988edo: Difference between revisions
→Theory: about the notablity of 988 & 2016... I also know the year 988 as the year of baptism of Kyiv Rus so I just casually stumbled upon 988 & 2016 temperament while composing, i think it's interesting |
→Theory: +quadritikleismic |
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988edo provides excellent tuning for the 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, 59th harmonics and a reasonable tuning for 23, 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 provides excellent tuning for the 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, 59th harmonics and a reasonable tuning for 23, 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}. | ||
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, 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 === | ||
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. | ||
| Line 17: | Line 17: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Note: temperaments represented 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 | ||
| Line 23: | Line 24: | ||
! Associated<br>Ratio | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |||
|1 | |||
|261\988 | |||
|317.004 | |||
|6/5 | |||
|[[Quadritikleismic]] (988ccd) | |||
|- | |- | ||
| 52 | | 52 | ||
| Line 28: | Line 35: | ||
| 394.736<br>(2.429) | | 394.736<br>(2.429) | ||
| 134560000/107132311<br>(?) | | 134560000/107132311<br>(?) | ||
| [[French deck]] | |[[French deck]] | ||
|} | |} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 20:58, 27 February 2023
| ← 987edo | 988edo | 989edo → |
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, 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}.
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
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}.
As an interval size measure
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
| 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) | |
Regular temperament properties
Rank-2 temperaments
Note: temperaments represented by 494edo are not included.
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 261\988 | 317.004 | 6/5 | Quadritikleismic (988ccd) |
| 52 | 325\988 (2\988) |
394.736 (2.429) |
134560000/107132311 (?) |
French deck |