1700edo: Difference between revisions
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{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1700edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the [[harmonic]] [[3/1|3]]. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 [[subgroup]], or if the harmonic [[5/1|5]] is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 [[subgroup]] (not including either 5 or 15). Nonetheless, it tunes the 323 & 2023 temperament [[leaves]] in the 17-limit on the [[patent val]]. | 1700edo is only [[consistent]] in the [[5-odd-limit]], and there is a large relative delta on the [[harmonic]] [[3/1|3]]. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 [[subgroup]], or if the harmonic [[5/1|5]] is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 [[subgroup]] (not including either 5 or 15). Nonetheless, it tunes the {{nowrap|323 & 2023}} temperament [[leaves]] in the 17-limit on the [[patent val]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 16: | Line 16: | ||
=== Rank-2 temperaments=== | === Rank-2 temperaments=== | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br>per 8ve | ! Periods<br />per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| 17 | | 17 | ||
| 121\1700<br>(21\1700) | | 121\1700<br />(21\1700) | ||
| 85.412<br>(14.824) | | 85.412<br />(14.824) | ||
| 1024/975<br>(8192/8125) | | 1024/975<br />(8192/8125) | ||
| [[Leaves]] | | [[Leaves]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Latest revision as of 12:33, 21 February 2025
| ← 1699edo | 1700edo | 1701edo → |
1700 equal divisions of the octave (abbreviated 1700edo or 1700ed2), also called 1700-tone equal temperament (1700tet) or 1700 equal temperament (1700et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1700 equal parts of about 0.706 ¢ each. Each step represents a frequency ratio of 21/1700, or the 1700th root of 2.
Theory
1700edo is only consistent in the 5-odd-limit, and there is a large relative delta on the harmonic 3. It has a reasonable approximation to the 2.9.15.21.11.13.17.23 subgroup, or if the harmonic 5 is desired, the 2.9.5.21.11.23 subgroup. Otherwise, it can be considered in the 2.9.21.11.23.31 subgroup (not including either 5 or 15). Nonetheless, it tunes the 323 & 2023 temperament leaves in the 17-limit on the patent val.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.308 | -0.196 | +0.351 | +0.090 | -0.024 | +0.178 | +0.202 | +0.221 | -0.337 | +0.043 | -0.039 |
| Relative (%) | -43.6 | -27.8 | +49.7 | +12.7 | -3.4 | +25.2 | +28.6 | +31.3 | -47.7 | +6.0 | -5.5 | |
| Steps (reduced) |
2694 (994) |
3947 (547) |
4773 (1373) |
5389 (289) |
5881 (781) |
6291 (1191) |
6642 (1542) |
6949 (149) |
7221 (421) |
7467 (667) |
7690 (890) | |
Subsets and supersets
Since 1700 factors into 22 × 52 × 17, 1700edo has subset edos 2, 4, 5, 10, 17, 20, 25, 34, 50, 68, 85, 100, 170, 340, 425, and 850.
One step of 1700edo is the relative cent for 17edo. It has been named iota by Margo Schulter and George Secor.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 17 | 121\1700 (21\1700) |
85.412 (14.824) |
1024/975 (8192/8125) |
Leaves |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct