# 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 2^{1/353}, 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 (%) | -49.2 | +35.9 | +0.4 | +1.6 | -17.9 | -25.5 | -13.2 | +12.6 | +48.2 | -48.8 | +18.3 | |

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