# 22L 1s

←21L 1s | 22L 1s | 23L 1s→ |

↙21L 2s | ↓22L 2s | 23L 2s↘ |

**Equal tunings**22L 1s is the scale that is most commonly produced by stacking the interval of 33/32.

A name **quartismoid** is proposed for this pattern since its harmonic entropy minimum corresponds to tempering out the quartisma - five 33/32s being equated with 7/6. In addition, both 22edo and 23edo, extreme ranges of the MOS temper out the quartisma, as well as a large portion of EDOs up to 100-200 which have this scale.

## Tuning ranges

### Mavila fifth and 91edo (Ultrasoft and supersoft)

Between 4\91 and 1\23, 13 steps amount to a pelog / mavila fifth, which corresponds to the ultrasoft step ratio range. In 91edo, the fifth produced by 13 steps of the quartismoid scale is the same as 4 steps of 7edo, and thus is the exact boundary between mavila and diatonic.

### Diatonic fifth (hard of supersoft)

From 1\22 to 4\91, 13 steps amount to a diatonic fifth.

If the pure 33/32 is used as a generator, the resulting fifth is 692.54826 cents, which puts it in the category around flattone.

#### 700-cent, just, and superpyth fifths (step ratio 7:2 and harder)

In 156edo, the fifth becomes the 12edo 700-cent fifth. In 200edo, the fifth comes incredibly close to just, as the number 200 is a semiconvergent denominator to the approximation of log2(3/2).

When the step ratio is greater than 4.472, then 13 generators amount to a superpyth fifth and the tuning approaches 22edo.

## Relation to other equal divisions

6 steps act as a pseudo-6/5, and when they actually act as 6/5 along with 5 steps being equal to 7/6, 385/384 is tempered out. If one were to instead tune in favour of 6/5 instead of 7/6, the resulting hardness would be around 1.233. 114edo and 137edo represent this the best.

## Modes

Eliora proposes naming the brightest mode Alpharabian, after the fact that 33/32 is called Al-Farabi quarter-tone, and the rest after Tarot Major Arcana adjectivals based on how many generators down there is.

Mode | Name |
---|---|

22|0 | Alpharabian |

21|1 | Magical |

20|2 | High Priestess's |

19|3 | Empress's |

... | ... |

2|20 | Judgemental |

1|21 | Worldwide |

0|22 | Foolish |

## Scale tree

Generator | L | s | L/s | Comments | |||||
---|---|---|---|---|---|---|---|---|---|

1\23 | 1 | 1 | 1.000 | ||||||

6\137 | 6 | 5 | 1.200 | ||||||

5\114 | 5 | 4 | 1.250 | ||||||

9\205 | 9 | 7 | 1.286 | ||||||

4\91 | 4 | 3 | 1.333 | 13 steps adding to lower bound of diatonic fifths (685.71c) is here | |||||

11\250 | 11 | 8 | 1.375 | ||||||

7\159 | 7 | 5 | 1.400 | ||||||

10\227 | 10 | 7 | 1.428 | ||||||

3\68 | 3 | 2 | 1.500 | Stretched 23edo is in this range | |||||

11\249 | 11 | 7 | 1.571 | ||||||

8\181 | 8 | 5 | 1.600 | ||||||

13\294 | 13 | 8 | 1.625 | ||||||

5\113 | 5 | 3 | 1.667 | ||||||

12\271 | 12 | 7 | 1.714 | ||||||

7\158 | 7 | 4 | 1.750 | ||||||

9\203 | 9 | 5 | 1.800 | ||||||

2\45 | 2 | 1 | 2.000 | Basic quartismoid | |||||

9\202 | 9 | 4 | 2.250 | ||||||

7\157 | 7 | 3 | 2.333 | ||||||

12\269 | 12 | 5 | 2.400 | ||||||

5\112 | 5 | 2 | 2.500 | 13 steps adding to 1/4 comma meantone fifth is around here | |||||

13\291 | 13 | 5 | 2.600 | ||||||

8\179 | 8 | 3 | 2.667 | ||||||

11\246 | 11 | 4 | 2.750 | ||||||

3\67 | 3 | 1 | 3.000 | ||||||

10\223 | 10 | 3 | 3.333 | ||||||

7\156 | 7 | 2 | 3.500 | 13 steps adding to a 700 cent fifth is here | |||||

11\245 | 11 | 3 | 3.667 | ||||||

4\89 | 4 | 1 | 4.000 | ||||||

9\200 | 9 | 2 | 4.500 | 13 steps adding to 3/2 perfect fifth is around here | |||||

5\111 | 5 | 1 | 5.000 | ||||||

6\133 | 6 | 1 | 6.000 | ||||||

1\22 | 1 | 0 | → inf |