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| es = 5 EDO
| es = 5 EDO
| ja = 5平均律
| ja = 5平均律
| ro = 5DEO
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 5 (prime)
{{ED intro}}
| Step size = 240¢
 
| Fifth = 3\5 = 720¢
5edo is notable for being the smallest [[edo]] containing xenharmonic intervals—1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]].
| Major 2nd = 1\5 = 240¢
| Minor 2nd = 0\5 = 0¢
| Augmented 1sn = 1\5 = 240¢
}}
'''5-edo''' divides the octave into 5 equal parts, making its smallest interval exactly 240 [[cent]]s, 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, and 4edo are all subsets of [[12edo]].


== Theory ==
== Theory ==
[[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]
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.
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.
5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.
{{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.
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).
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.
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 ===
{{Harmonics in equal|5}}
=== 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"
{| class="wikitable center-all"
! colspan="2" | <!-- empty cell -->
|+ style="font-size: 105%;" | Intervals of 5edo
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
|-
! rowspan="2" | error
! rowspan="2" | [[Degree]]
! absolute (¢)
! rowspan="2" | [[Cent]]s
| 0.0
! rowspan="2" | [[Interval region]]
| +18.0
! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])
| +93.7
! rowspan="2" | Audio
| -8.8
|-
| -71.3
! [[3-limit]]
| +119.5
! [[5-limit]]
| -105.0
! [[7-limit]]
| -57.5
! Other
|-
|-
! [[Relative error|relative]] (%)
| 0
| 0
| +8
| 0
| +39
| Unison (prime)
| -4
| [[1/1]] (just)
| -30
|
| +50
|
| -44
|
| -24
| [[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]]
|-
|-
! colspan="2" | [[nearest edomapping]]
| 5
| 3
| 3
| 2
| 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
| 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
| 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
| 4
| 0
| 960
| 1
| Augmented fifth (A5)<br>'''Major sixth (M6)'''<br>'''Minor seventh (m7)'''<br>Diminished octave (d8)
| A#<br>'''B'''<br>'''C'''<br>Db
|-
|-
! colspan="2" | [[fifthspan]]
| 5
|0
| 1200
| +1
| Augmented sixth (A6)<br>Major seventh (M7)<br>'''Perfect octave (P8)'''
| -1
| B#<br>C#<br>'''D'''
| -2
| -1
| -2
| 0
| +2
|}
|}


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]].
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>


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


Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent|consistently]], 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]].  Nevertheless, because the comma tempered out for this EDO's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second EDO to demonstrate 3-to-2 [[telicity]].
=== Alternative notations ===
* via Reinhard's cents notation
* a four-line hybrid treble/bass staff.


== Intervals ==
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.
 
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]].


== Solfege ==
{| class="wikitable center-all"
{| class="wikitable center-all"
! Steps
|+ style="font-size: 105%;" | Solfege of 5edo
! [[Cent]]s
|-
! Closest diatonic <br> interval name
! [[Degree]]
! The "neighborhood" of just intervals
! [[Cents]]
! Standard [[solfege]]<br>(movable do)
! [[Uniform solfege]]<br>(1 vowel)
|-
|-
| 0
| 0
| 0
| 0
| unison / prime
| Do (P1)
| '''1/1'''
| Da (P1)
|-
|-
| 1
| 1
| 240
| 240
| second, third
| Re (M2)<br>Me (m3)
| +8.826¢ from septimal second [[8/7]] <br> -4.969¢ from diminished third [[144/125]] <br> -13.076¢ from augmented second [[125/108]] <br> -26.871¢ from septimal minor third [[7/6]]
| Ra (M2)<br>Na (m3)
|-
|-
| 2
| 2
| 480
| 480
| fourth
| Mi (M3)<br>Fa (P4)
| +9.219¢ from narrow fourth [[21/16]] <br> -0.686¢ from smaller fourth [[33/25]] <br> -18.045¢ from just fourth [[4/3]]
| Ma (M3)<br>Fa (P4)
|-
|-
| 3
| 3
| 720
| 720
| fifth
| So (P5)<br>Le (m6)
| +18.045¢ from just fifth [[3/2]] <br> +0.686¢ from bigger fifth [[50/33]] <br> -9.219¢ from wide fifth [[32/21]]
| Sa (P5)<br>Fla (m6)
|-
|-
| 4
| 4
| 960
| 960
| sixth, seventh
| La (M6)<br>Te (m7)
| 26.871¢ from septimal major sixth [[12/7]] <br> 13.076¢ from diminished seventh [[216/125]] <br> 4.969¢ from augmented sixth [[125/72]] <br> -8.826¢ from septimal seventh [[7/4]]
| La (M6)<br>Tha (m7)
|-
|-
| 5
| 5
| 1200
| 1200
| octave
| Ti (M7)<br>Do (P8)
| '''2/1'''
| Da (P8)
|}
|}


