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{{Prime limit navigation|3}}
{{Prime limit navigation|3}}
{{Wikipedia|Pythagorean tuning}}
{{Wikipedia| Pythagorean tuning }}


A '''3-limit''' interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. Confining intervals to the 3-limit is known as [[Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it.
The '''3-limit''' consists of all [[just intonation]] intervals whose [[Ratio|numerators and denominators]] are both products of the primes 2 and 3. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. When octave-reduced, if b is non-zero, a and b are opposite signs. In other words, one number in the ratio is a power of 2 and the other number is a power of 3. Confining intervals to the 3-limit is known as [[Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. The 3-limit can be considered a [[Rank-2 temperament|rank-2]] [[temperament]] which [[Tempering out|tempers out]] no [[comma]]s.
 
== Terminology ==
A 3-limit interval is also known as a Pythagorean interval. Recently, composers [[Catherine Lamb]] and [[Marc Sabat]] have adopted ''tertial'' for intervals of [[harmonic class|HC3]]{{citation needed}}, not to be confused with ''tertian'' which is the adjective associated with the third [[5L 2s|diatonic]] degree.  


== Edo approximation ==
== Edo approximation ==
[[Edo]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[wikipedia: Continued fraction|continued fraction]] for the logarithm of 3 base 2. These are 1, 2, 3, [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[306edo|306]], ...
[[Edo]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[wikipedia: Continued fraction|continued fraction]] for the logarithm base 2 of 3. These are {{EDOs| 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, … }} ({{OEIS|A206788}})
 
Another approach is to find edos which have more accurate approximation to 3 than all smaller edos. This results in {{EDOs|1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867 }}, … ({{OEIS|A060528}})


Another approach is to find edos which have more accurate 3 than all smaller edos. This results in {{EDOs|1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867 }}, ...
A stricter approach is to find edos with an increasingly stronger [[consistent circle]] of 3/2. These are {{EDOs|1, 12, 53, 665, 190537, … }} (with strengths 1, 2, 3, 11, 28, … respectively)


== Table of intervals ==
== Table of intervals ==
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| C
| C
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== Music ==
; [[E8 Heterotic]]
* [https://youtu.be/NPoyCQ7aYY8?si=bnAq4FJ7f8s3AagZ "Elements - Metal"] from ''Elements'' (2019–2020)
; [[Francium]]
* [https://www.youtube.com/watch?v=tzFK7uzAR1g ''Pythagorean Metal''] (2023)
; [[John Doe]]
* [https://m.youtube.com/watch?v=GF7lTvOQ9r8 ''Building (A New Sun)''] (2017)
===== [[Charles Ives]] =====
[[Johnny Reinhard]]'s 2023 book, ''[https://www.visionedition.com/publication/the-transcendental-tuning-of-charles-ives/ The Transcendental Tuning of Charles Ives]'', lays the foundation for AFMM's realizations of some of Ives' works, employing chains of up to 29 perfect fifths.
* [https://johnnyreinhard.bandcamp.com/album/charles-ives-string-quartet-2-by-flux-quartet-three-quartone-pieces-for-2-pianos-played-by-pierce-jonas-the-unanswered-question-universe-symphony-realized-by-reinhard-michael-thorne-three-page-so String Quartet #2, The Unanswered Question, Three-Page Sonata, Universe Symphony]
* [https://johnnyreinhard.bandcamp.com/album/charles-ives-transcendental-concord-sonata-by-charles-ives-for-two-pianos-in-spiral-of-fifths-tuning-performed-by-pianists-gabriel-zucker-and-erika-dohi-american-festival-of-microtonal-music Concord Sonata]
* [https://www.youtube.com/watch?v=V8HkPie8y08 The Unanswered Question]
* [https://www.youtube.com/watch?v=OT2E13p3sLw Universe Symphony]
; [[Peter Kosmorsky|Peter 'Rush' Kosmorsky]]
* ''String Trio no. 2'' (2013) – [https://soundcloud.com/peter-rush-kosmorsky/string-trio-no-2-for-three-strings SoundCloud] | [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/__String_Trio_no__2_by_Peter__Rush__Kosmorsky.mp3 play] – in [[Pythagorean17|Pythagorean[17]]]
; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=ISHYKXPaL5o ''Circles of Indigo - Dreamsura''] (2023)


== See also ==
== See also ==
* [[Pythagorean tuning]]
* [[Harmonic limit]]
* [[Harmonic limit]]
* [[3-odd-limit]]
* [[3-odd-limit]]
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[[Category:3-limit| ]] <!-- main article -->
[[Category:3-limit| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/3-limit"