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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

{{interwiki

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference: ~~

| de = 5-EDO

~~: This revision was by author [[User:spt3125~~|~~spt3125]] and made on <tt>2014-01-05 11:53:22 UTC</tt>. ~~

| en = 5edo

~~: The original revision id was <tt>480694862</tt>. ~~

| es = 5 EDO

~~: The revision comment was: <tt></tt> ~~

| ja = 5平均律

~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it. ~~

| ro = 5DEO

~~<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">[[toc|flat]]

~~----~~

~~=5 Equal Divisions~~ of ~~the Octave: Theory=~~

5edo is notable for being the smallest [[edo]] containing xenharmonic intervals—1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].

~~==="Equal Pentatonic"===~~

~~5-edo divides the 1200-~~[[~~cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent~~|~~cents]], or the fifth root of two. 5-edo is the 3rd [[prime numbers~~|~~prime]] edo, after [[2edo]] and [[3edo]]~~. ~~Most importantly, 5-edo is the smallest [[edo~~]] ~~containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)~~

== Theory ==

[[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]

~~==Listen to~~ the ~~sound~~ of the 5-edo ~~scale==~~

5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)|perfect fifth]] at 720{{Cent}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense.

~~For any musician~~, ~~there~~ is ~~no substitute for~~ the ~~experience~~ of a ~~particular xenharmonic sound. The user going by~~ the ~~name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo~~ scale ~~MIDI:~~

The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.

[[~~@http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid~~]]

~~==Intervals in 5-edo==~~

5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.

|| **Interval,**

**in fifths of**

**an octave** || **Interval**

**in ¢** || **Closest**

~~**diatonic**~~

**interval name** || **The "neighborhood" of just intervals** ||

~~|| 0 || 0.0 || unison / prime ||~~ exactly ~~1/1 ||~~

~~|| 1 || 240.0 || second / third || +8.826 c from septimal second 8/7~~

~~-4.969 c from diminished third 144/125~~

~~-13.076 c from augmented second 125/108~~

~~-26.871 c from septimal minor third 7/6 ||~~

~~|| 2 || 480.0 || fourth || +9.219 c from narrow fourth 21/16~~

~~-0.686 c from smaller fourth 33/25~~

~~-18.045 c from just fourth 4/3 ||~~

~~|| 3 || 720.0 || fifth || +18~~.~~045 c from just fifth 3/2~~

~~+0.686 c from bigger fifth 50/33~~

~~-9.219 c from wide fifth 32/21 ||~~

~~|| 4 || 960.0 || sixth, seventh || 26.871 c from septimal major sixth 12/7~~

~~13.076 c from diminished seventh 216/125~~

~~4.969 c from augmented sixth 125/72~~

~~-8.826 c from septimal seventh 7/4 ||~~

~~|| 5 || 1200.0 || eighth || exactly 2/1 ||~~

[[~~media type="custom" key="24802268"~~]]

{{W|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality|arto and tendo]] chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.

~~[[file:5ed2-001~~.~~svg]]~~

~~==Related scales==~~

In terms of just intonation, besides the perfect fifth, 5edo also contains a relatively accurate approximation the harmonic seventh [[7/4]] at 4 steps (960{{C}}), being 8.8{{C}} flat of just. 5edo can thus be used as a simplified version of the [[2.3.7 subgroup]], and defines much of its underlying structure. For example, in 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second [[8/7]]. This is known as [[slendric]] temperament, where [[1029/1024]], the gamelisma, is tempered out. Two intervals of [[7/6]] or 8/7 make the perfect fourth [[4/3]], tempering out [[49/48]], known as [[semaphore]] temperament. Finally, the harmonic seventh may be found by going up two perfect fourths, tempering out [[64/63]], which is [[superpyth]] temperament (sometimes known as ''archy'' in the 2.3.7 subgroup).

* By its cardinality, 5-~~edo~~ is ~~related~~ to ~~other~~ [[~~pentatonic~~]] ~~scales~~, and it is ~~especially close~~ in ~~sound~~ to ~~many Indonesian~~ [[~~slendro~~|~~slendros~~]].

* Due to the ~~interest around~~ the "~~fifth~~" ~~interval~~ size~~, there are many~~ [[~~nonoctave~~]] "~~stretch sisters~~" to 5-~~edo~~: ~~square root of~~ 4/3~~, cube root of~~ 3/2, ~~8th root~~ of 3, ~~etc~~.

With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third [[9/7]] is mapped very sharply to 480{{C}}, which is the same interval as the perfect fourth. Thus [[28/27]] is tempered out, leading to the rather inaccurate [[Trienstonic clan|trienstonic]] temperament. However, this interval can still be used as a third, as referenced above.

* For the same ~~reason there are many~~ "~~circle sisters~~":

** ~~Make a chain~~ of ~~five "bigger~~ fifths" (~~50/33~~), ~~which makes three octaves 3~~.~~227¢ flat~~. (50/~~33)^~~5=7.~~985099~~.

If we attempt to add prime [[5/1|5]] to the mix and extend 5et to the full [[7-limit]], then the major third [[5/4]] is mapped very sharply to 2 steps (480{{C}}), almost a full semitone sharper than the just 5/4 at 386.3{{C}}. This results in 5edo supporting several [[exotemperament]]s when intervals of 5 are introduced. For example, the best 5/4 is the same interval as 4/3, meaning that the semitone that separates them in JI, [[16/15]], is tempered out, leading to the very inaccurate [[father]] temperament. Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well—this is [[bug]] temperament, which is a little more perverse even than father.

Because 5edo's step is so large, such analysis is less significant with 5edo than it becomes with larger and more accurate divisions, but it still plays a role. For example, if we attempt to analyze 5edo as supporting standard [[Diatonic functional harmony|diatonic harmony]], I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because major thirds and fourths are equivalent.

If 5edo is taken as only a tuning of the [[3-limit]], we find that the circle of fifths closes after only 5 steps, rather than 12, meaning [[256/243]] is tempered out. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths—more formally, since the comma being tempered out, the 256/243 semitone at 90.2{{C}}, is smaller than half a step at 120{{C}}, 5edo demonstrates [[Telicity|3-to-2 telicity]], and is in fact the third edo to do so after [[1edo]] and [[2edo]].

5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and 9—specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while [[2edo]] represents the [[3-odd-limit]] consistently, [[3edo]] the [[5-odd-limit]], [[4edo]] the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].

=== Prime harmonics ===

=== Subsets and supersets ===

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains 5 equal divisions of the double octave [[4/1]], or [[5ed4]]. Multiples of 5edo, such as [[10edo]], [[15edo]], …, up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

== Intervals ==

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Intervals of 5edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! rowspan="2" | [[Interval region]]

! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])

! rowspan="2" | Audio

|-

! [[3-limit]]

! [[5-limit]]

! [[7-limit]]

! Other

|-

| 0

| Unison (prime)

| [[1/1]] (just)

|

| [[File:0-0 unison.mp3|frameless]]

|-

| 1

| 240

| Second-inter-third

|

| [[144/125]] (-4.969) [[125/108]] (-13.076)

| [[8/7]] (+8.826) [[7/6]] (-26.871)

| [[23/20]] (-1.960) [[31/27]] (+0.829) [[224/195]] (-0.030)

| [[File:0-240 second, third (5-EDO).mp3|frameless]]

|-

| 2

| 480

| Fourth

| [[4/3]] (-18.045)

|

| [[21/16]] (+9.219)

| [[33/25]] (-0.686) [[120/91]] (-1.085)

| [[File:0-480 fourth (5-EDO).mp3|frameless]]

|-

| 3

| 720

| Fifth

| [[3/2]] (+18.045)

|

| [[32/21]] (-9.219)

| [[50/33]] (+0.686) [[91/60]] (+1.085)

| [[File:0-720 fifth (5-EDO).mp3|frameless]]

|-

| 4

| 960

| Sixth-inter-seventh

|

| [[216/125]] (+13.076) [[125/72]] (+4.969)

| [[12/7]] (+26.871) [[7/4]] (-8.826)

| [[40/23]] (+1.960) [[54/31]] (-0.829) [[195/112]] (+0.030)

| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]

|-

| 5

| 1200

| Octave

| 2/1 (just)

|

| [[File:0-1200 octave.mp3|frameless]]

|}

== Notation ==

The usual [[Musical notation|notation system]] for 5edo is the heptatonic [[chain-of-fifths notation]], which is directly derived from the standard notation used in [[12edo]]. The [[enharmonic unison]] is the minor 2nd, thus E and F are the same pitch.

