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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Prime limit navigation|3}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{Wikipedia| Pythagorean tuning }}
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-03-31 03:47:52 UTC</tt>.<br>
: The original revision id was <tt>215741072</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Some examples for 3-limit intervals are [[3_2|3/2]], [[4_3|4/3]], [[9_8|9/8]], ...


see [[Harmonic Limit]]</pre></div>
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.
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;3-limit&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Some examples for 3-limit intervals are &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;, &lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, ...&lt;br /&gt;
== Terminology ==
&lt;br /&gt;
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 &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;Harmonic Limit&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== 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:
{| 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]]
Retrieved from "https://en.xen.wiki/w/3-limit"