5-limit: Difference between revisions

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The '''5-limit''' consists of all [[just intonation]] intervals whose [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5~~; these are sometimes called {{w|regular number}}s~~. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]].

[[File:Lattice 5lim.png|alt=The lattice is a grid with 60 degree triangles and intervals at the vertices. The center vertex is 1/1, moving right is multiplication by 3/2, up-right is multiplication by 5/4, and up-left is multiplication by 5/3. Everything is octave-reduced.|thumb|5-limit hexagonal lattice]]

The '''5-limit''' (a.k.a. ''ya'' in [[color notation]]) consists of all [[just intonation]] intervals whose [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]].

These things are contained by the 5-limit, but not the 3-limit:

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== Rank-2 temperaments ==

=== Meantone ===

[[Meantone]] is the rank-2 temperament tempering out 81/80 in the 5-limit. It is generated by a flat perfect fifth, around ~~696 or 697~~ cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being ~~supported~~ by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious extension to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.

[[Meantone]] is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] [[tempering out]] [[81/80]] in the 5-limit. It is generated by a flat [[3/2|perfect fifth]], around 696–697 cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being [[support]]ed by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious [[extension]] to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.

=== Magic ===

[[Magic]] is the rank-2 temperament tempering out [[3125/3072]], the magic comma. It is generated by a flat major third of around 380 cents, and equates five of them with a [[3/1|perfect twelfth]]. It is one of the simplest 5-limit temperaments with decent accuracy to not temper out the syntonic comma, 81/80. It has an obvious extension to the 7-limit, tempering out [[225/224]] and [[245/243]], and mapping 7/4 to +12 generators. Edos supporting magic include [[19edo]], [[22edo]], and [[41edo]].

=== Diaschismic ===

[[Diaschismic]] tempers out [[2048/2025]], the diaschisma, which equates [[45/32]] with its octave complement [[64/45]]. This creates a half-octave period. This temperament is generated by a semitone of around 103–105 cents in optimal tunings, representing [[16/15]]. A half-octave plus a semitone reaches the perfect fifth, and a half-octave minus two semitones reaches the major third. This temperament has an obvious extension to prime [[17/1|17]] by equating the semitone to [[17/16]] and [[18/17]], known as [[srutal archagall]]. This temperament remarkably contains ''two'' relatively simple and accurate extensions to the full [[17-limit]]; one is simply called diaschismic, and the other is known as [[srutal]]. Another notable extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]].

=== Kleismic ===

[[Kleismic]]~~/Hanson~~ tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53.}}

[[Kleismic]] (a.k.a. hanson) tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53}}.

=== Schismic ===

[[Schismic]]~~/Schismatic/Helmholtz~~ tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[~~Garibaldi~~]], which maps 7/4 to the ~~doubly~~-diminished octave (-14 fifths), tempering out the [[~~Garischisma~~]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, ~~[[Garibaldi]]~~ is no longer a microtemperament unlike schismic. ~~EDOs~~ supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}

[[Schismic]] (a.k.a. schismatic and helmholtz) tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[garibaldi]], which maps 7/4 to the double-diminished octave (-14 fifths), tempering out the [[garischisma]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, garibaldi is no longer a microtemperament unlike schismic. Edos supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}

== Music ==

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; {{W|Paul Henning}}

* [http://web.archive.org/web/20201127013240/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/The%20Ballad%20of%20Jed%20Clampett.mp3 ''The Ballad of Jed Clampett''] (unknown arranger)

; {{W|Scott Joplin}}

* ''Maple Leaf Rag'' (1899) – rendered by [[Claudi Meneghin]]

** [https://www.youtube.com/shorts/N9psCtvbSQ8 harpsichord rendition] (2026-03-01)

** [https://www.youtube.com/shorts/N27o7IcOAJQ organ rendition (fragment) without swing] (2026-04-06)

** [https://www.youtube.com/watch?v=MSuSe4NRvCM fortepiano rendition without swing] (2026-04-30)

** harpsichord rendition [https://www.youtube.com/watch?v=95zyi0O7OE4 without swing] (2026-05-01) and [https://www.youtube.com/shorts/KuJzxepw6Ho with swing] (2026-05-02)

** organ rendition with swing ([https://www.youtube.com/shorts/dKXbvsL_7xw 2026-05-18], [https://www.youtube.com/watch?v=L6qF9gxT7pA 2026-06-08]) and without swing ([https://www.youtube.com/watch?v=a9Sz2aGurzw 2026-05-25]), with detailed description of syntonic chroma adjustment in video descriptions

