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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]], [[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 seed out of which grew the common-practice tradition of Western music.
{{Prime limit navigation|3}}
{{Wikipedia| Pythagorean tuning }}


[[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, [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[306edo|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, [[359edo|359]], [[665edo|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.  


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


[[Category:3-limit| ]] <!-- main article -->
[[Category:3-limit| ]] <!-- main article -->
[[Category:Example]]
[[Category:Rank-2 temperaments]]
[[Category:Interval]]
[[Category:Limit]]
[[Category:Prime limit]]
[[Category:Pythagorean]]
[[Category:Rank 2]]
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