7-limit: Difference between revisions

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The '''7-limit''' or 7-prime-limit consists of rational intervals where 7 is the highest allowable [[prime]] factor, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 7. ~~This~~ is ~~an infinite set~~ and ~~still infinite even if we restrict consideration to~~ a ~~single octave~~. Some examples ~~within the octave~~ include [[7/4]], [[7/5]], [[7/6]], [[9/7]], [[15/14]], [[21/16]], [[21/20]], [[35/27]], [[49/36]], and so on.

The '''7-limit''' or 7-prime-limit consists of [[just intonation|rational intervals]] where 7 is the highest allowable [[prime]] factor, so that every such interval may be written as a [[ratio]] of integers which are products of 2, 3, 5 and 7. The 7-limit is the fourth prime limit and is a superset of the [[5-limit]] and a subset of the [[11-limit]]. Some examples of 7-limit intervals include [[7/4]], [[7/5]], [[7/6]], [[9/7]], [[15/14]], [[21/16]], [[21/20]], [[35/27]], [[49/36]], and so on.

The [[7-odd-limit]] ~~refers to~~ a constraint on the selection of ~~[[Just intonation|~~just~~]] [[Interval class|~~intervals]] for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[6/5]], [[5/4]], [[4/3]], [[7/5]], [[10/7]], [[3/2]], [[8/5]], [[5/3]], [[12/7]], [[7/4]], [[2/1]], which is known as the ~~[[Wikipedia:Tonality diamond|~~7-limit tonality diamond]].

These things are contained by the 7-limit, but not the 5-limit:

* The [[7-odd-limit|7-]] and [[9-odd-limit]];

* Mode 4 and 5 of the harmonic or subharmonic series.

The 7-odd-limit is a constraint on the selection of just intervals for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[6/5]], [[5/4]], [[4/3]], [[7/5]], [[10/7]], [[3/2]], [[8/5]], [[5/3]], [[12/7]], [[7/4]], [[2/1]], which is known as the 7-odd-limit [[tonality diamond]].

The phrase "7-limit just intonation" usually refers to the 7-prime-limit and includes primes 2, 3, 5, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 5, and 7, which can be represented in [[The Seven Limit Symmetrical Lattices|3-dimensional lattice diagrams]], each prime represented by a different dimension. Lattices describing scales beyond the 7-limit require more than three dimensions, and in the 7-limit, such lattices have unique features which simplify the relations between 7-limit chords.

For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as [[11-~~limit|11-]]~~ or [[13-limit]], which usually sound much more exotic.

For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as 11- or [[13-limit]], which usually sound much more exotic.

== Edo approximation ==

Here is a list of [[edo]]s which tunes the 7-limit with more accuracy ([[monotonicity limit]] ≥ 7 and decreasing [[TE error]]): {{EDOs| 5, 8d, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, … }}. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

Here is a list of edos which tunes the 7-limit well relative to their size ([[TE relative error]] < 5%): {{EDOs| 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612, … }}.

~~A list of [[edo]]s which tunes the 7-limit with more accuracy:~~ {{~~EDOs~~| ~~5, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612 }} and so on. Another list~~ of ~~edos which tunes the~~ 7-limit well relative to their size ([[TE relative error|relative error]] < 5%): {{EDOs| 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612 }} ~~and so on.~~

== Intervals ==

{{See also| User:Lériendil/Table of 21-odd-limit 7-limit intervals }}

~~== List~~ of intervals in the 7-prime-limit and 81-odd-limit ==

Here is a table of intervals in the 7-prime-limit and 81-odd-limit.

{| class="wikitable center-1 right-3"

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== Subgroups of the 7-limit ==

* [[2.3.7 subgroup]]

* [[2.5.7 subgroup]]

* [[3.5.7 subgroup]]

== Music ==

=== Modern renderings ===

; {{W|Gustav Holst}}

* "Mars" from ''{{w|The Planets}}'' (1914–1917) – [http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ blog] | [http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3 play] – arranged by [[Chris Vaisvil]] (2012)

; {{W|Scott Joplin}}

* ''{{w|Maple Leaf Rag}}'' (1899) – [http://web.archive.org/web/20190412163308/http://soonlabel.com/xenharmonic/archives/2127 play] – arranged by [[Claudi Meneghin]] (2014)

; {{W|Franz Liszt}}

* {{W|Consolations (Liszt)|"Consolation No. 3"}} (1850) – [https://soundcloud.com/tallkite/liszt-consolation-3-by-ken-1 play] – Ken Stillwell performance, retuned by [[Kite Giedraitis]] to the [[kite33]] 7-limit JI scale

; {{W|Johann Pachelbel}}

* ''{{w|Pachelbel's Canon|Canon in D}}'' (''c''. 1680–1706) – [https://web.archive.org/web/20201127013008/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play] | [https://www.youtube.com/watch?v=HzQmaxDIxnc YouTube] – arranged by [[Claudi Meneghin]] (2011)

