Tuning | Harmonic Complexity | Largest Step Size (cents) | Smallest Step Size (cents) | Number of Step Sizes |
---|---|---|---|---|

5-limit Full | 6.5 | 133 | 71 | 4 |

5-limit Exact | 9.2 | 182 | 22 | 7 |

7-limit Full | 8.5 | 133 | 63 | 7 |

7-limit Exact | 8.5 | 182 | 27 | 10 |

11-limit Full | 10.9 | 151 | 53 | 9 |

11-limit Exact | 10.9 | 182 | 49 | 12 |

13-limit Full | 11.8 | 151 | 44 | 10 |

13-limit Exact | 13.4 | 182 | 53 | 11 |

17-limit Full | 14.6 | 151 | 44 | 11 |

17-limit Exact | 16.3 | 182 | 63 | 12 |

19-limit Full | 17.6 | 151 | 63 | 12 |

19-limit Exact | 18.1 | 151 | 68 | 12 |

23-limit Full | 20.7 | 165 | 63 | 12 |

23-limit Exact | 19.6 | 139 | 74 | 12 |

29-limit | 23.0 | 165 | 59 | 12 |

31-limit | 26.4 | 165 | 55 | 12 |

37-limit | 28.9 | 165 | 46 | 12 |

##### Overview

Prime-limit tunings are a kind of rational tuning. The **utonal** variety, or PLUTs, are constructed by placing in the denominators of the ratios a series of numbers containing prime factors no greater than the stated limit, while the numerators are the all same. That may sound far more complicated than it really is. To construct a 5-prime-limit utonal tuning, for example, simply take any set of numbers that factor wholly to 2s, 3s & 5s, and place them over the same numerator. Since octave equivalency is assumed, "same" includes all numbers related by powers of two alone. Thus, the numbers 5 and 9 may placed under 4 and 8, respectively; 4 and 8 are "the same" in terms of placement within the octave because with all factors of two removed, they are equal.

The pages in the table above address only 12-tone, octave-repeating versions of prime limit uotonal tunings. For simplicity's sake, I have spelled them out using only powers of two in the numerators.

##### Tuning Names

For the **Full** versions, the ratios chosen for each tuning are the simplest available near each key on a standard 7+5 keyboard; therefore, they are not always the 12 absolute simplest—often a few get skipped over because they most closely approximate a chromatic tone that is already "taken" by a simpler ratio.

For the **Exact** versions, the twelve absolute simplest ratios are used regardless of how inconveniently the result maps to a 7+5 keyboard. For both types, tuning tables are offset to average 0 cents difference from 12 EDO. Where no distinction is made, the exact version maps as well as is feasible. Above 23, mapping the 12 absolute simplest ratios would exclude the higher primes, so one or more of the simpler ratios are omitted to make room, usually the cubes and squares of the lowest primes. Since 37 is the 11th prime (starting with 3), that's the highest limit that will fit in 12 tones without omitting some.

##### Harmonic Complexity

I rate rational scales by *Harmonic Complexity*, as I understand it. In tuning theory, the harmonic complexity of a set of ratios, such as a tuning, is the *complexity* of the *Least Common Multiples* of the set. There are different ways to measure complexity. For this purpose, I prefer `LOG(LCM)`, the natural logarithm of the least common multiple of the prime factors. For purposes of comparing tunings that assume octave equivalence, I believe that powers of two can be removed. While that's not true when comparing individual ratios, for tunings that assume octave equivalence, to fail to remove powers of two provides different ratings for tunings that are sonically and musically identical, differing only in what portion of the harmonic lattice they cover. Therefore, for my purposes here, I have removed powers of two from the ratios of each tuning before calculating their HC.

The Harmonic Complexity rating should not be taken to indicate "better" or "worse," of course; it only gives some relative idea of the overall complexity of the scale members' harmonic relationships, for whatever that's worth.

Copyright © 2004 by David J. Finnamore

All rights reserved.

tuning@elvenminstrel.com