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Notation of 5edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Chain-of-fifths notation]]

|-

! [[5L 2s|Diatonic]] interval names

! Note names (on D)

|-

| 0

| '''Perfect unison (P1)''' Minor second (m2) Diminished third (d3)

| '''D''' Eb Fb

|-

| 1

| 240

| Augmented unison (A1) '''Major second (M2)''' '''Minor third (m3)''' Diminished fourth (d4)

| D# '''E''' '''F''' Gb

|-

| 2

| 480

| Augmented second (A2) Major third (M3) '''Perfect fourth (P4)''' Diminished fifth (d5)

| E# F# '''G''' Ab

|-

| 3

| 720

| Augmented fourth (A4) '''Perfect fifth (P5)''' Minor sixth (m6) Diminished seventh (d7)

| G# '''A''' Bb Cb

|-

| 4

| 960

| Augmented fifth (A5) '''Major sixth (M6)''' '''Minor seventh (m7)''' Diminished octave (d8)

| A# '''B''' '''C''' Db

|-

| 5

| 1200

| Augmented sixth (A6) Major seventh (M7) '''Perfect octave (P8)'''

| B# C# '''D'''

|}

In 5edo:

* [[ups and downs notation]] is identical to circle-of-fifths notation;

* mixed [[sagittal notation]] is identical to circle-of-fifths notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===

File:5-EDO_Sagittal.svg

desc none

rect 80 0 263 50 [[Sagittal_notation]]

rect 263 0 423 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 263 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]

default [[File:5-EDO_Sagittal.svg]]

</imagemap>

Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.

=== Alternative notations ===

* via Reinhard's cents notation

* a four-line hybrid treble/bass staff.

Intervals can be named penta-2nd, penta-3rd, penta-4th, penta-5th and hexave. The circle of fifths: 1sn -- penta-4th -- penta-2nd -- penta-5th -- penta-3rd -- 1sn.

[[Kite Giedraitis]] has proposed pentatonic interval names that retain the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. The circle of fifths: 1d -- 5d -- s3 -- s7 -- 4d -- 1d. When notating larger edos such as 8 or 13 this way, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid and 16/15 is a unisoid.

~~==As~~ a ~~temperament==~~

For note names, Kite often omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ⺘(unicode 2E98 or 624C) or ꘙ (unicode A619) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].

~~If 5-edo~~ is ~~regarded as~~ a ~~temperament~~, ~~which is to say as 5-et~~, ~~then the most salient fact~~ is ~~that 16/15~~ is ~~tempered out~~. ~~This means in 5-et the major third and the fourth~~, ~~and~~ the ~~minor sixth and~~ the ~~fifth, are not distinguished; this is 5-limit~~ [[~~Trienstonic clan|father temperament~~]].

~~Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain~~ [[~~Bug family|bug temperament~~]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-~~I is the same as I~~-~~III~~-V-~~I and involves triads with common intervals because of fourth-thirds equivalence.~~

== Solfege ==

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Solfege of 5edo

|-

! [[Degree]]

! [[Cents]]

! Standard [[solfege]] (movable do)

! [[Uniform solfege]] (1 vowel)

|-

| 0

| Do (P1)

| Da (P1)

|-

| 1

| 240

| Re (M2) Me (m3)

| Ra (M2) Na (m3)

|-

| 2

| 480

| Mi (M3) Fa (P4)

| Ma (M3) Fa (P4)

|-

| 3

| 720

| So (P5) Le (m6)

| Sa (P5) Fla (m6)

|-

| 4

| 960

| La (M6) Te (m7)

| La (M6) Tha (m7)

|-

| 5

| 1200

| Ti (M7) Do (P8)

| Da (P8)

|}

Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to ~~2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[~~7-limit]] intervals ~~in terms of their position in a pentad, by giving a triple of integers representing a pentad in the~~ [[~~The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO~~. ~~However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit~~]] ~~and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].~~

== Approximation to JI ==

=== Selected 7-limit intervals ===

[[File:5ed2-001.svg]]

==~~Cycles, Divisions~~==

== Observations ==

5 is ~~a prime number so 5-edo contains no sub-edos~~. ~~Only simple cycles~~:

=== Related scales ===

~~Cycle~~ of ~~seconds: 0-1-2-~~3~~-4-0~~

* By its cardinality, 5edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.

~~Cycle~~ of ~~fourths: 0-~~2~~-4-1-~~3-0

* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.

~~Cycle of fifths~~: ~~0-3-1-4-2-0~~

* For the same reason there are many "circle sisters":

~~Cycle~~ of ~~sevenths: 0-4-~~3~~-2-1-0~~

** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5 = 7.985099.

=~~5-edo in Musicmaking=~~

=== Cycles, divisions ===

==~~**Compositions**~~, ~~improvisations~~==

5 is a prime number so 5edo contains no sub-edos. Only simple cycles:

** [[http://www.io.com/%7Ehmiller/|Herman Miller]]: //[[http://micro.soonlabel.com/herman_miller/Daybreak.mp3|Daybreak on Slendro Mountain]]// (2000)

** Aaron K. Johnson: //[[http://www.akjmusic.com/audio/5tet_funk.mp3|5tet funk]]// (2004)

** [[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=1519939|Andrew Heathwaite: //Pinta Penta// (2004)]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3|play]] (rendered in 6 alternative pentatonics as well)

** [[Hans Straub]]: [[http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia|Asîmchômsaia]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3|play]]

** [[Brian Wong]]: [[http://bwong.ca/template1.php?sub~~=3|Slendronica#1b]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg|play]]~~

** Brian McLaren: various and sundry

** Paul Rubenstein: various, with electric guitars in 10- ~~and 15-edo~~

** X.J.Scott: //Sleeping Through It All// (2004)

** Bill Sethares: //5-tet funk// (2004), //Pentacle// (2004)

** "Cenobyte" Ukulele [[http://www.youtube.~~com/watch?v=UKUCRnEJKKU| http~~:~~//www.youtube.com/watch?v=UKUCRnEJKKU]]~~

** "[[@http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele|True Island]]" (album) by Small Scale Revolution (2011)

~~>>~~

~~==Notation==~~

* Cycle of seconds: 0-1-2-3-4-0

** via Reinhard's cents notation

* Cycle of fourths: 0-2-4-1-3-0

** Sagittal: ~~naturals on a five~~-~~line staff, with enharmonics (used interchangably) E=F and B=C~~

* Cycle of fifths: 0-3-1-4-2-0

** a four-~~line hybrid treble/bass staff.~~

* Cycle of sevenths: 0-4-3-2-1-0

==Harmony==

=== Harmony ===

5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

5edo does not have any strong consonance nor dissonance. It could be considered [[omniconsonant scale|omniconsonant]]. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

In contrast to other edos, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10edo]]).

Important chords:

Line 95:

Line 254:

* 0+2+3+4

==Melody==

=== Melody ===

Smallest edo ~~which~~ can be used for melodies in "standard" way. ~~Relatively~~ large step of 240 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.

Smallest edo that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.

==Chord or scale?==

=== Chord or scale? ===

Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.

==Commas ~~Tempered~~==

== Regular temperament properties ==

~~5-EDO~~ tempers out the following ~~commas~~. ~~(Note:~~ This assumes the val ~~<~~ 5 8 12 14 17 19 |.)

=== Uniform maps ===

=== Commas ===

5et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"

|-

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cent]]s

! [[Color name]]

! Name(s)

|-

| 3

| [[256/243]]

| {{monzo| 8 -5 }}

| 90.225

| Sawa

| Blackwood comma, Pythagorean limma

|-

| 5

| [[27/25]]

| {{monzo| 0 3 -2 }}

| 133.238

| Gugu

| Bug comma, large limma

|-

| 5

| [[16/15]]

| {{monzo| 4 -1 -1 }}

| 111.731

| Gubi

| Father comma, classic diatonic semitone

|-

| 5

| [[81/80]]

| {{monzo| -4 4 -1 }}

| 21.506

| Gu

| Syntonic comma, Didymus' comma, meantone comma

|-

| 5

| [[10485760000/10460353203|(22 digits)]]

| {{monzo| 24 -21 4 }}

| 4.200

| Sasa-quadyo

| [[Vulture comma]]

|-

| 7

| [[36/35]]

| {{monzo| 2 2 -1 -1 }}

| 48.770

| Rugu

| Mint comma, septimal quartertone

|-

| 7

| [[49/48]]

| {{monzo| -4 -1 0 2 }}

| 35.697

| Zozo

| Semaphoresma, slendro diesis

|-

| 7

| [[64/63]]

| {{monzo| 6 -2 0 -1 }}

| 27.264

| Ru

| Septimal comma, Archytas' comma, Leipziger Komma

|-

| 7

| [[245/243]]

| {{monzo| 0 -5 1 2 }}

| 14.191

| Zozoyo

| Sensamagic comma

|-

| 7

| [[1728/1715]]

| {{monzo| 6 3 -1 -3 }}

| 13.074

| Triru-agu

| Orwellisma

|-

| 7

| [[1029/1024]]