=== 20th century ===

; [[Ben Johnston]]

* ''String Quartet No. 4'' (1973) – [https://newworldrecords.bandcamp.com/track/crossings-the-ascent-string-quartet-no-4-amazing-grace Bandcamp] | [https://www.youtube.com/watch?v=ReHIe0WDvNs YouTube] – performed by Kepler Quartet

=== 21st century ===

; [[Abnormality]]

* [https://www.youtube.com/watch?v=WuW5COnfOlE ''Just Elevation''] (2023)

; [[Jacob Adler]]

* [https://m.youtube.com/watch?v=IUePyH2C9Y0 ''7-Limit Harmony''] (2024)

; [[Ivor Darreg]]

* [http://web.archive.org/web/20201127014610/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 ''Prelude #1 in 7-limit JI'']

; [[dotuXil]]

* [https://dotuxil.bandcamp.com/track/waterpad "waterpad"] from [https://dotuxil.bandcamp.com/album/collected-refractions ''Collected Refractions''] (2024)

; [[E8 Heterotic]]

* [https://youtu.be/mecOmJbqbxU?si=ILByj2hvPIMKRjJT ''Justicar''] (2020)

; [[Francium]]

* [https://www.youtube.com/watch?v=NANoBRyxll8 ''Too Happy For My Mood''] (2023)

* [https://www.youtube.com/watch?v=YcMcychEAoE ''The Bazillionth Party Track''] (2023)

* [https://www.youtube.com/watch?v=qDfIzd_Q-Hc ''Counting to Infinity''] (2025)

* "You Geese" from ''Holy Carp'' (2025) – [https://open.spotify.com/track/5xcKZqwgw2SXSZRf1NQsyT Spotify] | [https://francium223.bandcamp.com/track/you-geese Bandcamp] | [https://www.youtube.com/watch?v=jlLQYfHp69A YouTube]

; [[Kite Giedraitis]]

* [http://tallkite.com/music/IHearNumbers.html ''I Hear Numbers'']

; [[Ralph Lewis]]

* [http://micro.soonlabel.com/0-praxis/audio/August/august_12_Ruckus.mp3 ''Ruckus From the Quiet Zone'']

; [[Kaiveran Lugheidh]]

* [https://soundcloud.com/vale-10/nostalgic-blue Nostalgic Blue] (2017) – in 2.3.7 subgroup

; [[Nick, The NRG]]

* [https://www.youtube.com/watch?v=6IBM_JX52ck ''Cloudy Dreams''] (2022)

; [[Juhani Nuorvala]]

* [https://www.youtube.com/watch?v=0bXhY83asUo ''The tap dance scene from Flash Flash''] (2019)

; [[Gene Ward Smith]]

* ''Clinton Variations'' (2010) – [http://www.archive.org/details/ClintonVariations detail] | [http://www.archive.org/download/ClintonVariations/clinton.mp3 play]

; [[Chris Vaisvil]]

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* [http://micro.soonlabel.com/centaur_tuning/Prelude_For_Centaur_Tuned_Piano.mp3 ''Prelude for Centaur Tuned Piano'']

; [[~~Ivor Darreg~~]]

; [[Randy Wells]]

* [~~http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 ''Prelude #1 in 7-limit JI'']{{dead link}}~~

* [https://www.youtube.com/watch?v=rTvMMwkH2Z8 ''The Antidote for Entropy''] (2022)

~~; [[Gene Ward Smith]]~~

* [http://www.archive.org/details/ClintonVariations ''Clinton Variations''] [http://www.archive.org/download/ClintonVariations/clinton.mp3 play]

* [http://www.youtube.com/watch?v=~~HzQmaxDIxnc&feature=channel_video_title~~ ''Pachelbel's Canon in D in 7-limit JI''] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play]

* [http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3 ''~~Mars in 7-Limit JI''] from [[Wikipedia: The Planets|The Planets~~]~~] – arrangement of Gustav Holst's orchestral suite~~ (~~[http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ blog entry]~~)

~~; [[Kite Giedraitis]]~~

* [http://micro.soonlabel.com/gene_ward_smith/Others/Kite/Consolation%20%233%20by%20Ken%20Stillwell%20retuned.mp3 Liszt Consolation #3]{{dead link}} Ken Stillwell performance, retuned to the [[kite33]] 7-limit JI scale

* [http://tallkite.com/music/IHearNumbers.html ''I Hear Numbers'']

== See also ==

* [[Harmonic limit]]

* [[7-odd-limit]]

* [[Wikipedia: Highly composite number]]

