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~~ [[~~wikipedia: Regular number|regular numbers~~]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80~~]]. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit~~]].

[[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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The 5-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3]], [[3/2]], [[8/5]], [[5/3]], [[2/1]]. Approximating these ratios has been basic to Western common-practice music since the Renaissance.

The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a ~~[[wikipedia: Hexagonal lattice~~|hexagonal lattice]] or as a ~~[[wikipedia: Square lattice~~|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a ~~[[wikipedia:Hexagonal tiling~~|hexagonal tiling]].

The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a {{w|hexagonal lattice}} or as a {{w|square lattice}}; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a {{w|hexagonal tiling}}.

== Terminology ==

Due to their historical significance, 5-limit intervals go by various names, including '''classic(al)'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref>, '''pental'''<ref>[https://chrisvaisvil.com/microtonal-theory-pages/gallery-of-just-intervals/ Gallery of Just Intervals « Music & Techniques by Chris Vaisvil]</ref>~~, or '''quinquimal'''{{citation needed}}~~.

Due to their historical significance, 5-limit intervals go by various names, including '''classic(al)'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref> or '''pental'''<ref>[https://chrisvaisvil.com/microtonal-theory-pages/gallery-of-just-intervals/ Gallery of Just Intervals « Music & Techniques by Chris Vaisvil]</ref>.

Recently, composers [[Catherine Lamb]] and [[Marc Sabat]] have adopted ''quintal'' for the ~~HC5{{citation needed}}~~ since the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit. ''Pental'' is less consistent due to its Greek origins. However, that creates a conflict of usage as ''quintal'' has been the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.) [[User:Lériendil|Lériendil]] suggests the term ''quinary'' as opposed to ''quintal'' (seeing as the pent- root is still overloaded with various terms referring to fifths and pentatonic scales), though there is a minor conflict in naming with [[Arity|quinary]] scales.

Recently, composers [[Catherine Lamb]] and [[Marc Sabat]] have adopted ''quintal'' for the [[harmonic class]] 5 since the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit.<ref>See e.g. liner notes for Lambs' ''Curva Triangulus'', "Curva Triangulus is an ever-fluctuating modal space - shifting between Tertial (3 prime qualitative), Quintal (5 prime qualitative), and Septimal (7 prime qualitative)."</ref> ''Pental'' is less consistent due to its Greek origins. However, that creates a conflict of usage as ''quintal'' has been the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.) [[User:Lériendil|Lériendil]] suggests the term ''quinary'' as opposed to ''quintal'' (seeing as the pent- root is still overloaded with various terms referring to fifths and pentatonic scales), though there is a minor conflict in naming with [[Arity|quinary]] scales.

A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://~~marsbat~~.~~space~~/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.

A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://masa.plainsound.org/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.

The term '''ptolemaic''' could also refer to the 5-limit<ref>[https://~~marsbat~~.~~space~~/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.

The term '''ptolemaic''' could also refer to the 5-limit<ref>[https://masa.plainsound.org/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.

== Edo approximation ==

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{| class="wikitable"

|-

! colspan="2" rowspan="2" | ~~interval~~

! colspan="2" rowspan="2" | Interval<br>category

category

! colspan="4" | Wa (3-limit) interval

! colspan="4" | wa (3-limit) interval

! colspan="4" | Yo or gu (5-limit) interval (81/80)

! colspan="4" | yo or gu (5-limit) interval (81/80)

! colspan="4" | Yoyo or gugu interval (6561/6400)

! colspan="4" | ~~yoyo~~ or gugu interval (6561/6400)

|-

! ~~ratio~~

! Ratio

! ~~cents~~

! Cents

! colspan="2" |[[~~Kite's color notation|~~Color name]]

! colspan="2" | [[Color name]]

! ~~ratio~~

! Ratio

! ~~cents~~

! Cents

! colspan="2" ~~|[[Kite's color notation~~|Color name]]

! colspan="2" | Color name

! ~~ratio~~

! Ratio

! ~~cents~~

! Cents

! colspan="2" ~~| [[Kite's color notation~~|Color name]]

! colspan="2" | Color name

|-

| unison

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It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means allowing for ''both'' 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit ''includes'' the 3-limit – a work in 5-limit JI will utilize intervals from both sides of the chart above.