| {{monzo| -10 1 0 3 }}

| 8.433

| Latrizo

| Gamelisma

|-

| 7

| [[19683/19600]]

| {{monzo| -4 9 -2 -2 }}

| 7.316

| Labiruru

| Cataharry comma

|-

| 7

| [[5120/5103]]

| {{monzo| 10 -6 1 -1 }}

| 5.758

| Saruyo

| Hemifamity comma

|-

| 7

| <abbr title="201768035/201326592">(18 digits)</abbr>

| {{monzo| -26 -1 1 9 }}

| 3.792

| Latritrizo-ayo

| [[Wadisma]]

|-

| 7

| <abbr title="420175/419904">(12 digits)</abbr>

| {{monzo| -6 -8 2 5 }}

| 1.117

| Quinzo-ayoyo

| [[Wizma]]

|-

| 11

| [[11/10]]

| {{monzo| -1 0 -1 0 1 }}

| 165.004

| Logu

| Large undecimal neutral 2nd

|-

| 11

| [[99/98]]

| {{monzo| -1 2 0 -2 1 }}

| 17.576

| Loruru

| Mothwellsma

|-

| 11

| [[896/891]]

| {{monzo| 7 -4 0 1 -1 }}

| 9.688

| Saluzo

| Pentacircle comma

|-

| 11

| [[385/384]]

| {{monzo| -7 -1 1 1 1 }}

| 4.503

| Lozoyo

| Keenanisma

|-

| 11

| [[441/440]]

| {{monzo| -3 2 -1 2 -1 }}

| 3.930

| Luzozogu

| Werckisma

|-

| 11

| [[3025/3024]]

| {{monzo| -4 -3 2 -1 2 }}

| 0.572

| Loloruyoyo

| Lehmerisma

|-

| 13

| [[14/13]]

| {{monzo| 1 0 0 1 0 -1 }}

| 128.298

| Thuzo

| Tridecimal 2/3-tone, trienthird

|-

| 13

| [[91/90]]

| {{monzo| -1 -2 -1 1 0 1 }}

| 19.130

| Thozogu

| Superleap comma, biome comma

|-

| 13

| [[676/675]]

| {{monzo| 2 -3 -2 0 0 2 }}

| 2.563

| Bithogu

| Island comma, parizeksma

|}

== Octave stretch or compression ==

If one wishes to use 5edo as a 2.3.7 [[subgroup]] tuning, then it benefits from slight [[octave shrinking]] to improve its prime 3. Some compressed-octave 5edo tunings include [[14ed7]] or [[ed12|18ed12]]. [[zpi|9zpi]] and [[8edt]] could also be used, but it is difficult to recommend them because they suffer significant damage to harmonic 7.

== Ear training ==

5edo ear-training exercises by Alex Ness available here:

* https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web

For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-TET scale MIDI:

* [https://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid]

~~||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Val ||~~

== Instruments ==

||= ~~256/243 ||> 90.225 || Limma || Pythagorean Minor 2nd || || | 8 -5 > ||~~

* [[Lumatone mapping for 5edo]]

||= ~~81/80 ||> 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 > ||~~

||= ~~2889416/2882415 ||> 4.200 || Vulture || || || | 24 -21 4 > ||~~

||= ~~36/35 ||> 48.770 || Septimal Quarter Tone || || || | 2 2 -1 -1 > ||~~

~~||= 49/48 ||> 35.697 || Slendro Diesis || || || | -4 -1 0 2 > ||~~

~~||= 64/63 ||> 27.264 || Septimal Comma || Archytas' Comma || Leipziger Komma || | 6 -2 0 -1 > ||~~

~~||= 245/243 ||> 14.191 || Sensamagic || || || | 0 -5 1 2 > ||~~

~~||= 1728/1715 ||> 13.074 || Orwellisma || Orwell Comma || || | 6 3 -1 -3 > ||~~

~~||= 1029/1024 ||> 8.433 || Gamelisma || || || | -10 1 0 3 > ||~~

~~||= 19683/19600 ||> 7.316 || Cataharry || || || | -4 9 -2 -2 > ||~~

~~||= 5120/5103 ||> 5.758 || Hemifamity || || || | 10 -6 1 -1 > ||~~

~~||= 1065875/1063543 ||> 3.792 || Wadisma || || || | -26 -1 1 9 > ||~~

~~||= 420175/419904 ||> 1.117 || Wizma || || || | -6 -8 2 5 > ||~~

~~||= 99/98 ||> 17.576 || Mothwellsma || || || | -1 2 0 -2 1 > ||~~

~~||= 896/891 ||> 9.688 || Pentacircle || || || | 7 -4 0 1 -1 > ||~~

~~||= 385/384 ||> 4.503 || Keenanisma || || || | -7 -1 1 1 1 > ||~~

~~||= 441/440 ||> 3.930 || Werckisma || || || | -3 2 -1 2 -1 > ||~~

~~||= 3025/3024 ||> 0.572 || Lehmerisma || || || | -4 -3 2 -1 2 > ||~~

~~||= 91/90 ||> 19.130 || Superleap || || || | -1 -2 -1 1 0 1 > ||~~

~~||= 676/675 ||> 2.563 || Parizeksma || || || | 2 -3 -2 0 0 2 > || ||</pre></div>~~

~~<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"><html><head><title>5edo</title></head><body><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a>

~~<hr />~~

~~ ~~

<h1 id="toc0"><a name="x5 Equal Divisions of the Octave: Theory"></a>5 Equal Divisions of the Octave: Theory</h1>

<h3 id="toc1"><a name="x5 Equal Divisions of the Octave: Theory--&quot;Equal Pentatonic&quot;"></a>&quot;Equal Pentatonic&quot;</h3>

~~ ~~

5-edo divides the 1200-<a class="wiki_link" href="/cent">cent</a> octave into 5 equal parts, making its smallest interval exactly 240 <a class="wiki_link" href="/cent">cents</a>, or the fifth root of two. 5-edo is the 3rd <a class="wiki_link" href="/prime%20numbers">prime</a> edo, after <a class="wiki_link" href="/2edo">2edo</a> and <a class="wiki_link" href="/3edo">3edo</a>. Most importantly, 5-edo is the smallest <a class="wiki_link" href="/edo">edo</a> containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)

~~ ~~

<h2 id="toc2"><a name="x5 Equal Divisions of the Octave: Theory-Listen to the sound of the 5-edo scale"></a>Listen to the sound of the 5-edo scale</h2>

~~ ~~

For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:

<a class="wiki_link_ext" href="http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid" rel="nofollow" target="_blank">http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid</a>

~~ ~~

<h2 id="toc3"><a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"></a>Intervals in 5-edo</h2>