@@ Line 1: / Line 1: @@
 {{Prime limit navigation|7}}
 {{Wikipedia|7-limit tuning}}
-The '''7-limit''' or 7-prime-limit consists of rational intervals where 7 is the highest allowable [[prime]] factor, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the octave include [[7/4]], [[7/5]], [[7/6]], [[9/7]], [[15/14]], [[21/16]], [[21/20]], [[35/27]], [[49/36]], and so on.
+The '''7-limit''' or 7-prime-limit consists of [[just intonation|rational intervals]] where 7 is the highest allowable [[prime]] factor, so that every such interval may be written as a [[ratio]] of integers which are products of 2, 3, 5 and 7. The 7-limit is the fourth prime limit and is a superset of the [[5-limit]] and a subset of the [[11-limit]]. Some examples of 7-limit intervals include [[7/4]], [[7/5]], [[7/6]], [[9/7]], [[15/14]], [[21/16]], [[21/20]], [[35/27]], [[49/36]], and so on.
-The [[7-odd-limit]] refers to a constraint on the selection of [[Just intonation|just]] [[Interval class|intervals]] for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[6/5]], [[5/4]], [[4/3]], [[7/5]], [[10/7]], [[3/2]], [[8/5]], [[5/3]], [[12/7]], [[7/4]], [[2/1]], which is known as the [[Wikipedia:Tonality diamond|7-limit tonality diamond]].
+These things are contained by the 7-limit, but not the 5-limit:
+* The [[7-odd-limit|7-]] and [[9-odd-limit]];
+* Mode 4 and 5 of the harmonic or subharmonic series.
+The 7-odd-limit is a constraint on the selection of just intervals for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[6/5]], [[5/4]], [[4/3]], [[7/5]], [[10/7]], [[3/2]], [[8/5]], [[5/3]], [[12/7]], [[7/4]], [[2/1]], which is known as the 7-odd-limit [[tonality diamond]].
 The phrase "7-limit just intonation" usually refers to the 7-prime-limit and includes primes 2, 3, 5, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 5, and 7, which can be represented in [[The Seven Limit Symmetrical Lattices|3-dimensional lattice diagrams]], each prime represented by a different dimension. Lattices describing scales beyond the 7-limit require more than three dimensions, and in the 7-limit, such lattices have unique features which simplify the relations between 7-limit chords.
-For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as [[11-limit|11-]] or [[13-limit]], which usually sound much more exotic.
+For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as 11- or [[13-limit]], which usually sound much more exotic.
+== Edo approximation ==
+Here is a list of [[edo]]s which tunes the 7-limit with more accuracy ([[monotonicity limit]] ≥ 7 and decreasing [[TE error]]): {{EDOs| 5, 8d, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, … }}. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
+Here is a list of edos which tunes the 7-limit well relative to their size ([[TE relative error]] < 5%): {{EDOs| 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612, … }}.
-A list of [[edo]]s which tunes the 7-limit with more accuracy: {{EDOs| 5, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612 }} and so on. Another list of edos which tunes the 7-limit well relative to their size ([[TE relative error|relative error]] < 5%): {{EDOs| 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612 }} and so on.
+== Intervals ==
+{{See also| User:Lériendil/Table of 21-odd-limit 7-limit intervals }}
-== List of intervals in the 7-prime-limit and 81-odd-limit ==
+Here is a table of intervals in the 7-prime-limit and 81-odd-limit.
 {| class="wikitable center-1 right-3"
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 | wa 8ve
 |}
+== Subgroups of the 7-limit ==
+* [[2.3.7 subgroup]]
+* [[2.5.7 subgroup]]
+* [[3.5.7 subgroup]]
 == Music ==
+=== Modern renderings ===
+; {{W|Gustav Holst}}
+* "Mars" from ''{{w|The Planets}}'' (1914–1917) – [http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ blog] | [http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3 play] – arranged by [[Chris Vaisvil]] (2012)
+; {{W|Scott Joplin}}
+* ''{{w|Maple Leaf Rag}}'' (1899) – [http://web.archive.org/web/20190412163308/http://soonlabel.com/xenharmonic/archives/2127 play] – arranged by [[Claudi Meneghin]] (2014)
+; {{W|Franz Liszt}}
