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The ''7-limit'' or "7 prime-limit" refers to a constraint on rational intervals such that 7 is the highest allowable [[prime_number|prime number]], 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 [[9/7|9/7]], [[14/9|14/9]], [[15/14|15/14]], [[28/15|28/15]], [[21/16|21/16]], [[32/21|32/21]], [[25/14|25/14]], [[28/25|28/25]], [[25/21|25/21]], [[42/25|42/25]], [[28/27|28/27]], [[27/14|27/14]], [[35/27|35/27]], 54/35, 45/28, 56/45, 49/32, 64/49, 49/36, 72/49, 49/30, 60/49, 49/25, 50/49, 49/27, 54/49, 49/35, 70/49, 49/45, 90/49.
The '''7-limit''' or "7 prime-limit" refers to a constraint on rational intervals such that 7 is the highest allowable [[prime_number|prime number]], 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 [[9/7]], [[14/9]], [[15/14]], [[28/15]], [[21/16]], [[32/21]], [[25/14]], [[28/25]], [[25/21]], [[42/25]], [[28/27]], [[27/14]], [[35/27]], 54/35, 45/28, 56/45, 49/32, 64/49, 49/36, 72/49, 49/30, 60/49, 49/25, 50/49, 49/27, 54/49, 49/35, 70/49, 49/45, 90/49.


"7 odd-limit" refers to a constraint on the selection of [[JustIntonation|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|1/1]], [[8/7|8/7]], [[7/6|7/6]], [[6/5|6/5]], [[5/4|5/4]], [[4/3|4/3]], [[7/5|7/5]], [[10/7|10/7]], [[3/2|3/2]], [[8/5|8/5]], [[5/3|5/3]], [[12/7|12/7]], [[7/4|7/4]], [[2/1|2/1]], which is known as the 7-limit [http://en.wikipedia.org/wiki/Tonality_diamond tonality diamond].
"7 odd-limit" refers to a constraint on the selection of [[JustIntonation|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|1/1]], [[8/7|8/7]], [[7/6|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 7-limit [http://en.wikipedia.org/wiki/Tonality_diamond 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 a pitch class representing that interval in every possible octave. 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.
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 a pitch class representing that interval in every possible octave. 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|13-limit]], which usually sound much more exotic.
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.


Relative to their size, the equal divisions [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[19edo|19edo]], [[21edo|21edo]], [[22edo|22edo]], [[31edo|31edo]], [[53edo|53edo]], [[84edo|84edo]], [[87edo|87edo]], [[94edo|94edo]], [[99edo|99edo]], [[118edo|118edo]], [[130edo|130edo]], [[140edo|140edo]], [[171edo|171edo]], [[270edo|270edo]], [[410edo|410edo]], [[441edo|441edo]] and [[612edo|612edo]] provide good approximations to the 7-limit.
Relative to their size, the equal divisions [[1edo]], [[2edo]], [[3edo]], [[4edo]], [[5edo]], [[7edo]], [[9edo]], [[10edo]], [[12edo]], [[15edo|15edo]], [[19edo]], [[21edo]], [[22edo]], [[31edo]], [[53edo]], [[84edo]], [[87edo]], [[94edo]], [[99edo]], [[118edo]], [[130edo]], [[140edo]], [[171edo]], [[270edo]], [[410edo]], [[441edo]] and [[612edo]] provide good approximations to the 7-limit.


==List of Intervals in the 7-Prime Limit and 81-Odd Limit==
== List of Intervals in the 7-Prime Limit and 81-Odd Limit ==