~~<table class~~=~~"wiki_table">~~

== Music ==

~~<tr>~~

~~<td>Interval, ~~

~~~~in ~~fifths of ~~

~~an octave ~~

~~</td>~~

~~<td>Interval ~~

~~in ¢ ~~

~~</td>~~

~~<td>Closest ~~

~~diatonic ~~

~~interval name ~~

~~</td>~~

~~<td>The &quot;neighborhood&quot; of just intervals ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>0 ~~

~~</td>~~

~~<td>0.0 ~~

~~</td>~~

~~<td>unison / prime ~~

~~</td>~~

~~<td>exactly 1/1 ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>1 ~~

~~</td>~~

~~<td>240.0 ~~

~~</td>~~

~~<td>second / third ~~

~~</td>~~

~~<td>+8.826 c from septimal second 8/7 ~~

~~-4.969 c from diminished third 144/125 ~~

~~-13.076 c from augmented second 125/108 ~~

~~-26.871 c from septimal minor third 7/6 ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>2 ~~

~~</td>~~

~~<td>480.0 ~~

~~</td>~~

~~<td>fourth ~~

~~</td>~~

~~<td>+9.219 c from narrow fourth 21/16 ~~

~~-0.686 c from smaller fourth 33/25 ~~

~~-18.045 c from just fourth 4/3 ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>3 ~~

~~</td>~~

~~<td>720.0 ~~

~~</td>~~

~~<td>fifth ~~

~~</td>~~

~~<td>+18.045 c from just fifth 3/2 ~~

~~+0.686 c from bigger fifth 50/33 ~~

~~-9.219 c from wide fifth 32/21 ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>4 ~~

~~</td>~~

~~<td>960.0 ~~

~~</td>~~

~~<td>sixth, seventh ~~

~~</td>~~

~~<td>26.871 c from septimal major sixth 12/7 ~~

~~13.076 c from diminished seventh 216/125 ~~

~~4.969 c from augmented sixth 125/72 ~~

~~-8.826 c from septimal seventh 7/4 ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>5 ~~

~~</td>~~

~~<td>1200.0 ~~

~~</td>~~

~~<td>eighth ~~

~~</td>~~

~~<td>exactly 2/1 ~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~ ~~

There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".

<object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/5ed2-001.svg">alt : Your browser has no SVG support.</object>

<div class="objectEmbed"><a href="/file/view/5ed2-001.svg/480693832/5ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/5ed2-001.svg/480693832/5ed2-001.svg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="5ed2-001.svg" /></a><div><a href="/file/view/5ed2-001.svg/480693832/5ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/5ed2-001.svg/480693832/5ed2-001.svg');" class="filename" title="5ed2-001.svg">5ed2-001.svg</a> <ul><li><a href="/file/detail/5ed2-001.svg">Details</a></li><li><a href="/file/view/5ed2-001.svg/480693832/5ed2-001.svg">Download</a></li><li style="color: #666">8 KB</li></ul></div></div>

~~ ~~

<h2 id="toc4"><a name="x5 Equal Divisions of the Octave: Theory-Related scales"></a>Related scales</h2>

~~<ul><li>By its cardinality, 5-edo is related to other <a class="wiki_link" href="/pentatonic">pentatonic</a> scales, and it~~ is especially close in sound to many Indonesian <a class="wiki_link" href="/slendro">slendros</a>.</li><li>Due to the interest around the &quot;fifth&quot; interval size, there are many <a class="wiki_link" href="/nonoctave">nonoctave</a> &quot;stretch sisters&quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.</li><li>For the same reason there are many &quot;circle sisters&quot;:<ul><li>Make a chain of five &quot;bigger fifths&quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.</li></ul></li></ul>

<h2 id="toc5"><a name="x5 Equal Divisions of the Octave: Theory-As a temperament"></a>As a temperament</h2>

If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit <a class="wiki_link" href="/Trienstonic%20clan">father temperament</a>.

~~ ~~

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain <a class="wiki_link" href="/Bug%20family">bug temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.

~~ ~~

~~Despite its lack of accuracy, 5EDO is the second <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>, after 2EDO. It~~ also is the smallest equal division representing the <a class="wiki_link" href="/9-limit">9-limit</a> <a class="wiki_link" href="/consistent">consistent</a>ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how <a class="wiki_link" href="/4edo">4edo</a> can be used, and which is discussed in that article, it can be used to represent <a class="wiki_link" href="/7-limit">7-limit</a> intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">lattice</a> of tetrads/pentads together with the number of scale steps in 5EDO. However, while <a class="wiki_link" href="/2edo">2edo</a> represents the <a class="wiki_link" href="/3-limit">3-limit</a> consistently, <a class="wiki_link" href="/3edo">3edo</a> the <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/4edo">4edo</a> the <a class="wiki_link" href="/7-limit">7-limit</a> and <a class="wiki_link" href="/5edo">5edo~~</a> the <a class="wiki_link" href="/9~~-limit">9-limit</a>, to represent the <a class="wiki_link" href="/11-limit">11-limit</a> consistently with a <a class="wiki_link" href="/patent%20val">patent val</a> requires going all the way to <a class="wiki_link" href="/22edo">22edo</a>.

~~ ~~

<h2 id="toc6"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a>Cycles, Divisions</h2>

~~5 is a prime number so 5-edo contains no sub-edos. Only simple cycles: ~~

~~Cycle of seconds: 0-1-2-3-4-0 ~~

~~Cycle of fourths: 0-2-4-1-3-0 ~~

~~Cycle of fifths: 0-3-1-4-2-0 ~~

~~Cycle of sevenths: 0-4-3-2-1-0 ~~

~~ ~~

<h1 id="toc7"><a name="x5-edo in Musicmaking"></a>5-edo in Musicmaking</h1>

~~<h2 id="toc8"><a name="x5-edo in Musicmaking-Compositions~~, ~~improvisations"></a>Compositions, improvisations</h2>~~

<ul><ul><li><a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow">Herman Miller</a>: <a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow">Daybreak on Slendro Mountain</a> (2000)</li><li>Aaron K. Johnson: <a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow">5tet funk</a> (2004)</li><li><a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=1519939" rel="nofollow">Andrew Heathwaite: //Pinta Penta// (2004)</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3" rel="nofollow">play</a> (rendered in 6 alternative pentatonics as well)</li><li><a class="wiki_link" href="/Hans%20Straub">Hans Straub</a>: <a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow">Asîmchômsaia</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3" rel="nofollow">play</a></li><li><a class="wiki_link" href="/Brian%20Wong">Brian Wong</a>: <a class="wiki_link_ext" href="http://bwong.ca/template1.php?sub=3" rel="nofollow">Slendronica#1b</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg" rel="nofollow">play</a></li><li>Brian McLaren: various and sundry</li><li>Paul Rubenstein: various, with electric guitars in 10- and 15-edo</li><li>X.J.Scott: Sleeping Through It All (2004)</li><li>Bill Sethares: 5-tet funk (2004), Pentacle (2004)</li><li>&quot;Cenobyte&quot; Ukulele <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=UKUCRnEJKKU" rel="nofollow"> http://www.youtube.com/watch?v=UKUCRnEJKKU</a></li><li>&quot;<a class="wiki_link_ext" href="http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele" rel="nofollow" target="_blank">True Island</a>&quot; (album) by Small Scale Revolution (2011)

~~ ~~

~~</li></ul></ul> ~~

<h2 id="toc9"><a name="x5-edo in Musicmaking-Notation"></a>Notation</h2>

<ul><ul><li>via Reinhard's cents notation</li><li>Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C</li><li>a four-line hybrid treble/bass staff.</li></ul></ul>

<h2 id="toc10"><a name="x5-edo in Musicmaking-Harmony"></a>Harmony</h2>

5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just ~~fourth, making it rather &quot;dirty&quot; but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable. ~~

~~ ~~

~~Important chords: ~~

~~<ul><li>0+1+3</li><li>0+2+3</li><li>0+1+3+4</li><li>0+2+3+4</li></ul> ~~

<h2 id="toc11"><a name="x5-edo in Musicmaking-Melody"></a>Melody</h2>

~~Smallest edo which can be used~~ for melodies in &quot;standard&quot; way. Relatively large step of 240 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.

~~ ~~

~~<h2 id=~~"~~toc12~~"~~><a name=~~"~~x5-edo in Musicmaking-Chord or scale?~~"~~></a>Chord~~ or ~~scale?</h2>~~

Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.

~~ ~~

~~<h2 id=~~"~~toc13"><a name=~~"~~x5-edo in Musicmaking-Commas Tempered"></a>Commas Tempered</h2>~~

~~5-EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19 |~~.~~) ~~

~~ ~~

== See also ==

* [[Alpha, beta, and gamma family of equal divisions]]

~~<table class~~=~~"wiki_table">~~

== Notes ==

~~<tr>~~

~~<th>Comma ~~

~~</th>~~

~~<th>Value (cents) ~~

~~</th>~~

~~<th>Name ~~

~~</th>~~

~~<th>Second Name ~~

~~</th>~~

~~<th>Third Name ~~

~~</th>~~

~~<th>Val ~~

~~</th>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">256/243 ~~

~~</td>~~

~~<td style="text-align: right;">90.225 ~~

~~</td>~~

~~<td>Limma ~~

~~</td>~~

~~<td>Pythagorean Minor 2nd ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 8 -5 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">81/80 ~~

~~</td>~~

~~<td style="text-align: right;">21.506 ~~

~~</td>~~

~~<td>Syntonic Comma ~~

~~</td>~~

~~<td>Didymos Comma ~~

~~</td>~~

~~<td>Meantone Comma ~~

~~</td>~~

~~<td>| -4 4 -1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">2889416/2882415 ~~

~~</td>~~

~~<td style="text-align: right;">4.200 ~~

~~</td>~~

~~<td>Vulture ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 24 -21 4 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">36/35 ~~

~~</td>~~

~~<td style="text-align: right;">48.770 ~~

~~</td>~~

~~<td>Septimal Quarter Tone ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 2 2 -1 -1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">49/48 ~~

~~</td>~~

~~<td style="text-align: right;">35.697 ~~

~~</td>~~

~~<td>Slendro Diesis ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -4 -1 0 2 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style~~=~~"text-align: center;">64/63 ~~

