# Pajara

**Pajara** (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a temperament with a half-octave period that represents both 7/5 and 10/7, so 50/49 is tempered out and it is in the jubilismic clan. The generator is in the neighborhood of 105-110 cents, so that period + generator represents 3/2. Period minus 2 generators is 5/4, which, if you work it out, implies that 2048/2025 is tempered out, so pajara is also in the diaschismic family. Finally, two 4/3s (or a 2/1 minus two generators) represents 7/4 as well as 16/9, so 64/63 is tempered out and pajara is in the Archytas clan. Tempering out any two of these commas (among others) produces the unique temperament, pajara.

The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by Paul Erlich and Gene Ward Smith. They are often thought of as subsets of 22edo, without much loss of generality and accuracy.

## Interval chains

There are two different mappings of the 11-limit. One is just called "pajara" and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. The other, called "pajarous" to avoid confusion, loses some accuracy and there's little reason to use it unless you're using 22edo, which is the intersection of both systems.

**Intervals of pajara (12&22)**

Generator | -11 | -10 | -9 | -8 | -7 | -6 |
---|---|---|---|---|---|---|

Cents* | 24.26 | 131.15 | 238.03 | 344.92 | 451.80 | 558.69 |

Ratios | 11/9 | 11/8 | ||||

Generator | -5 | -4 | -3 | -2 | -1 | 0 |

Cents* | 65.57 | 172.46 | 279.34 | 386.23 | 493.11 | 600.00 |

Ratios | 11/10, 10/9 | 7/6 | 5/4 | 4/3 | 7/5, 10/7 | |

Generator | 0 | 1 | 2 | 3 | 4 | 5 |

Cents* | 0.00 | 106.89 | 213.77 | 320.66 | 427.54 | 534.43 |

Ratios | 1/1 | 16/15, 15/14 | 9/8, 8/7 | 6/5 | 14/11, 9/7 | 15/11 |

Generator | 6 | 7 | 8 | 9 | 10 | 11 |

Cents* | 41.31 | 148.20 | 255.08 | 361.97 | 468.85 | 575.74 |

Ratios | 12/11 |

* in 11-limit POTE tuning

**Intervals of pajarous (10&22)**

Generator | -10 | -9 | -8 | -7 | -6 | |
---|---|---|---|---|---|---|

Cents* | 104.22 | 213.80 | 323.38 | 432.96 | 542.53 | |

Ratios | 14/11 | 15/11 | ||||

Generator | -5 | -4 | -3 | -2 | -1 | 0 |

Cents* | 52.11 | 161.69 | 271.27 | 380.84 | 490.42 | 600.00 |

Ratios | 12/11, 10/9 | 7/6 | 5/4 | 4/3 | 7/5, 10/7 | |

Generator | 0 | 1 | 2 | 3 | 4 | 5 |

Cents* | 0.00 | 109.58 | 219.16 | 328.73 | 438.31 | 547.89 |

Ratios | 1/1 | 16/15, 15/14 | 9/8, 8/7 | 6/5, 11/9 | 9/7 | 11/8 |

Generator | 6 | 7 | 8 | 9 | 10 | |

Cents* | 57.47 | 167.04 | 276.62 | 386.20 | 495.78 | |

Ratios | 11/10 |

* in 11-limit POTE tuning

## MOSes

### 10-note (proper)

See 2L 8s.

The true MOS is called the "symmetric" decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-MOS, LsssLsssss, in which only the 5-step interval violates the "no more than 2 intervals per class" rule, is called the "pentachordal" decatonic, because it consists of two identical "pentachords" plus a split 9/8~8/7 whole tone to complete the octave.

### 12-note (proper)

See 10L 2s.

## Tuning spectrum

Gencom: [7/5 3/2; 50/49 64/63 65/63 99/98]

Gencom mapping: [⟨2 2 7 8 14 5], ⟨0 1 -2 -2 -6 2]]

ET generator |
eigenmonzo (unchanged-interval) |
decatonic seventh (¢) |
comments |
---|---|---|---|

7\12 | 700.000 | ||

4/3 | 701.955 | ||

41\70 | 702.857 | ||

34\58 | 703.448 | ||

61\104 | 703.846 | ||

27\46 | 704.348 | ||

14/11 | 704.377 | ||

10/9 | 704.399 | ||

74\126 | 704.762 | ||

47\80 | 705.000 | ||

114\194 | 705.155 | ||

6/5 | 705.214 | 5 and 15-odd-limit minimax | |

67\114 | 705.263 | ||

87\148 | 705.405 | ||

20\34 | 705.882 | ||

93\158 | 706.329 | ||

73\124 | 706.452 | ||

126\214 | 706.542 | ||

11/9 | 706.574 | ||

53\90 | 706.667 | ||

139\236 | 706.780 | ||

5/4 | 706.843 | 7 and 11-limit POTT | |

86\146 | 706.849 | ||

119\202 | 706.931 | ||

33\56 | 707.143 | ||

12/11 | 707.234 | ||

112\190 | 707.368 | ||

15/11 | 707.390 | ||

79\134 | 707.463 | ||

125\212 | 707.547 | ||

46\78 | 707.692 | ||

105\178 | 707.865 | ||

59\100 | 708.000 | ||

11/8 | 708.114 | ||

72\122 | 708.196 | ||

11/10 | 708.749 | 11-odd-limit minimax | |

9/7 | 708.771 | ||

13\22 | 709.091 | ||

58\98 | 710.204 | ||

45\76 | 710.526 | ||

122\206 | 710.680 | ||

77\130 | 710.769 | ||

109\184 | 710.870 | ||

7/6 | 711.043 | 7-odd-limit minimax | |

32\54 | 711.111 | ||

13/11 | 711.151 | 13-odd-limit minimax | |

83\140 | 711.429 | ||

51\86 | 711.628 | ||

16/15 | 711.731 | ||

70\118 | 711.864 | ||

19\32 | 712.500 | ||

44\74 | 713.5135 | ||

13/10 | 713.553 | ||

25\42 | 714.286 | ||

31\52 | 715.385 | ||

8/7 | 715.587 | ||

6\10 | 720.000 |

## References

- Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf

## Music

- Pieces by Joel Grant Taylor, in the hexachordal dodecatonic MODMOS:
- Smoke Filled Bar by Chris Vaisvil, also in 12-22h.
- Chord Sequence in Paul Erlich's Decatonic Major by Jake Freivald

## See also

- Pajara extensions - 13-limit extensions for pajara
- Lumatone mapping for diaschismic