== Rank-2 temperaments ==

=== Meantone ===

[[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]] (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]] (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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; [[mannfishh]]

* [https://www.youtube.com/watch?v=CSL_Axohw94 ''Microtonal Tetris''] (2023)

; [[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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* [https://www.youtube.com/watch?v=I1qpZeswcCM ''Impede''] (2022)

* [https://www.youtube.com/watch?v=TCyfQtF2x_U ''Icicle''] (2023)

~~== See also ==~~

* [[Table of 5-limit commas]]

== Notes ==

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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 [[wikipedia: Regular number|regular numbers]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]].
+[[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 9: / Line 10: @@
 The 5-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3]], [[3/2]], [[8/5]], [[5/3]], [[2/1]]. Approximating these ratios has been basic to Western common-practice music since the Renaissance.
-The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [[wikipedia: Hexagonal lattice|hexagonal lattice]] or as a [[wikipedia: Square lattice|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[wikipedia:Hexagonal tiling|hexagonal tiling]].
+The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a {{w|hexagonal lattice}} or as a {{w|square lattice}}; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a {{w|hexagonal tiling}}.
 == Terminology ==
-Due to their historical significance, 5-limit intervals go by various names, including '''classic(al)'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref>, '''pental'''<ref>[https://chrisvaisvil.com/microtonal-theory-pages/gallery-of-just-intervals/ Gallery of Just Intervals « Music & Techniques by Chris Vaisvil]</ref>, or '''quinquimal'''{{citation needed}}.
+Due to their historical significance, 5-limit intervals go by various names, including '''classic(al)'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref> or '''pental'''<ref>[https://chrisvaisvil.com/microtonal-theory-pages/gallery-of-just-intervals/ Gallery of Just Intervals « Music & Techniques by Chris Vaisvil]</ref>.
-Recently, composers [[Catherine Lamb]] and [[Marc Sabat]] have adopted ''quintal'' for the HC5{{citation needed}} since the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit. ''Pental'' is less consistent due to its Greek origins. However, that creates a conflict of usage as ''quintal'' has been the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.) [[User:Lériendil|Lériendil]] suggests the term ''quinary'' as opposed to ''quintal'' (seeing as the pent- root is still overloaded with various terms referring to fifths and pentatonic scales), though there is a minor conflict in naming with [[Arity|quinary]] scales.
+Recently, composers [[Catherine Lamb]] and [[Marc Sabat]] have adopted ''quintal'' for the [[harmonic class]] 5 since the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit.<ref>See e.g. liner notes for Lambs' ''Curva Triangulus'', "Curva Triangulus is an ever-fluctuating modal space - shifting between Tertial (3 prime qualitative), Quintal (5 prime qualitative), and Septimal (7 prime qualitative)."</ref> ''Pental'' is less consistent due to its Greek origins. However, that creates a conflict of usage as ''quintal'' has been the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.) [[User:Lériendil|Lériendil]] suggests the term ''quinary'' as opposed to ''quintal'' (seeing as the pent- root is still overloaded with various terms referring to fifths and pentatonic scales), though there is a minor conflict in naming with [[Arity|quinary]] scales.
-A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://marsbat.space/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.
+A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://masa.plainsound.org/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.
-The term '''ptolemaic''' could also refer to the 5-limit<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.
+The term '''ptolemaic''' could also refer to the 5-limit<ref>[https://masa.plainsound.org/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contain a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.
 == Edo approximation ==
@@ Line 32: / Line 33: @@
 {| class="wikitable"
 |-
-! colspan="2" rowspan="2" | interval
+! colspan="2" rowspan="2" | Interval<br>category
-category
+! colspan="4" | Wa (3-limit) interval
-! colspan="4" | wa (3-limit) interval
+! colspan="4" | Yo or gu (5-limit) interval (81/80)
-! colspan="4" | yo or gu (5-limit) interval (81/80)
+! colspan="4" | Yoyo or gugu interval (6561/6400)
-! colspan="4" | yoyo or gugu interval (6561/6400)
 |-
-! ratio
+! Ratio
-! cents
+! Cents
-! colspan="2" |[[Kite's color notation|Color name]]
+! colspan="2" | [[Color name]]
-! ratio
+! Ratio
-! cents
+! Cents
-! colspan="2" |[[Kite's color notation|Color name]]
+! colspan="2" | Color name
-! ratio
+! Ratio
-! cents
+! Cents
-! colspan="2" | [[Kite's color notation|Color name]]
+! colspan="2" | Color name
 |-
 | unison
@@ Line 350: / Line 350: @@
 It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means allowing for ''both'' 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit ''includes'' the 3-limit – a work in 5-limit JI will utilize intervals from both sides of the chart above.
+== Rank-2 temperaments ==
+=== Meantone ===
+[[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]] (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]] (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 358: / 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 371: / 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 382: / Line 411: @@
 ; [[mannfishh]]
 * [https://www.youtube.com/watch?v=CSL_Axohw94 ''Microtonal Tetris''] (2023)
+; [[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 395: / Line 427: @@
 * [https://www.youtube.com/watch?v=I1qpZeswcCM ''Impede''] (2022)
 * [https://www.youtube.com/watch?v=TCyfQtF2x_U ''Icicle''] (2023)
-== See also ==
-* [[Table of 5-limit commas]]
 == Notes ==
@@ Line 403: / Line 432: @@
 [[Category:5-limit| ]] <!-- main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Lattice]]
 [[Category:Listen]]
-[[Category:Rank 3]]