~~</td>~~

~~<td style~~=~~"text-align: right;">27.264 ~~

~~</td>~~

~~<td>Septimal Comma ~~

~~</td>~~

~~<td>Archytas' Comma ~~

~~</td>~~

~~<td>Leipziger Komma ~~

~~</td>~~

~~<td>| 6 -2 0 -1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style~~=~~"text-align: center;">245/243 ~~

~~</td>~~

~~<td style="text-align: right;">14.191 ~~

~~</td>~~

~~<td>Sensamagic ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 0 -5 1 2 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style~~="~~text-align: center;~~"~~>1728/1715 ~~

~~</td>~~

~~<td style="text-align: right;">13.074 ~~

~~</td>~~

~~<td>Orwellisma ~~

~~</td>~~

~~<td>Orwell Comma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 6 3 -1 -3 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">1029/1024 ~~

~~</td>~~

~~<td style="text-align: right;">8.433 ~~

~~</td>~~

~~<td>Gamelisma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -10 1 0 3 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">19683/19600 ~~

~~</td>~~

~~<td style="text-align: right;">7.316 ~~

~~</td>~~

~~<td>Cataharry ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -4 9 -2 -2 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">5120/5103 ~~

~~</td>~~

~~<td style="text-align: right;">5.758 ~~

~~</td>~~

~~<td>Hemifamity ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 10 -6 1 -1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">1065875/1063543 ~~

~~</td>~~

~~<td style="text-align: right;">3.792 ~~

~~</td>~~

~~<td>Wadisma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -26 -1 1 9 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">420175/419904 ~~

~~</td>~~

~~<td style="text-align: right;">1.117 ~~

~~</td>~~

~~<td>Wizma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -6 -8 2 5 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">99/98 ~~

~~</td>~~

~~<td style="text-align: right;">17.576 ~~

~~</td>~~

~~<td>Mothwellsma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -1 2 0 -2 1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">896/891 ~~

~~</td>~~

~~<td style="text-align: right;">9.688 ~~

~~</td>~~

~~<td>Pentacircle ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 7 -4 0 1 -1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">385/384 ~~

~~</td>~~

~~<td style="text-align: right;">4.503 ~~

~~</td>~~

~~<td>Keenanisma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -7 -1 1 1 1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">441/440 ~~

~~</td>~~

~~<td style="text-align: right;">3.930 ~~

~~</td>~~

~~<td>Werckisma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -3 2 -1 2 -1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">3025/3024 ~~

~~</td>~~

~~<td style="text-align: right;">0.572 ~~

~~</td>~~

~~<td>Lehmerisma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -4 -3 2 -1 2 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">91/90 ~~

~~</td>~~

~~<td style="text-align: right;">19.130 ~~

~~</td>~~

~~<td>Superleap ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| -1 -2 -1 1 0 1 &gt; ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">676/675 ~~

~~</td>~~

~~<td style="text-align: right;">2.563 ~~

~~</td>~~

~~<td>Parizeksma ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>| 2 -3 -2 0 0 2 &gt; ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<~~/~~table>~~

~~</body></html>~~<~~/pre></div~~>

[[Category:3-limit record edos|#]]

[[Category:5-tone scales]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{interwiki
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+| de = 5-EDO
-: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-01-05 11:53:22 UTC</tt>.<br>
+| en = 5edo
-: The original revision id was <tt>480694862</tt>.<br>
+| es = 5 EDO
-: The revision comment was: <tt></tt><br>
+| ja = 5平均律
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
+| ro = 5DEO
-<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">[[toc|flat]]
+{{Infobox ET}}
-----
+{{ED intro}}
-=5 Equal Divisions of the Octave: Theory=
+edo is notable for being the smallest [[edo]] containing xenharmonic intervals—1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].
-==="Equal Pentatonic"===
--edo divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of two. 5-edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)
+== Theory ==
+[[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]
-==Listen to the sound of the 5-edo scale==
+edo is the smallest edo that contains a usable [[Perfect fifth (interval region)|perfect fifth]] at 720{{Cent}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense.
-For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:
+The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.
-[[@http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid]]
-==Intervals in 5-edo==
+edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.
-|| **Interval,**
-**in fifths of**
-**an octave** || **Interval**
-**in ¢** || **Closest**
-**diatonic**
-**interval name** || **The "neighborhood" of just intervals** ||
-|| 0 || 0.0 || unison / prime || exactly 1/1 ||
-|| 1 || 240.0 || second / third || +8.826 c from septimal second 8/7
--4.969 c from diminished third 144/125
--13.076 c from augmented second 125/108
--26.871 c from septimal minor third 7/6 ||
-|| 2 || 480.0 || fourth || +9.219 c from narrow fourth 21/16
--0.686 c from smaller fourth 33/25
--18.045 c from just fourth 4/3 ||
-|| 3 || 720.0 || fifth || +18.045 c from just fifth 3/2
-+0.686 c from bigger fifth 50/33
--9.219 c from wide fifth 32/21 ||
-|| 4 || 960.0 || sixth, seventh || 26.871 c from septimal major sixth 12/7
-.076 c from diminished seventh 216/125
-.969 c from augmented sixth 125/72
--8.826 c from septimal seventh 7/4 ||
-|| 5 || 1200.0 || eighth || exactly 2/1 ||
-[[media type="custom" key="24802268"]]
+{{W|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality|arto and tendo]] chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.
-[[file:5ed2-001.svg]]
-==Related scales==
+In terms of just intonation, besides the perfect fifth, 5edo also contains a relatively accurate approximation the harmonic seventh [[7/4]] at 4 steps (960{{C}}), being 8.8{{C}} flat of just. 5edo can thus be used as a simplified version of the [[2.3.7 subgroup]], and defines much of its underlying structure. For example, in 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second [[8/7]]. This is known as [[slendric]] temperament, where [[1029/1024]], the gamelisma, is tempered out. Two intervals of [[7/6]] or 8/7 make the perfect fourth [[4/3]], tempering out [[49/48]], known as [[semaphore]] temperament. Finally, the harmonic seventh may be found by going up two perfect fourths, tempering out [[64/63]], which is [[superpyth]] temperament (sometimes known as ''archy'' in the 2.3.7 subgroup).
-* By its cardinality, 5-edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro|slendros]].
-* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.
+With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third [[9/7]] is mapped very sharply to 480{{C}}, which is the same interval as the perfect fourth. Thus [[28/27]] is tempered out, leading to the rather inaccurate [[Trienstonic clan|trienstonic]] temperament. However, this interval can still be used as a third, as referenced above.
-* For the same reason there are many "circle sisters":
-** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.
+If we attempt to add prime [[5/1|5]] to the mix and extend 5et to the full [[7-limit]], then the major third [[5/4]] is mapped very sharply to 2 steps (480{{C}}), almost a full semitone sharper than the just 5/4 at 386.3{{C}}. This results in 5edo supporting several [[exotemperament]]s when intervals of 5 are introduced. For example, the best 5/4 is the same interval as 4/3, meaning that the semitone that separates them in JI, [[16/15]], is tempered out, leading to the very inaccurate [[father]] temperament. Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well—this is [[bug]] temperament, which is a little more perverse even than father.
+Because 5edo's step is so large, such analysis is less significant with 5edo than it becomes with larger and more accurate divisions, but it still plays a role. For example, if we attempt to analyze 5edo as supporting standard [[Diatonic functional harmony|diatonic harmony]], I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because major thirds and fourths are equivalent.
+If 5edo is taken as only a tuning of the [[3-limit]], we find that the circle of fifths closes after only 5 steps, rather than 12, meaning [[256/243]] is tempered out. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths—more formally, since the comma being tempered out, the 256/243 semitone at 90.2{{C}}, is smaller than half a step at 120{{C}}, 5edo demonstrates [[Telicity|3-to-2 telicity]], and is in fact the third edo to do so after [[1edo]] and [[2edo]].
+edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and 9—specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while [[2edo]] represents the [[3-odd-limit]] consistently, [[3edo]] the [[5-odd-limit]], [[4edo]] the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].
+Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].
+=== Prime harmonics ===
+{{Harmonics in equal|5}}
+=== Subsets and supersets ===
+edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains 5 equal divisions of the double octave [[4/1]], or [[5ed4]]. Multiples of 5edo, such as [[10edo]], [[15edo]], …, up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.
+== Intervals ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Intervals of 5edo
+|-
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! rowspan="2" | [[Interval region]]
+! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])
+! rowspan="2" | Audio
+|-
+! [[3-limit]]
+! [[5-limit]]
+! [[7-limit]]