Warning: No 49/48. Recalculate and expand! And don't blindly add this interval because this warning will just change to no 10/9!
Warning: No 49/48. Recalculate and expand! And don't blindly add this interval because this warning will just change to no 10/9!
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{| class="wikitable"
{| class="wikitable"
|-
|-
| | [[Ratio|Ratio]]
| [[Ratio|Ratio]]
| | [[monzo|Monzo]]
| [[monzo|Monzo]]
| | [[cents|Cents]] Value
| [[cents|Cents]] Value
|-
|-
| | 1/1
| 1/1
| | |0>
| {{Monzo| 0 }}
| | 0.000
| 0.000
|-
|-
| | 81/80
| 81/80
| | |-4 4 -1>
| {{Monzo| -4 4 -1 }}
| | 21.506
| 21.506
|-
|-
| | 64/63
| 64/63
| | |6 -2 0 -1>
| {{Monzo| 6 -2 0 -1 }}
| | 27.264
| 27.264
|-
|-
| | 50/49
| 50/49
| | |1 0 2 -2>
| {{Monzo| 1 0 2 -2 }}
| | 34.976
| 34.976
|-
|-
| | 36/35
| 36/35
| | |2 2 -1 -1>
| {{Monzo| 2 2 -1 -1 }}
| | 48.770
| 48.770
|-
|-
| | 28/27
| 28/27
| | |2 -3 0 1>
| {{Monzo| 2 -3 0 1 }}
| | 62.961
| 62.961
|-
|-
| | 25/24
| 25/24
| | |-3 -1 2>
| {{Monzo| -3 -1 2 }}
| | 70.672
| 70.672
|-
|-
| | 21/20
| 21/20
| | |-2 1 -1 1>
| {{Monzo| -2 1 -1 1 }}
| | 84.467
| 84.467
|-
|-
| | 16/15
| 16/15
| | |4 -1 -1>
| {{Monzo| 4 -1 -1 }}
| | 111.731
| 111.731
|-
|-
| | 15/14
| 15/14
| | |-1 1 1 -1>
| {{Monzo| -1 1 1 -1 }}
| | 119.443
| 119.443
|-
|-
| | 27/25
| 27/25
| | |0 3 -2>
| | |0 3 -2>
| | 133.238
| 133.238
|-
|-
| | 49/45
| 49/45
| | |0 -2 -1 2>
| | |0 -2 -1 2>
| | 147.428
| 147.428
|-
|-
| | 35/32
| 35/32
| | |-5 0 1 1>
| | |-5 0 1 1>
| | 155.140
| 155.140
|-
|-
| | 54/49
| 54/49
| | |1 3 0 -2>
| | |1 3 0 -2>
| | 168.213
| 168.213
|-
|-
| | 28/25
| 28/25
| | |2 0 -2 1>
| | |2 0 -2 1>
| | 196.198
| 196.198
|-
|-
| | 9/8
| 9/8
| | |-3 2>
| | |-3 2>
| | 203.910
| 203.910
|-
|-
| | 8/7
| 8/7
| | |3 0 0 -1>
| | |3 0 0 -1>
| | 231.174
| 231.174
|-
|-
| | 81/70
| 81/70
| | |-1 4 -1 -1>
| | |-1 4 -1 -1>
| | 252.68
| 252.68
|-
|-
| | 7/6
| 7/6
| | |-1 -1 0 1>
| | |-1 -1 0 1>
| | 266.871
| 266.871
|-
|-
| | 75/64
| 75/64
| | |-6 1 2>
| | |-6 1 2>
| | 274.582
| 274.582
|-
|-
| | 32/27
| 32/27
| | |5 -3>
| | |5 -3>
| | 294.135
| 294.135
|-
|-
| | 25/21
| 25/21
| | |0 -1 2 -1>
| | |0 -1 2 -1>
| | 301.847
| 301.847
|-
|-
| | 6/5
| 6/5
| | |1 1 -1>
| | |1 1 -1>
| | 315.641
| 315.641
|-
|-
| | 98/81
| 98/81
| | |1 -4 0 2>
| | |1 -4 0 2>
| | 329.832
| 329.832
|-
|-
| | 60/49
| 60/49
| | |2 1 1 -2>
| | |2 1 1 -2>
| | 350.617
| 350.617
|-
|-
| | 49/40
| | 49/40
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|}
|}


