHoRkKxYNSYm0F Spotify] | [https://birdshitestalactite.bandcamp.com/track/clagworks Bandcamp] | [https://www.youtube.com/watch?v=S0zS0rYtT2Y YouTube]
 
; [[Eliora]]
* [https://www.youtube.com/watch?v=c7BW2xnQBb4 ''Alien ethnic motive in 13edo and 12rdo''] (2023)


(?)
[[Category:Listen]]
|[[French deck]]
|}<!-- 3-digit number -->

Latest revision as of 13:32, 13 March 2026

← 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

988 equal divisions of the octave (abbreviated 988edo or 988ed2), also called 988-tone equal temperament (988tet) or 988 equal temperament (988et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 988 equal parts of about 1.21 ¢ each. Each step represents a frequency ratio of 21/988, or the 988th root of 2.

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. A 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 provides a tuning that is extremely close to the POTE 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

Since 988 factors into 22 × 13 × 19, 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.

Table of rank-2 temperaments by generator
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 distinct

Music

birdshite stalactite
Eliora
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