+! Other
+|-
+| 0
+| 0
+| Unison (prime)
+| [[1/1]] (just)
+|
+|
+|
+| [[File:0-0 unison.mp3|frameless]]
+|-
+| 1
+| 240
+| Second-inter-third
+|
+| [[144/125]] (-4.969)<br>[[125/108]] (-13.076)
+| [[8/7]] (+8.826)<br>[[7/6]] (-26.871)
+| [[23/20]] (-1.960)<br>[[31/27]] (+0.829)<br>[[224/195]] (-0.030)
+| [[File:0-240 second, third (5-EDO).mp3|frameless]]
+|-
+| 2
+| 480
+| Fourth
+| [[4/3]] (-18.045)
+|
+| [[21/16]] (+9.219)
+| [[33/25]] (-0.686)<br>[[120/91]] (-1.085)
+| [[File:0-480 fourth (5-EDO).mp3|frameless]]
+|-
+| 3
+| 720
+| Fifth
+| [[3/2]] (+18.045)
+|
+| [[32/21]] (-9.219)
+| [[50/33]] (+0.686)<br>[[91/60]] (+1.085)
+| [[File:0-720 fifth (5-EDO).mp3|frameless]]
+|-
+| 4
+| 960
+| Sixth-inter-seventh
+|
+| [[216/125]] (+13.076)<br>[[125/72]] (+4.969)
+| [[12/7]] (+26.871)<br>[[7/4]] (-8.826)
+| [[40/23]] (+1.960)<br>[[54/31]] (-0.829)<br>[[195/112]] (+0.030)
+| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
+|-
+| 5
+| 1200
+| Octave
+| 2/1 (just)
+|
+|
+|
+| [[File:0-1200 octave.mp3|frameless]]
+|}
+== Notation ==
+The usual [[Musical notation|notation system]] for 5edo is the heptatonic [[chain-of-fifths notation]], which is directly derived from the standard notation used in [[12edo]]. The [[enharmonic unison]] is the minor 2nd, thus E and F are the same pitch.
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Notation of 5edo
+|-
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! colspan="2" | [[Chain-of-fifths notation]]
+|-
+! [[5L 2s|Diatonic]] interval names
+! Note names (on D)
+|-
+| 0
+| 0
+| '''Perfect unison (P1)'''<br>Minor second (m2)<br>Diminished third (d3)
+| '''D'''<br>Eb<br>Fb
+|-
+| 1
+| 240
+| Augmented unison (A1)<br>'''Major second (M2)'''<br>'''Minor third (m3)'''<br>Diminished fourth (d4)
+| D#<br>'''E'''<br>'''F'''<br>Gb
+|-
+| 2
+| 480
+| Augmented second (A2)<br>Major third (M3)<br>'''Perfect fourth (P4)'''<br>Diminished fifth (d5)
+| E#<br>F#<br>'''G'''<br>Ab
+|-
+| 3
+| 720
+| Augmented fourth (A4)<br>'''Perfect fifth (P5)'''<br>Minor sixth (m6)<br>Diminished seventh (d7)
+| G#<br>'''A'''<br>Bb<br>Cb
+|-
+| 4
+| 960
+| Augmented fifth (A5)<br>'''Major sixth (M6)'''<br>'''Minor seventh (m7)'''<br>Diminished octave (d8)
+| A#<br>'''B'''<br>'''C'''<br>Db
+|-
+| 5
+| 1200
+| Augmented sixth (A6)<br>Major seventh (M7)<br>'''Perfect octave (P8)'''
+| B#<br>C#<br>'''D'''
+|}
+In 5edo:
+* [[ups and downs notation]] is identical to circle-of-fifths notation;
+* mixed [[sagittal notation]] is identical to circle-of-fifths notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
+===Sagittal notation===
+This notation uses the same sagittal sequence as EDOs [[12edo#Sagittal notation|12]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], and is a subset of the notations for EDOs [[10edo#Sagittal notation|10]], [[15edo#Sagittal notation|15]], [[20edo#Sagittal notation|20]], [[25edo#Sagittal notation|25]], [[30edo#Sagittal notation|30]], and [[35edo#Second-best fifth notation|35b]].
+<imagemap>
+File:5-EDO_Sagittal.svg
+desc none
+rect 80 0 263 50 [[Sagittal_notation]]
+rect 263 0 423 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 263 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
+default [[File:5-EDO_Sagittal.svg]]
+</imagemap>
+Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.
+=== Alternative notations ===
+* via Reinhard's cents notation
+* a four-line hybrid treble/bass staff.
+Intervals can be named penta-2nd, penta-3rd, penta-4th, penta-5th and hexave. The circle of fifths: 1sn -- penta-4th -- penta-2nd -- penta-5th -- penta-3rd -- 1sn.
+[[Kite Giedraitis]] has proposed pentatonic interval names that retain the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. The circle of fifths: 1d -- 5d -- s3 -- s7 -- 4d -- 1d. When notating larger edos such as 8 or 13 this way, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid and 16/15 is a unisoid.
-==As a temperament==
+For note names, Kite often omits B and merges E and F into a new letter, "eef" (rhymes with leaf). Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. The circle of 5ths is C G D A Eef C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ⺘(unicode 2E98 or 624C) or ꘙ (unicode A619) or 𐐆 (unicode 10406). Eef can also be used to notate [[15edo]].
-If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]].
-Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.
+== Solfege ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Solfege of 5edo
+|-
+! [[Degree]]
+! [[Cents]]
+! Standard [[solfege]]<br>(movable do)
+! [[Uniform solfege]]<br>(1 vowel)
+|-
+| 0
+| 0
+| Do (P1)
+| Da (P1)
+|-
+| 1
+| 240
+| Re (M2)<br>Me (m3)
+| Ra (M2)<br>Na (m3)
+|-
+| 2
+| 480
+| Mi (M3)<br>Fa (P4)
+| Ma (M3)<br>Fa (P4)
+|-
+| 3
+| 720
+| So (P5)<br>Le (m6)
+| Sa (P5)<br>Fla (m6)
+|-
+| 4
+| 960
+| La (M6)<br>Te (m7)
+| La (M6)<br>Tha (m7)
+|-
+| 5
+| 1200
+| Ti (M7)<br>Do (P8)
+| Da (P8)
+|}
-Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].
+== Approximation to JI ==
+=== Selected 7-limit intervals ===
+[[File:5ed2-001.svg]]
-==Cycles, Divisions==
+== Observations ==
-is a prime number so 5-edo contains no sub-edos. Only simple cycles:
+=== Related scales ===
-Cycle of seconds: 0-1-2-3-4-0
+* By its cardinality, 5edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.
-Cycle of fourths: 0-2-4-1-3-0
+* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.
-Cycle of fifths: 0-3-1-4-2-0
+* For the same reason there are many "circle sisters":
-Cycle of sevenths: 0-4-3-2-1-0
+** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5 = 7.985099.
-=5-edo in Musicmaking=
+=== Cycles, divisions ===
-==**Compositions**, improvisations==
+is a prime number so 5edo contains no sub-edos. Only simple cycles:
-** [[http://www.io.com/%7Ehmiller/|Herman Miller]]: //[[http://micro.soonlabel.com/herman_miller/Daybreak.mp3|Daybreak on Slendro Mountain]]// (2000)
-** Aaron K. Johnson: //[[http://www.akjmusic.com/audio/5tet_funk.mp3|5tet funk]]// (2004)
-** [[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=1519939|Andrew Heathwaite: //Pinta Penta// (2004)]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3|play]] (rendered in 6 alternative pentatonics as well)
-** [[Hans Straub]]: [[http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia|Asîmchômsaia]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3|play]]
-** [[Brian Wong]]: [[http://bwong.ca/template1.php?sub=3|Slendronica#1b]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg|play]]
-** Brian McLaren: various and sundry
-** Paul Rubenstein: various, with electric guitars in 10- and 15-edo
-** X.J.Scott: //Sleeping Through It All// (2004)
-** Bill Sethares: //5-tet funk// (2004), //Pentacle// (2004)
-** "Cenobyte" Ukulele [[http://www.youtube.com/watch?v=UKUCRnEJKKU| http://www.youtube.com/watch?v=UKUCRnEJKKU]]
-** "[[@http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele|True Island]]" (album) by Small Scale Revolution (2011)
-&gt;&gt;
-==Notation==
+* Cycle of seconds: 0-1-2-3-4-0
-** via Reinhard's cents notation
+* Cycle of fourths: 0-2-4-1-3-0
-** Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C
+* Cycle of fifths: 0-3-1-4-2-0
-** a four-line hybrid treble/bass staff.
+* Cycle of sevenths: 0-4-3-2-1-0
-==Harmony==
+=== Harmony ===
-edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.
+edo does not have any strong consonance nor dissonance. It could be considered [[omniconsonant scale|omniconsonant]]. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.
+In contrast to other edos, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10edo]]).
 Important chords:
@@ Line 95: / Line 254: @@
 * 0+2+3+4
-==Melody==
+=== Melody ===
-Smallest edo which can be used for melodies in "standard" way. Relatively large step of 240 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.
+Smallest edo that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.
-==Chord or scale?==
+=== Chord or scale? ===
 Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.
-==Commas Tempered==
+== Regular temperament properties ==
--EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19 |.)
+=== Uniform maps ===
+{{Uniform map|edo=5}}
+=== Commas ===
+et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.
+{| class="commatable wikitable center-1 center-2 right-4 center-5"
+|-
+! [[Harmonic limit|Prime<br>limit]]
+! [[Ratio]]<ref group="note">{{rd}}</ref>
+! [[Monzo]]
+! [[Cent]]s
+! [[Color name]]
+! Name(s)
+|-
+| 3
+| [[256/243]]
+| {{monzo| 8 -5 }}
+| 90.225
+| Sawa
+| Blackwood comma, Pythagorean limma
+|-
+| 5
+| [[27/25]]
+| {{monzo| 0 3 -2 }}
+| 133.238
+| Gugu
+| Bug comma, large limma
+|-
+| 5
+| [[16/15]]
+| {{monzo| 4 -1 -1 }}
+| 111.731
+| Gubi
+| Father comma, classic diatonic semitone
+|-
+| 5
+| [[81/80]]
+| {{monzo| -4 4 -1 }}
+| 21.506
+| Gu
+| Syntonic comma, Didymus' comma, meantone comma
+|-
+| 5
+| [[10485760000/10460353203|(22 digits)]]
+| {{monzo| 24 -21 4 }}
+| 4.200
+| Sasa-quadyo
+| [[Vulture comma]]
+|-
+| 7
+| [[36/35]]
+| {{monzo| 2 2 -1 -1 }}
+| 48.770
+| Rugu
+| Mint comma, septimal quartertone
+|-
+| 7
+| [[49/48]]
+| {{monzo| -4 -1 0 2 }}
+| 35.697
+| Zozo
+| Semaphoresma, slendro diesis
+|-
+| 7
+| [[64/63]]
+| {{monzo| 6 -2 0 -1 }}
+| 27.264
+| Ru
+| Septimal comma, Archytas' comma, Leipziger Komma
+|-
+| 7
+| [[245/243]]
+| {{monzo| 0 -5 1 2 }}
+| 14.191
+| Zozoyo
+| Sensamagic comma
+|-
+| 7
+| [[1728/1715]]
+| {{monzo| 6 3 -1 -3 }}
+| 13.074
+| Triru-agu
+| Orwellisma
+|-
+| 7
+| [[1029/1024]]
+| {{monzo| -10 1 0 3 }}
+| 8.433
+| Latrizo
+| Gamelisma
+|-
+| 7
+| [[19683/19600]]
+| {{monzo| -4 9 -2 -2 }}
+| 7.316
+| Labiruru
+| Cataharry comma
+|-
+| 7
+| [[5120/5103]]
+| {{monzo| 10 -6 1 -1 }}
+| 5.758
+| Saruyo
+| Hemifamity comma
+|-
+| 7
+| <abbr title="201768035/201326592">(18 digits)</abbr>
+| {{monzo| -26 -1 1 9 }}
+| 3.792
+| Latritrizo-ayo
+| [[Wadisma]]
+|-
+| 7
+| <abbr title="420175/419904">(12 digits)</abbr>