=== 20th century ===

; [[David B Doty]]

* ''[https://soundcloud.com/user-238628374/pop-1 Pop#1]'' (composed c. 1985, remastered 2026)

; [[Ben Johnston]]

* ''String Quartet No. 2'' (1964)

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; [[Axoid Music]]

* [https://m.youtube.com/watch?v=D-5f11nR2BI ''Alien Harmonies - Pure Data Generative Microtonal Ambient Music''] (2022)

; [[John Robert Bagby]] ([https://www.youtube.com/@SocraticSwansongs YouTube])

* [https://www.youtube.com/watch?v=F1vYmdaNHHE ''53 unequal divisions of the octave Just Intonation lumatone experiment''] (2024) – uses [https://sevish.com/scaleworkshop/?n=Harmonic+series+12+tone&l=29F28_3kF3h_pFo_3rF3k_gFf_rFp_m8Fk9_aF9_9F8_74F69_40F3h_23F1s_wFr_6F5_6rF5k_2sF29_5F4_29F1s_wFp_4iF3h_8wF6r_4F3_rFk_e8Faf_pFi_19Fw_1sF19_10Fp_afF74_14Fr_3F2_6rF4g_3hF29_pFg_3kF29_8F5_29F1e_b4F6r_5F3_rFg_3kF23_3hF20_69F3k_gF9_9F5_k9Fb4_1eFr_fF8_74F3r_1cFp_3hF1s_4gF29_2F1&version=2.4.1 this 53-tone subset]

; [[William Copper]] ([http://www.williamcopper.com site 1] [http://www.hartenshield.com/william_copper.html site 2]{{dead link}})

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; [[Juhani Nuorvala]]

* [https://www.youtube.com/watch?v=6nQ_TCRoXbA ''Five Preludes for Kantele in Just Intonation''] (2009; performed by Eija Kankaanranta, kantele, October New Music Festival (Uuden musiikin lokakuu), Oulu, Finland, September 30, 2018)

; [[Carlo Serafini]]

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[[Category:5-limit| ]]

[[Category:Rank-3 temperaments]]

[[Category:Lattice]]

[[Category:Listen]]