=Music=
== Music ==
[http://micro.soonlabel.com/0-praxis/audio/August/august_12_Ruckus.mp3 Ruckus From the Quiet Zone] by Ralph Lewis
* [http://micro.soonlabel.com/0-praxis/audio/August/august_12_Ruckus.mp3 Ruckus From the Quiet Zone] by Ralph Lewis
* [http://micro.soonlabel.com/blue-tuning/blue-ji-excluded-by-peers.mp3 Excluded by Peers] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/centaur_tuning/Prelude_For_Centaur_Tuned_Piano.mp3 Prelude for Centaur Tuned Piano] by Chris Vaisvil
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 Prelude #1 in 7-limit JI] by [[Ivor_Darreg|Ivor Darreg]] <-- are there any notations for it?
* [http://www.archive.org/details/ClintonVariations Clinton Variations] [http://www.archive.org/download/ClintonVariations/clinton.mp3 play] by [[Gene Ward Smith]]
* [http://www.youtube.com/watch?v=HzQmaxDIxnc&feature=channel_video_title Pachelbel's Canon in D in 7-limit JI] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play]
* [http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3  Mars in 7-Limit JI] from [http://en.wikipedia.org/wiki/The_Planets The Planets] the orchestral suite by Gustav Holst arranged by [[Chris Vaisvil]] (Blog entry: [http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ Gustav Holst’s Mars arranged for 7-limit JI Orchestra « Music & Techniques by Chris Vaisvil])
* [http://micro.soonlabel.com/gene_ward_smith/Others/Kite/Consolation%20%233%20by%20Ken%20Stillwell%20retuned.mp3 Liszt Consolation #3] Ken Stillwell performance, retuned by Kite Giedraitis to the [[kite33]] 7-limit JI scale


[http://micro.soonlabel.com/blue-tuning/blue-ji-excluded-by-peers.mp3 Excluded by Peers] by [[Chris_Vaisvil|Chris Vaisvil]]
== See also ==
* [[Harmonic Limit]]
* [https://en.wikipedia.org/wiki/7-limit 7-limit - Wikipedia]
* [https://en.wikipedia.org/wiki/Highly_composite_number Highly composite number - Wikipedia]


[http://micro.soonlabel.com/centaur_tuning/Prelude_For_Centaur_Tuned_Piano.mp3 Prelude for Centaur Tuned Piano] by Chris Vaisvil
[[Category:7-limit| ]] <!-- main page -->
 
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 Prelude #1 in 7-limit JI] by [[Ivor_Darreg|Ivor Darreg]] &lt;-- are there any notations for it?
 
[http://www.archive.org/details/ClintonVariations Clinton Variations] [http://www.archive.org/download/ClintonVariations/clinton.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]]
 
[http://www.youtube.com/watch?v=HzQmaxDIxnc&feature=channel_video_title Pachelbel's Canon in D in 7-limit JI] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play]
 
[http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3  Mars in 7-Limit JI] from [http://en.wikipedia.org/wiki/The_Planets The Planets] the orchestral suite by Gustav Holst arranged by [[Chris_Vaisvil|Chris Vaisvil]] (Blog entry: [http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ Gustav Holst’s Mars arranged for 7-limit JI Orchestra « Music &amp; Techniques by Chris Vaisvil])
 
[http://micro.soonlabel.com/gene_ward_smith/Others/Kite/Consolation%20%233%20by%20Ken%20Stillwell%20retuned.mp3 Liszt Consolation #3] Ken Stillwell performance, retuned by Kite Giedraitis to the [[kite33|kite33]] 7-limit JI scale
 
=see also=
<ul><li>[[Harmonic_Limit|Harmonic Limit]]</li><li>[http://en.wikipedia.org/wiki/7-limit 7-limit - Wikipedia]</li><li>[http://en.wikipedia.org/wiki/Highly_composite_number Highly composite number - Wikipedia]</li></ul>
 
[[Category:7-limit]]
[[Category:example]]
[[Category:example]]
[[Category:interval]]
[[Category:interval]]
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