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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 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2|3/2]], [[4/3|4/3]], [[9/8|9/8]]. Confining intervals to the 3-limit is known as [http://en.wikipedia.org/wiki/Pythagorean_tuning Pythagorean tuning], and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music.
{{Prime limit navigation|3}}
{{Wikipedia| Pythagorean tuning }}


[[EDO|EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [http://en.wikipedia.org/wiki/Continued_fraction continued fraction] for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306..., ...
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.


Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 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, ...
== 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.  


See [[Harmonic_Limit|Harmonic Limit]].
== Edo approximation ==
[[Category:3-limit]]
[[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}})
[[Category:example]]
 
[[Category:interval]]
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}})
[[Category:limit]]
 
[[Category:prime_limit]]
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)
[[Category:pythagorean]]
 
[[Category:rank_2]]
== Table of intervals ==
3-limit intervals up to [[odd-limit]] 19683:
{| class="wikitable center-1 right-3 center-6 center-7"
|-
! [[Ratio]]
! [[Monzo]]
! Size ([[Cent|¢]])
! colspan="2" | [[Kite's color notation|Color Name]]
! colspan="2" | Diatonic Category
|-
| [[1/1]]
| {{Monzo| 0 }}
| 0.000
| w1
| wa unison
| P1
| C
|-
| [[2187/2048]]
| {{Monzo| -11 7 }}
| 113.685
| Lw1
| lawa 1sn
| A1
| C#
|-
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.225
| sw2
| sawa 2nd
| m2
| Db
|-
| [[9/8]]
| {{Monzo| -3 2 }}
| 203.910
| w2
| wa 2nd
| M2
| D
|-
| [[19683/16384]]
| {{Monzo| -14 9 }}
| 317.595
| Lw2
| lawa 2nd
| A2
| D#
|-
| [[32/27]]
| {{Monzo| 5 -3 }}
| 294.135
| w3
| wa 3rd
| m3
| Eb
|-
| [[81/64]]
| {{Monzo| -6 4 }}
| 407.820
| Lw3
| lawa 3rd
| M3
| E
|-
| [[8192/6561]]
| {{Monzo| 13 -8 }}
| 384.360
| sw4
| sawa 4th
| d4
| Fb
|-
| [[4/3]]
| {{Monzo| 2 -1 }}
| 498.045
| w4
| wa 4th
| P4
| F
|-
| [[729/512]]
| {{Monzo| -9 6 }}
| 611.730
| Lw4
| lawa 4th
| A4
| F#
|-
| [[1024/729]]
| {{Monzo| 10 -6 }}
| 588.270
| sw5
| sawa 5th
| d5
| Gb
|-
| [[3/2]]
| {{Monzo| -1 1 }}
| 701.955
| w5
| wa 5th
| P5
| G
|-
| [[6561/4096]]
| {{Monzo| -12 8 }}
| 815.640
| Lw5
| lawa 5th
| A5
| G#
|-
| [[128/81]]
| {{Monzo| 7 -4 }}
| 792.180
| sw6
| sawa 6th
| m6
| Ab
|-
| [[27/16]]
| {{Monzo| -4 3 }}
| 905.865
| w6
| wa 6th
| M6
| A
|-
| [[32768/19683]]
| {{Monzo| 15 -9 }}
| 882.405
| sw7
| sawa 7th
| d7
| Bbb
|-
| [[16/9]]
| {{Monzo| 4 -2 }}
| 996.090
| w7
| wa 7th
| m7
| Bb
|-
| [[243/128]]
| {{Monzo| -7 5 }}
| 1109.775
| Lw7
| lawa 7th
| M7
| B
|-
| [[4096/2187]]
| {{Monzo| 12 -7 }}
| 1086.315
| sw8
| sawa 8ve
| d8
| Cb
|-
| [[2/1]]
| {{Monzo| 1 }}
| 1200.000
| w8
| wa 8ve
| P8
| C
|}
 
== 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 ==
* [[Pythagorean tuning]]
* [[Harmonic limit]]
* [[3-odd-limit]]
* [[Gallery of just intervals]]
 
[[Category:3-limit| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
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