~~[[Category:Rank 3]]~~

@@ Line 1: / Line 1: @@
 {{Prime limit navigation|5}}
 {{Wikipedia|Five-limit tuning}}
-The '''5-limit''' consists of all [[just intonation]] intervals whose [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5; these are sometimes called {{w|regular number}}s. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]].
+[[File:Lattice 5lim.png|alt=The lattice is a grid with 60 degree triangles and intervals at the vertices. The center vertex is 1/1, moving right is multiplication by 3/2, up-right is multiplication by 5/4, and up-left is multiplication by 5/3. Everything is octave-reduced.|thumb|5-limit hexagonal lattice]]
+The '''5-limit''' (a.k.a. ''ya'' in [[color notation]]) consists of all [[just intonation]] intervals whose [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]].
 These things are contained by the 5-limit, but not the 3-limit:
@@ Line 352: / Line 353: @@
 == Rank-2 temperaments ==
 === Meantone ===
-[[Meantone]] is the rank-2 temperament tempering out 81/80 in the 5-limit. It is generated by a flat perfect fifth, around 696 or 697 cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being supported by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious extension to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.
+[[Meantone]] is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] [[tempering out]] [[81/80]] in the 5-limit. It is generated by a flat [[3/2|perfect fifth]], around 696–697 cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being [[support]]ed by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious [[extension]] to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.
 === Magic ===
 [[Magic]] is the rank-2 temperament tempering out [[3125/3072]], the magic comma. It is generated by a flat major third of around 380 cents, and equates five of them with a [[3/1|perfect twelfth]]. It is one of the simplest 5-limit temperaments with decent accuracy to not temper out the syntonic comma, 81/80. It has an obvious extension to the 7-limit, tempering out [[225/224]] and [[245/243]], and mapping 7/4 to +12 generators. Edos supporting magic include [[19edo]], [[22edo]], and [[41edo]].
+=== Diaschismic ===
+[[Diaschismic]] tempers out [[2048/2025]], the diaschisma, which equates [[45/32]] with its octave complement [[64/45]]. This creates a half-octave period. This temperament is generated by a semitone of around 103–105 cents in optimal tunings, representing [[16/15]]. A half-octave plus a semitone reaches the perfect fifth, and a half-octave minus two semitones reaches the major third. This temperament has an obvious extension to prime [[17/1|17]] by equating the semitone to [[17/16]] and [[18/17]], known as [[srutal archagall]]. This temperament remarkably contains ''two'' relatively simple and accurate extensions to the full [[17-limit]]; one is simply called diaschismic, and the other is known as [[srutal]]. Another notable extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]].
 === Kleismic ===
-[[Kleismic]]/Hanson tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53.}}
+[[Kleismic]] (a.k.a. hanson) tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53}}.
 === Schismic ===
-[[Schismic]]/Schismatic/Helmholtz tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[Garibaldi]], which maps 7/4 to the doubly-diminished octave (-14 fifths), tempering out the [[Garischisma]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, [[Garibaldi]] is no longer a microtemperament unlike schismic. EDOs supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}
+[[Schismic]] (a.k.a. schismatic and helmholtz) tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[garibaldi]], which maps 7/4 to the double-diminished octave (-14 fifths), tempering out the [[garischisma]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, garibaldi is no longer a microtemperament unlike schismic. Edos supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}
 == Music ==
@@ Line 370: / Line 374: @@
 ; {{W|Paul Henning}}
 * [http://web.archive.org/web/20201127013240/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/The%20Ballad%20of%20Jed%20Clampett.mp3 ''The Ballad of Jed Clampett''] (unknown arranger)
+; {{W|Scott Joplin}}
+* ''Maple Leaf Rag'' (1899) – rendered by [[Claudi Meneghin]]
+** [https://www.youtube.com/shorts/N9psCtvbSQ8 harpsichord rendition] (2026-03-01)
+** [https://www.youtube.com/shorts/N27o7IcOAJQ organ rendition (fragment) without swing] (2026-04-06)
+** [https://www.youtube.com/watch?v=MSuSe4NRvCM fortepiano rendition without swing] (2026-04-30)
+** harpsichord rendition [https://www.youtube.com/watch?v=95zyi0O7OE4 without swing] (2026-05-01) and [https://www.youtube.com/shorts/KuJzxepw6Ho with swing] (2026-05-02)
+** organ rendition with swing ([https://www.youtube.com/shorts/dKXbvsL_7xw 2026-05-18], [https://www.youtube.com/watch?v=L6qF9gxT7pA 2026-06-08]) and without swing ([https://www.youtube.com/watch?v=a9Sz2aGurzw 2026-05-25]), with detailed description of syntonic chroma adjustment in video descriptions
 === 20th century ===
+; [[David B Doty]]
+* ''[https://soundcloud.com/user-238628374/pop-1 Pop#1]'' (composed c. 1985, remastered 2026)
 ; [[Ben Johnston]]
 * ''String Quartet No. 2'' (1964)
@@ Line 383: / Line 397: @@
 ; [[Axoid Music]]
 * [https://m.youtube.com/watch?v=D-5f11nR2BI ''Alien Harmonies - Pure Data Generative Microtonal Ambient Music''] (2022)
+; [[John Robert Bagby]] ([https://www.youtube.com/@SocraticSwansongs YouTube])
+* [https://www.youtube.com/watch?v=F1vYmdaNHHE ''53 unequal divisions of the octave Just Intonation lumatone experiment''] (2024) – uses [https://sevish.com/scaleworkshop/?n=Harmonic+series+12+tone&l=29F28_3kF3h_pFo_3rF3k_gFf_rFp_m8Fk9_aF9_9F8_74F69_40F3h_23F1s_wFr_6F5_6rF5k_2sF29_5F4_29F1s_wFp_4iF3h_8wF6r_4F3_rFk_e8Faf_pFi_19Fw_1sF19_10Fp_afF74_14Fr_3F2_6rF4g_3hF29_pFg_3kF29_8F5_29F1e_b4F6r_5F3_rFg_3kF23_3hF20_69F3k_gF9_9F5_k9Fb4_1eFr_fF8_74F3r_1cFp_3hF1s_4gF29_2F1&version=2.4.1 this 53-tone subset]
 ; [[William Copper]] ([http://www.williamcopper.com site 1] [http://www.hartenshield.com/william_copper.html site 2]{{dead link}})
@@ Line 397: / Line 414: @@
 ; [[Juhani Nuorvala]]
 * [https://www.youtube.com/watch?v=6nQ_TCRoXbA ''Five Preludes for Kantele in Just Intonation''] (2009; performed by Eija Kankaanranta, kantele, October New Music Festival (Uuden musiikin lokakuu), Oulu, Finland, September 30, 2018)
 ; [[Carlo Serafini]]
@@ Line 416: / Line 432: @@
 [[Category:5-limit| ]] <!-- main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Lattice]]
 [[Category:Listen]]
-[[Category:Rank 3]]