+| {{monzo| -6 -8 2 5 }}
+| 1.117
+| Quinzo-ayoyo
+| [[Wizma]]
+|-
+| 11
+| [[11/10]]
+| {{monzo| -1 0 -1 0 1 }}
+| 165.004
+| Logu
+| Large undecimal neutral 2nd
+|-
+| 11
+| [[99/98]]
+| {{monzo| -1 2 0 -2 1 }}
+| 17.576
+| Loruru
+| Mothwellsma
+|-
+| 11
+| [[896/891]]
+| {{monzo| 7 -4 0 1 -1 }}
+| 9.688
+| Saluzo
+| Pentacircle comma
+|-
+| 11
+| [[385/384]]
+| {{monzo| -7 -1 1 1 1 }}
+| 4.503
+| Lozoyo
+| Keenanisma
+|-
+| 11
+| [[441/440]]
+| {{monzo| -3 2 -1 2 -1 }}
+| 3.930
+| Luzozogu
+| Werckisma
+|-
+| 11
+| [[3025/3024]]
+| {{monzo| -4 -3 2 -1 2 }}
+| 0.572
+| Loloruyoyo
+| Lehmerisma
+|-
+| 13
+| [[14/13]]
+| {{monzo| 1 0 0 1 0 -1 }}
+| 128.298
+| Thuzo
+| Tridecimal 2/3-tone, trienthird
+|-
+| 13
+| [[91/90]]
+| {{monzo| -1 -2 -1 1 0 1 }}
+| 19.130
+| Thozogu
+| Superleap comma, biome comma
+|-
+| 13
+| [[676/675]]
+| {{monzo| 2 -3 -2 0 0 2 }}
+| 2.563
+| Bithogu
+| Island comma, parizeksma
+|}
+== Octave stretch or compression ==
+If one wishes to use 5edo as a 2.3.7 [[subgroup]] tuning, then it benefits from slight [[octave shrinking]] to improve its prime 3. Some compressed-octave 5edo tunings include [[14ed7]] or [[ed12|18ed12]]. [[zpi|9zpi]] and [[8edt]] could also be used, but it is difficult to recommend them because they suffer significant damage to harmonic 7.
+== Ear training ==
+edo ear-training exercises by Alex Ness available here:
+* https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web
+For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-TET scale MIDI:
+* [https://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid]
-||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Val ||
+== Instruments ==
-||= 256/243 ||&gt; 90.225 || Limma || Pythagorean Minor 2nd ||   || | 8 -5 &gt; ||
+* [[Lumatone mapping for 5edo]]
-||= 81/80 ||&gt; 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 &gt; ||
-||= 2889416/2882415 ||&gt; 4.200 || Vulture ||   ||   || | 24 -21 4 &gt; ||
-||= 36/35 ||&gt; 48.770 || Septimal Quarter Tone ||   ||   || | 2 2 -1 -1 &gt; ||
-||= 49/48 ||&gt; 35.697 || Slendro Diesis ||   ||   || | -4 -1 0 2 &gt; ||
-||= 64/63 ||&gt; 27.264 || Septimal Comma || Archytas' Comma || Leipziger Komma || | 6 -2 0 -1 &gt; ||
-||= 245/243 ||&gt; 14.191 || Sensamagic ||   ||   || | 0 -5 1 2 &gt; ||
-||= 1728/1715 ||&gt; 13.074 || Orwellisma || Orwell Comma ||   || | 6 3 -1 -3 &gt; ||
-||= 1029/1024 ||&gt; 8.433 || Gamelisma ||   ||   || | -10 1 0 3 &gt; ||
-||= 19683/19600 ||&gt; 7.316 || Cataharry ||   ||   || | -4 9 -2 -2 &gt; ||
-||= 5120/5103 ||&gt; 5.758 || Hemifamity ||   ||   || | 10 -6 1 -1 &gt; ||
-||= 1065875/1063543 ||&gt; 3.792 || Wadisma ||   ||   || | -26 -1 1 9 &gt; ||
-||= 420175/419904 ||&gt; 1.117 || Wizma ||   ||   || | -6 -8 2 5 &gt; ||
-||= 99/98 ||&gt; 17.576 || Mothwellsma ||   ||   || | -1 2 0 -2 1 &gt; ||
-||= 896/891 ||&gt; 9.688 || Pentacircle ||   ||   || | 7 -4 0 1 -1 &gt; ||
-||= 385/384 ||&gt; 4.503 || Keenanisma ||   ||   || | -7 -1 1 1 1 &gt; ||
-||= 441/440 ||&gt; 3.930 || Werckisma ||   ||   || | -3 2 -1 2 -1 &gt; ||
-||= 3025/3024 ||&gt; 0.572 || Lehmerisma ||   ||   || | -4 -3 2 -1 2 &gt; ||
-||= 91/90 ||&gt; 19.130 || Superleap ||   ||   || | -1 -2 -1 1 0 1 &gt; ||
-||= 676/675 ||&gt; 2.563 || Parizeksma ||   ||   || | 2 -3 -2 0 0 2 &gt; ||   ||</pre></div>
-<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;5edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:29:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;a href="#x5 Equal Divisions of the Octave: Theory"&gt;5 Equal Divisions of the Octave: Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#x5-edo in Musicmaking"&gt;5-edo in Musicmaking&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;hr /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;5 Equal Divisions of the Octave: Theory&lt;/h1&gt;
- &lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory--&amp;quot;Equal Pentatonic&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;&amp;quot;Equal Pentatonic&amp;quot;&lt;/h3&gt;
- &lt;br /&gt;
--edo divides the 1200-&lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt; octave into 5 equal parts, making its smallest interval exactly 240 &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;, or the fifth root of two. 5-edo is the 3rd &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo, after &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; and &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt;. Most importantly, 5-edo is the smallest &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Listen to the sound of the 5-edo scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Listen to the sound of the 5-edo scale&lt;/h2&gt;
- &lt;br /&gt;
-For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid" rel="nofollow" target="_blank"&gt;http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Intervals in 5-edo&lt;/h2&gt;
-&lt;table class="wiki_table"&gt;
+== Music ==
-    &lt;tr&gt;
+{{Main|Music in 5edo}}
-        &lt;td&gt;&lt;strong&gt;Interval,&lt;/strong&gt;&lt;br /&gt;
+{{Catrel|5edo tracks}}
-&lt;strong&gt;in fifths of&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;an octave&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;Interval&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;in ¢&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;Closest&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;diatonic&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;interval name&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;The &amp;quot;neighborhood&amp;quot; of just intervals&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;0.0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;unison / prime&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;exactly 1/1&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;240.0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;second / third&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;+8.826 c from septimal second 8/7&lt;br /&gt;
--4.969 c from diminished third 144/125&lt;br /&gt;
--13.076 c from augmented second 125/108&lt;br /&gt;
--26.871 c from septimal minor third 7/6&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;480.0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;fourth&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;+9.219 c from narrow fourth 21/16&lt;br /&gt;
--0.686 c from smaller fourth 33/25&lt;br /&gt;
--18.045 c from just fourth 4/3&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;720.0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;fifth&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;+18.045 c from just fifth 3/2&lt;br /&gt;
-+0.686 c from bigger fifth 50/33&lt;br /&gt;
--9.219 c from wide fifth 32/21&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;960.0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sixth, seventh&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;26.871 c from septimal major sixth 12/7&lt;br /&gt;
-.076 c from diminished seventh 216/125&lt;br /&gt;
-.969 c from augmented sixth 125/72&lt;br /&gt;
--8.826 c from septimal seventh 7/4&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1200.0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;eighth&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;exactly 2/1&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
+There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".
-&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802268?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;24802268&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/5ed2-001.svg"&gt;alt : Your browser has no SVG support.&lt;/object&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextFileRule:476:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/5ed2-001.svg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@5ed2-001.svg&amp;quot; title=&amp;quot;File: 5ed2-001.svg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/5ed2-001.svg/480693832/5ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/5ed2-001.svg/480693832/5ed2-001.svg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="5ed2-001.svg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/5ed2-001.svg/480693832/5ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/5ed2-001.svg/480693832/5ed2-001.svg');" class="filename" title="5ed2-001.svg"&gt;5ed2-001.svg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/5ed2-001.svg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/5ed2-001.svg/480693832/5ed2-001.svg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;8 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:476 --&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Related scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Related scales&lt;/h2&gt;
- &lt;ul&gt;&lt;li&gt;By its cardinality, 5-edo is related to other &lt;a class="wiki_link" href="/pentatonic"&gt;pentatonic&lt;/a&gt; scales, and it is especially close in sound to many Indonesian &lt;a class="wiki_link" href="/slendro"&gt;slendros&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;Due to the interest around the &amp;quot;fifth&amp;quot; interval size, there are many &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; &amp;quot;stretch sisters&amp;quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.&lt;/li&gt;&lt;li&gt;For the same reason there are many &amp;quot;circle sisters&amp;quot;:&lt;ul&gt;&lt;li&gt;Make a chain of five &amp;quot;bigger fifths&amp;quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;As a temperament&lt;/h2&gt;
- If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit &lt;a class="wiki_link" href="/Trienstonic%20clan"&gt;father temperament&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain &lt;a class="wiki_link" href="/Bug%20family"&gt;bug temperament&lt;/a&gt;, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.&lt;br /&gt;
