Harmonic limit: Difference between revisions
Wikispaces>FREEZE No edit summary |
|||
| Line 4: | Line 4: | ||
With increasing limits, the tonal space becomes more dense. | With increasing limits, the tonal space becomes more dense. | ||
<ul><li>[[2-limit|2-limit]] contains only multiples of the [[Octave|octave]] (2/1), see [[1edo|1edo]]</li><li>[[3-limit|3-limit]] contains [[3/2|3/2]], the [[just_perfect_fifth|just perfect fifth]]</li><li>[[5-limit|5-limit]] contains [[5/4|5/4]], the just major third</li><li>[[7-limit|7-limit]] contains [[7/4|7/4]], the | <ul><li>[[2-limit|2-limit]] contains only multiples of the [[Octave|octave]] (2/1), see [[1edo|1edo]]</li><li>[[3-limit|3-limit]] contains [[3/2|3/2]], the [[just_perfect_fifth|just perfect fifth]]</li><li>[[5-limit|5-limit]] contains [[5/4|5/4]], the just major third</li><li>[[7-limit|7-limit]] contains [[7/4|7/4]], the just augmented sixth</li><li>[[11-limit|11-limit]] contains [[11/8|11/8]], the just augmented augmented third</li><li>[[13-limit|13-limit]]</li><li>[[17-limit|17-limit]]</li><li>[[19-limit|19-limit]]</li><li>[[23-limit|23-limit]]</li><li>[[29-limit|29-limit]]</li><li>[[31-limit|31-limit]]</li><li>[[37-limit|37-limit]]</li><li>[[41-limit|41-limit]]</li><li>[[43-limit|43-limit]]</li><li>[[47-limit|47-limit]]</li><li>[[53-limit|53-limit]]</li><li>[[59-limit|59-limit]]</li><li>[[61-limit|61-limit]]</li></ul> | ||
==See also== | ==See also== | ||
Revision as of 16:43, 27 September 2018
A positive rational number q belongs to the p-limit, called the p harmonic or prime limit, for a given prime number p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a finitely generated free abelian group. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the 7-limit is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of p-smooth numbers, where a p-smooth number is an integer with prime factors no larger than p.
List of small p-limits
With increasing limits, the tonal space becomes more dense.
- 2-limit contains only multiples of the octave (2/1), see 1edo
- 3-limit contains 3/2, the just perfect fifth
- 5-limit contains 5/4, the just major third
- 7-limit contains 7/4, the just augmented sixth
- 11-limit contains 11/8, the just augmented augmented third
- 13-limit
- 17-limit
- 19-limit
- 23-limit
- 29-limit
- 31-limit
- 37-limit
- 41-limit
- 43-limit
- 47-limit
- 53-limit
- 59-limit
- 61-limit
See also
- Odd limit
- Harmonic Class (HC)
- consistency
- Limit (music) - Wikipedia (covers also the distinction between odd-limit and prime-limit)
- Størmer's theorem - Wikipedia