7-limit: Difference between revisions

The '''7-limit''' or 7-prime-limit consists of rational intervals where 7 is the highest allowable [[prime]] factor, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the octave include [[7/4]], [[7/5]], [[7/6]], [[9/7]], [[15/14]], [[21/16]], [[21/20]], [[35/27]], [[49/36]], and so on.

The [[7-odd-limit]] refers to a constraint on the selection of [[Just intonation|just]] [[Interval class|intervals]] for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[6/5]], [[5/4]], [[4/3]], [[7/5]], [[10/7]], [[3/2]], [[8/5]], [[5/3]], [[12/7]], [[7/4]], [[2/1]], which is known as the [[Wikipedia:Tonality diamond|7-limit tonality diamond]].

The phrase "7-limit just intonation" usually refers to the 7-prime-limit and includes primes 2, 3, 5, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 5, and 7, which can be represented in [[The Seven Limit Symmetrical Lattices|3-dimensional lattice diagrams]], each prime represented by a different dimension. Lattices describing scales beyond the 7-limit require more than three dimensions, and in the 7-limit, such lattices have unique features which simplify the relations between 7-limit chords.

For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as [[11-limit|11-]] or [[13-limit]], which usually sound much more exotic.

A list of [[edo]]s which tunes the 7-limit with more accuracy: {{EDOs| 5, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612 }} and so on. Another list of edos which tunes the 7-limit well relative to their size ([[TE relative error|relative error]] < 5%): {{EDOs| 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612 }} and so on.  

== List of intervals in the 7-prime-limit and 81-odd-limit ==

| {{Monzo| 1 0 2 -2 }}

; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/blue-tuning/blue-ji-excluded-by-peers.mp3 ''Excluded by Peers'']

* [http://micro.soonlabel.com/centaur_tuning/Prelude_For_Centaur_Tuned_Piano.mp3 ''Prelude for Centaur Tuned Piano'']

; [[Ivor Darreg]]

* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 ''Prelude #1 in 7-limit JI'']{{dead link}}

* [http://www.archive.org/details/ClintonVariations ''Clinton Variations''] [http://www.archive.org/download/ClintonVariations/clinton.mp3 play]

; [[Kite Giedraitis]]

* [http://micro.soonlabel.com/gene_ward_smith/Others/Kite/Consolation%20%233%20by%20Ken%20Stillwell%20retuned.mp3 Liszt Consolation #3]{{dead link}} Ken Stillwell performance, retuned to the [[kite33]] 7-limit JI scale

* [http://tallkite.com/music/IHearNumbers.html ''I Hear Numbers'']

* [[Harmonic limit]]

* [[7-odd-limit]]

* [[Wikipedia: Highly composite number]]

[[Category:Interval collection]]