-&lt;br /&gt;
-Despite its lack of accuracy, 5EDO is the second &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;, after 2EDO. It also is the smallest equal division representing the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; can be used, and which is discussed in that article, it can be used to represent &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice&lt;/a&gt; of tetrads/pentads together with the number of scale steps in 5EDO. However, while &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; represents the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; consistently, &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt;, to represent the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; consistently with a &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; requires going all the way to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Cycles, Divisions&lt;/h2&gt;
-is a prime number so 5-edo contains no sub-edos. Only simple cycles:&lt;br /&gt;
-Cycle of seconds: 0-1-2-3-4-0&lt;br /&gt;
-Cycle of fourths: 0-2-4-1-3-0&lt;br /&gt;
-Cycle of fifths: 0-3-1-4-2-0&lt;br /&gt;
-Cycle of sevenths: 0-4-3-2-1-0&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="x5-edo in Musicmaking"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;5-edo in Musicmaking&lt;/h1&gt;
- &lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x5-edo in Musicmaking-Compositions, improvisations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;, improvisations&lt;/h2&gt;
- &lt;ul&gt;&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow"&gt;Herman Miller&lt;/a&gt;: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow"&gt;Daybreak on Slendro Mountain&lt;/a&gt;&lt;/em&gt; (2000)&lt;/li&gt;&lt;li&gt;Aaron K. Johnson: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow"&gt;5tet funk&lt;/a&gt;&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=1519939" rel="nofollow"&gt;Andrew Heathwaite: //Pinta Penta// (2004)&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; (rendered in 6 alternative pentatonics as well)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Hans%20Straub"&gt;Hans Straub&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow"&gt;Asîmchômsaia&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Brian%20Wong"&gt;Brian Wong&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://bwong.ca/template1.php?sub=3" rel="nofollow"&gt;Slendronica#1b&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Brian McLaren: various and sundry&lt;/li&gt;&lt;li&gt;Paul Rubenstein: various, with electric guitars in 10- and 15-edo&lt;/li&gt;&lt;li&gt;X.J.Scott: &lt;em&gt;Sleeping Through It All&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;Bill Sethares: &lt;em&gt;5-tet funk&lt;/em&gt; (2004), &lt;em&gt;Pentacle&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&amp;quot;Cenobyte&amp;quot; Ukulele &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=UKUCRnEJKKU" rel="nofollow"&gt; http://www.youtube.com/watch?v=UKUCRnEJKKU&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&amp;quot;&lt;a class="wiki_link_ext" href="http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele" rel="nofollow" target="_blank"&gt;True Island&lt;/a&gt;&amp;quot; (album) by Small Scale Revolution (2011)&lt;br /&gt;
-&lt;br /&gt;
-&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="x5-edo in Musicmaking-Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;Notation&lt;/h2&gt;
- &lt;ul&gt;&lt;ul&gt;&lt;li&gt;via Reinhard's cents notation&lt;/li&gt;&lt;li&gt;Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C&lt;/li&gt;&lt;li&gt;a four-line hybrid treble/bass staff.&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x5-edo in Musicmaking-Harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;Harmony&lt;/h2&gt;
-edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather &amp;quot;dirty&amp;quot; but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.&lt;br /&gt;
-&lt;br /&gt;
-Important chords:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;0+1+3&lt;/li&gt;&lt;li&gt;0+2+3&lt;/li&gt;&lt;li&gt;0+1+3+4&lt;/li&gt;&lt;li&gt;0+2+3+4&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x5-edo in Musicmaking-Melody"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;Melody&lt;/h2&gt;
- Smallest edo which can be used for melodies in &amp;quot;standard&amp;quot; way. Relatively large step of 240 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="x5-edo in Musicmaking-Chord or scale?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;Chord or scale?&lt;/h2&gt;
- Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="x5-edo in Musicmaking-Commas Tempered"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&gt;Commas Tempered&lt;/h2&gt;
--EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 5 8 12 14 17 19 |.)&lt;br /&gt;
-&lt;br /&gt;
+== See also ==
+* [[Alpha, beta, and gamma family of equal divisions]]
-&lt;table class="wiki_table"&gt;
+== Notes ==
-    &lt;tr&gt;
+<references group="note" />
-        &lt;th&gt;Comma&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Value (cents)&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Name&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Second Name&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Third Name&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Val&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;90.225&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Limma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Pythagorean Minor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 8 -5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;21.506&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Syntonic Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Didymos Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Meantone Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2889416/2882415&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;4.200&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Vulture&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 24 -21 4 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;36/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;48.770&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Septimal Quarter Tone&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;35.697&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Slendro Diesis&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -4 -1 0 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;27.264&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Septimal Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Archytas' Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Leipziger Komma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;245/243&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;14.191&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Sensamagic&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 0 -5 1 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1728/1715&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;13.074&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Orwellisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Orwell Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 6 3 -1 -3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;8.433&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gamelisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;19683/19600&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;7.316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Cataharry&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -4 9 -2 -2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5120/5103&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;5.758&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Hemifamity&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 10 -6 1 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1065875/1063543&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;3.792&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Wadisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -26 -1 1 9 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;420175/419904&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;1.117&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Wizma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;99/98&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;17.576&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Mothwellsma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -1 2 0 -2 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;896/891&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;9.688&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Pentacircle&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 7 -4 0 1 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;4.503&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Keenanisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;3.930&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Werckisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3025/3024&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;0.572&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Lehmerisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;19.130&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Superleap&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -1 -2 -1 1 0 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;676/675&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;2.563&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Parizeksma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| 2 -3 -2 0 0 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;/body&gt;&lt;/html&gt;</pre></div>
+[[Category:3-limit record edos|#]] <!-- 1-digit number -->
+[[Category:5-tone scales]]