353edo
| ← 352edo | 353edo | 354edo → |
353 equal divisions of the octave (abbreviated 353edo or 353ed2), also called 353-tone equal temperament (353tet) or 353 equal temperament (353et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 353 equal parts of about 3.4 ¢ each. Each step represents a frequency ratio of 21/353, or the 353rd root of 2.
Theory
353edo is inconsistent in the 5-odd-limit and harmonic 3 is about halfway between its steps. It is suitable for use with the 2.9.15.7.11.13.17.23.29.31.37 subgroup. This makes 353edo an "upside-down" edo – poor approximation of the low harmonics, but an improvement over the high ones. Nonetheless, it provides the optimal patent val for didacus, the 2.5.7 subgroup temperament tempering out 3136/3125.
Using the patent val nonetheless, 353edo supports apparatus, marvo and zarvo.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -1.67 | +1.22 | +0.01 | +0.06 | -0.61 | -0.87 | -0.45 | +0.43 | +1.64 | -1.66 | +0.62 |
| relative (%) | -49 | +36 | +0 | +2 | -18 | -26 | -13 | +13 | +48 | -49 | +18 | |
| Steps (reduced) |
559 (206) |
820 (114) |
991 (285) |
1119 (60) |
1221 (162) |
1306 (247) |
1379 (320) |
1443 (31) |
1500 (88) |
1550 (138) |
1597 (185) | |
Subsets and supersets
353edo is the 71st prime edo.
Miscellaneous properties
Eliora associates 353edo with a reformed Hebrew calendar. In the original Hebrew calendar, years number 3, 6, 8, 11, 14, 17, and 19 within a 19-year pattern (makhzor (מחזור), plural: makhzorim) are leap. When converted to 19edo, this results in 5L 2s mode, and simply the diatonic major scale. Following this logic, a temperament (→ rectified hebrew) can be constructed for the Rectified Hebrew calendar. The 11-step perfect fifth in this scale becomes 209\353, and it corresponds to 98/65, which is sharp of 3/2 by 196/195.
In addition, every sub-pattern in a 19-note generator is actually a Hebrew makhzor, that is a mini-19edo on its own, until it is truncated to an 11-note pattern. Just as the original calendar reform consists of 18 makhzorim with 1 hendecaeteris, Hebrew[130] scale consists of a stack of naively 18 "major scales" finished with one 11-edo tetratonic.
The number 353 in this version of the Hebrew calendar must not be confused with the number of days in shanah chaserah (שנה חסרה), the deficient year.
It is possible to use a superpyth-ish fifth of Rectified Hebrew fifth, 209\353, as a generator. In this case, 76 & 353 temperament is obtained. In the 2.5.7.13 subgroup, this results in the fifth being equal to 98/65 and the comma basis of 10985/10976, [-103 0 -38 51 0 13⟩.
Table of intervals
| Step | Note name
(diatonic Hebrew[19] version) |
Associated ratio
(2.5.7.13 subgroup) |
|---|---|---|
| 0 | C | 1/1 |
| 1 | C-C# | |
| 2 | C-Db | |
| 3 | C-D | 196/195 |
| 4 | C-D# | |
| 19 | C# | 26/25 |
| 38 | Db | 14/13 |
| 41 | Db-D | 13/12 |
| 46 | Db-F | 35/32 |
| 57 | D | |
| 76 | D# | |
| 95 | Eb | |
| 114 | E | 5/4 |
| 133 | E# | 13/10 I (patent val approximation) |
| 134 | E#-C# | 13/10 II (direct approximation) |
| 152 | F | |
| 171 | F# | 7/5 |
| 190 | Gb | |
| 206 | Gb-Bb | 3/2 |
| 209 | G | 98/65 |
| 228 | G# | |
| 247 | Ab | 13/8 |
| 266 | A | |
| 285 | A# | 7/4 |
| 304 | Bb | |
| 323 | B | |
| 342 | B#/Cb | |
| 353 | C | 2/1 |
Regular temperament properties
Assuming 353edo is treated as the 2.5.7.11.13.17 subgroup temperament.
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.5 | [820 -353⟩ | [⟨353 820]] | -0.263 | 0.263 | 7.74 |
| 2.5.7 | 3136/3125, [209 -9 -67⟩ | [⟨353 820 991]] | -0.177 | 0.247 | 7.26 |
| 2.5.7.11 | 3136/3125, 5767168/5764801, [-20 -6 1 9⟩ | [⟨353 820 991 1221]] | -0.089 | 0.263 | 7.73 |
| 2.5.7.11.13 | 3136/3125, 4394/4375, 6656/6655, 5767168/5764801 | [⟨353 820 991 1221 1306]] | -0.024 | 0.268 | 7.89 |
| 2.5.7.11.13.17 | 3136/3125, 4394/4375, 7744/7735, 60112/60025, 64141/64000 | [⟨353 820 991 1221 1306 1443]] | -0.037 | 0.247 | 7.26 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 19\353 | 64.59 | 26/25 | Rectified hebrew |
| 1 | 34\353 | 115.58 | 77/72 | Apparatus |
| 1 | 152\353 | 516.71 | 27/20 | Marvo (353c) / zarvo (353cd) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
- RectifiedHebrew[19] - 18L 1s
- RectifiedHebrew[130] - 93L 37s
- Austro-Hungarian Minor[9] - 57 38 38 38 38 38 38 38 30
See also
Music
- Snow On My City (2022) - cover of Naomi Shemer in Rectified Hebrew and apparatus
- Bottom Text (2022) in Rectified Hebrew