3-limit - Revision history

Fredg999: Interwiki

2026-07-17T04:13:34Z

Interwiki

← Older revision		Revision as of 04:13, 17 July 2026
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			{{interwiki
			\| de =
			\| en = 3-limit
			\| es =
			\| ja = 3リミット
			}}
	{{Prime limit navigation\|3}}		{{Prime limit navigation\|3}}
	{{Wikipedia\| Pythagorean tuning }}		{{Wikipedia\| Pythagorean tuning }}

	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.		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.

Lucius Chiaraviglio: /* Music */ Add Bill Wesley's ''Familly Across the Border'' (2023); also fix formatting for Charles Ives entries

2026-07-10T04:02:25Z

Music: Add Bill Wesley's ''Familly Across the Border'' (2023); also fix formatting for Charles Ives entries

← Older revision		Revision as of 04:02, 10 July 2026
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	* [https://m.youtube.com/watch?v=GF7lTvOQ9r8 ''Building (A New Sun)''] (2017)		* [https://m.youtube.com/watch?v=GF7lTvOQ9r8 ''Building (A New Sun)''] (2017)

	~~=====~~ [[Charles Ives]] ~~=====~~		; [[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.		* [[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-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://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]
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	; [[Peter Kosmorsky\|Peter 'Rush' Kosmorsky]]		; [[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]]]		* ''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]]]

			; [[Bill Wesley]]
			* [https://www.youtube.com/watch?v=-L-jKEmSTh0 ''Familly Across the Border''] (2023, performed on [[mbira]] tuned to 30-note extended Pythagorean)

	; [[Zhea Erose]]		; [[Zhea Erose]]

Xenjacob: /* Music */ +Charles Ives extended Pyth works as recorded by JR/AFMM

2026-03-28T16:58:27Z

Music: +Charles Ives extended Pyth works as recorded by JR/AFMM

← Older revision		Revision as of 16:58, 28 March 2026
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	* [https://m.youtube.com/watch?v=GF7lTvOQ9r8 ''Building (A New Sun)''] (2017)		* [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]]		; [[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]]]		* ''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]]]

MisterShafXen at 10:28, 20 March 2026

2026-03-20T10:28:50Z

← Older revision		Revision as of 10:28, 20 March 2026
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	== 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 {{EDOs\| 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, … }} ({{OEIS\|A206788}})		[[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 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}})

Overthink: /* Edo approximation */ 2edo isn't increasing

2026-03-15T17:01:40Z

Edo approximation: 2edo isn't increasing

← Older revision		Revision as of 17:01, 15 March 2026
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	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 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~~, 2~~, 12, 53, 665, 190537, … }} (with strengths 1, 1, 2, 3, 11, 28, … respectively)		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 ==

FloraC: Recategorize

2026-02-06T08:19:39Z

Recategorize

← Older revision		Revision as of 08:19, 6 February 2026
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	[[Category:3-limit\| ]] <!-- main article -->		[[Category:3-limit\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank-2 temperaments]]

FloraC: Undo revision 211141 by M1k1 H2tsun2 (talk)

2025-09-27T11:53:18Z

Undo revision 211141 by M1k1 H2tsun2 (talk)

← Older revision		Revision as of 11:53, 27 September 2025
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	== Terminology ==		== Terminology ==
	A 3-limit interval is also known as a Pythagorean ~~interval, or Threerish~~ 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.		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 ==

M1k1 H2tsun2: /* Terminology */ add it them

2025-09-27T10:55:43Z

Terminology: add it them

← Older revision		Revision as of 10:55, 27 September 2025
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	== Terminology ==		== 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.		A 3-limit interval is also known as a Pythagorean interval, or Threerish 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 ==

BudjarnLambeth: /* Music */ add 1

2025-09-27T00:29:58Z

Music: add 1

← Older revision		Revision as of 00:29, 27 September 2025
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	; [[Francium]]		; [[Francium]]
	* [https://www.youtube.com/watch?v=tzFK7uzAR1g ''Pythagorean Metal''] (2023)		* [https://www.youtube.com/watch?v=tzFK7uzAR1g ''Pythagorean Metal''] (2023)

			; [[John Doe]]
			* [https://m.youtube.com/watch?v=GF7lTvOQ9r8 ''Building (A New Sun)''] (2017)

	; [[Peter Kosmorsky\|Peter 'Rush' Kosmorsky]]		; [[Peter Kosmorsky\|Peter 'Rush' Kosmorsky]]

Fredg999: /* Edo approximation */ Add 3rd list, add OEIS link to 1st list

2025-08-14T21:50:10Z

Edo approximation: Add 3rd list, add OEIS link to 1st list

← Older revision		Revision as of 21:50, 14 August 2025
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	== 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 {{EDOs\| 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, … }}		[[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 {{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 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, 2, 12, 53, 665, 190537, … }} (with strengths 1, 1, 2, 3, 11, 28, … respectively)

	== Table of intervals ==		== Table of intervals ==