9edt: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>JosephRuhf **Imported revision 591725430 - Original comment: ** |
m Removing from Category:Edt using Cat-a-lot |
||
| (21 intermediate revisions by 11 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6{{c}} flat. However, the [[5/1]] is 39{{c}} flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845{{c}}, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]]. | |||
Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music. | |||
9edt is the third [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]]. | |||
=== Relation to edos === | |||
9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]]. | |||
=== Harmonics === | |||
{{Harmonics in equal|9|3|1|}} | |||
{{Harmonics in equal|9|3|1|intervals=prime}} | |||
{| class="wikitable" | |||
|- | |||
! rowspan="2" | Steps | |||
! colspan="2" | Size | |||
! rowspan="2" | Comparable intervals (¢) | |||
|- | |||
! Cents | |||
! [[Hekt]]s | |||
0 | |- | ||
1 | ! colspan="3" | 0 | ||
2 | | [[1/1]] | ||
3 | |- | ||
4 | | 1 | ||
5 | | 211.328 | ||
6 | | 144.444 | ||
7 | | [[9/8]] (204) | ||
8 | |- | ||
9 | | 2 | ||
| 422.657 | |||
| 288.889 | |||
| [[9/7]] (435) | |||
|- | |||
| 3 | |||
| 633.985 | |||
| 433.333 | |||
| [[13/9]] (637) | |||
|- | |||
| 4 | |||
| 845.313 | |||
| 577.778 | |||
| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814) | |||
|- | |||
| 5 | |||
| 1056.642 | |||
| 722.222 | |||
| [[9/5]] (1018), [[11/6]] (1049) | |||
|- | |||
| 6 | |||
| 1267.970 | |||
| 866.667 | |||
| [[27/13]] (1265) | |||
|- | |||
| 7 | |||
| 1479.298 | |||
| 1011.111 | |||
| [[7/3]] (1467) | |||
|- | |||
| 8 | |||
| 1690.627 | |||
| 1155.556 | |||
| [[8/3]] (1698) | |||
|- | |||
| 9 | |||
| 1901.955 | |||
| 1300 | |||
| [[3/1]] | |||
|} | |||
== Music == | |||
* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]] | |||
[[Category:Macrotonal]] | |||
Latest revision as of 15:31, 31 July 2025
| ← 8edt | 9edt | 10edt → |
9 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 9edt or 9ed3), is a nonoctave tuning system that divides the interval of 3/1 into 9 equal parts of about 211 ¢ each. Each step represents a frequency ratio of 31/9, or the 9th root of 3.
Theory
It has a decent seventh harmonic (7/1) which is 12.4¢ sharp, and an excellent 13/1 inherited from 3edt which is only 2.6 ¢ flat. However, the 5/1 is 39 ¢ flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 ¢, which is between 8/5 and 5/3; if this interval is also taken as an approximation to 13/8, it would temper out 40/39—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 subgroup, of which it tempers out 351/343 and 2197/2187.
Following 4edt, this is the next edt that supports BPS temperament. For small edts, this property is virtually the same as supporting a 3/1-equivalent "lambda" scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to 7edo in diatonic (5L 2s) music.
9edt is the third no-twos zeta peak edt.
Relation to edos
9edt is related to 17edo, by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to 3/1.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68.0 | +0.0 | -75.4 | -39.0 | +68.0 | +12.4 | -7.4 | +0.0 | +28.9 | +75.2 | -75.4 |
| Relative (%) | +32.2 | +0.0 | -35.7 | -18.5 | +32.2 | +5.9 | -3.5 | +0.0 | +13.7 | +35.6 | -35.7 | |
| Steps (reduced) |
6 (6) |
9 (0) |
11 (2) |
13 (4) |
15 (6) |
16 (7) |
17 (8) |
18 (0) |
19 (1) |
20 (2) |
20 (2) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68.0 | +0.0 | -39.0 | +12.4 | +75.2 | -2.6 | -44.4 | -25.6 | +66.3 | +87.6 | -27.8 |
| Relative (%) | +32.2 | +0.0 | -18.5 | +5.9 | +35.6 | -1.2 | -21.0 | -12.1 | +31.4 | +41.5 | -13.2 | |
| Steps (reduced) |
6 (6) |
9 (0) |
13 (4) |
16 (7) |
20 (2) |
21 (3) |
23 (5) |
24 (6) |
26 (8) |
28 (1) |
28 (1) | |
| Steps | Size | Comparable intervals (¢) | |
|---|---|---|---|
| Cents | Hekts | ||
| 0 | 1/1 | ||
| 1 | 211.328 | 144.444 | 9/8 (204) |
| 2 | 422.657 | 288.889 | 9/7 (435) |
| 3 | 633.985 | 433.333 | 13/9 (637) |
| 4 | 845.313 | 577.778 | 13/8 (841), 5/3 (884), 8/5 (814) |
| 5 | 1056.642 | 722.222 | 9/5 (1018), 11/6 (1049) |
| 6 | 1267.970 | 866.667 | 27/13 (1265) |
| 7 | 1479.298 | 1011.111 | 7/3 (1467) |
| 8 | 1690.627 | 1155.556 | 8/3 (1698) |
| 9 | 1901.955 | 1300 | 3/1 |