+* {{W|Consolations (Liszt)|"Consolation No. 3"}} (1850) – [https://soundcloud.com/tallkite/liszt-consolation-3-by-ken-1 play] – Ken Stillwell performance, retuned by [[Kite Giedraitis]] to the [[kite33]] 7-limit JI scale
+; {{W|Johann Pachelbel}}
+* ''{{w|Pachelbel's Canon|Canon in D}}'' (''c''. 1680–1706) – [https://web.archive.org/web/20201127013008/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play] | [https://www.youtube.com/watch?v=HzQmaxDIxnc YouTube] – arranged by [[Claudi Meneghin]] (2011)
+=== 20th century ===
+; [[Ben Johnston]]
+* ''String Quartet No. 4'' (1973) – [https://newworldrecords.bandcamp.com/track/crossings-the-ascent-string-quartet-no-4-amazing-grace Bandcamp] | [https://www.youtube.com/watch?v=ReHIe0WDvNs YouTube] – performed by Kepler Quartet
+=== 21st century ===
+; [[Abnormality]]
+* [https://www.youtube.com/watch?v=WuW5COnfOlE ''Just Elevation''] (2023)
+; [[Jacob Adler]]
+* [https://m.youtube.com/watch?v=IUePyH2C9Y0 ''7-Limit Harmony''] (2024)
+; [[Ivor Darreg]]
+* [http://web.archive.org/web/20201127014610/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 ''Prelude #1 in 7-limit JI'']
+; [[dotuXil]]
+* [https://dotuxil.bandcamp.com/track/waterpad "waterpad"] from [https://dotuxil.bandcamp.com/album/collected-refractions ''Collected Refractions''] (2024)
+; [[E8 Heterotic]]
+* [https://youtu.be/mecOmJbqbxU?si=ILByj2hvPIMKRjJT ''Justicar''] (2020)
+; [[Francium]]
+* [https://www.youtube.com/watch?v=NANoBRyxll8 ''Too Happy For My Mood''] (2023)
+* [https://www.youtube.com/watch?v=YcMcychEAoE ''The Bazillionth Party Track''] (2023)
+* [https://www.youtube.com/watch?v=qDfIzd_Q-Hc ''Counting to Infinity''] (2025)
+* "You Geese" from ''Holy Carp'' (2025) – [https://open.spotify.com/track/5xcKZqwgw2SXSZRf1NQsyT Spotify] | [https://francium223.bandcamp.com/track/you-geese Bandcamp] | [https://www.youtube.com/watch?v=jlLQYfHp69A YouTube]
+; [[Kite Giedraitis]]
+* [http://tallkite.com/music/IHearNumbers.html ''I Hear Numbers'']
 ; [[Ralph Lewis]]
 * [http://micro.soonlabel.com/0-praxis/audio/August/august_12_Ruckus.mp3 ''Ruckus From the Quiet Zone'']
+; [[Kaiveran Lugheidh]]
+* [https://soundcloud.com/vale-10/nostalgic-blue Nostalgic Blue] (2017) – in 2.3.7 subgroup
+; [[Nick, The NRG]]
+* [https://www.youtube.com/watch?v=6IBM_JX52ck ''Cloudy Dreams''] (2022)
+; [[Juhani Nuorvala]]
+* [https://www.youtube.com/watch?v=0bXhY83asUo ''The tap dance scene from Flash Flash''] (2019)
+; [[Gene Ward Smith]]
+* ''Clinton Variations'' (2010) – [http://www.archive.org/details/ClintonVariations detail] | [http://www.archive.org/download/ClintonVariations/clinton.mp3 play]
 ; [[Chris Vaisvil]]
@@ Line 617: / Line 686: @@
 * [http://micro.soonlabel.com/centaur_tuning/Prelude_For_Centaur_Tuned_Piano.mp3 ''Prelude for Centaur Tuned Piano'']
-; [[Ivor Darreg]]
+; [[Randy Wells]]
-* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prelude%20%231%20for%207-limit%20JI.mp3 ''Prelude #1 in 7-limit JI'']{{dead link}}
+* [https://www.youtube.com/watch?v=rTvMMwkH2Z8 ''The Antidote for Entropy''] (2022)
-; [[Gene Ward Smith]]
-* [http://www.archive.org/details/ClintonVariations ''Clinton Variations''] [http://www.archive.org/download/ClintonVariations/clinton.mp3 play]
-* [http://www.youtube.com/watch?v=HzQmaxDIxnc&feature=channel_video_title ''Pachelbel's Canon in D in 7-limit JI''] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Pachelbel_s%20Canon%20in%20D%20-%20Relaxing%20music,%20with%20mountain%20views.mp3 play]
-* [http://clones.soonlabel.com/public/classical-music/mars-7-limit-kontakt5.mp3 ''Mars in 7-Limit JI''] from [[Wikipedia: The Planets|The Planets]] – arrangement of Gustav Holst's orchestral suite ([http://chrisvaisvil.com/gustav-holsts-mars-arranged-for-7-limit-ji-orchestra/ blog entry])
-; [[Kite Giedraitis]]
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Kite/Consolation%20%233%20by%20Ken%20Stillwell%20retuned.mp3 Liszt Consolation #3]{{dead link}} Ken Stillwell performance, retuned to the [[kite33]] 7-limit JI scale
-* [http://tallkite.com/music/IHearNumbers.html ''I Hear Numbers'']
 == See also ==
-* [[Harmonic limit]]
-* [[7-odd-limit]]
 * [[Wikipedia: Highly composite number]]