[[File:5ed2-001.svg|alt=alt : Your browser has no SVG support.]]
== Approximation to JI ==
 
=== Selected 7-limit intervals ===
[[:File:5ed2-001.svg|5ed2-001.svg]]
[[File:5ed2-001.svg]]
 
== Notation ==
 
* via Reinhard's cents notation
* naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C
* a four-line hybrid treble/bass staff.
[[Kite Giedraitis]] has proposed a pentatonic notation that retains 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. When notating larger edos such as 8 or 13, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid.


== Observations ==
== Observations ==
=== Related scales ===
=== Related scales ===
 
* By its cardinality, 5edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.
* By its cardinality, 5-edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.
* 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.
* 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.
* For the same reason there are many "circle sisters":
* 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.
** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5 = 7.985099.
 
=== Cycles, Divisions ===


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


* Cycle of seconds: 0-1-2-3-4-0
* Cycle of seconds: 0-1-2-3-4-0
Line 144: Line 244:


=== Harmony ===
=== Harmony ===
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.


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.
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]]).
 
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]]).


Important chords:
Important chords:
Line 156: Line 255:


=== Melody ===
=== Melody ===
 
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.
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.
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 ==
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=5}}


5-EDO [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.  
=== 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"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 181: Line 281:
| 90.225
| 90.225
| Sawa
| Sawa
| Limma, Pythagorean diatonic semitone
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
Line 188: Line 288:
| 133.238
| 133.238
| Gugu
| Gugu
| Large limma
| Bug comma, large limma
|-
|-
| 5
| 5
Line 195: Line 295:
| 111.731
| 111.731
| Gubi
| Gubi
| Classic diatonic semitone
| Father comma, classic diatonic semitone
|-
|-
| 5
| 5
Line 202: Line 302:
| 21.506
| 21.506
| Gu
| Gu
| Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| 5
| 5
Line 209: Line 309:
| 4.200
| 4.200
| Sasa-quadyo
| Sasa-quadyo
| [[Vulture]]
| [[Vulture comma]]
|-
|-
| 7
| 7
Line 216: Line 316:
| 48.770
| 48.770
| Rugu
| Rugu
| Septimal quarter tone
| Mint comma, septimal quartertone
|-
|-
| 7
| 7
Line 223: Line 323:
| 35.697
| 35.697
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 237: Line 337:
| 14.191
| 14.191
| Zozoyo
| Zozoyo
| Sensamagic
| Sensamagic comma
|-
|-
| 7
| 7
Line 244: Line 344:
| 13.074
| 13.074
| Triru-agu
| Triru-agu
| Orwellisma, Orwell comma
| Orwellisma
|-
|-
| 7
| 7
Line 258: Line 358:
| 7.316
| 7.316
| Labiruru
| Labiruru
| Cataharry
| Cataharry comma
|-
|-
| 7
| 7
Line 265: Line 365:
| 5.758
| 5.758
| Saruyo
| Saruyo
| Hemifamity
| Hemifamity comma
|-
|-
| 7
| 7
Line 300: Line 400:
| 9.688
| 9.688
| Saluzo
| Saluzo
| Pentacircle
| Pentacircle comma
|-
|-
| 11
| 11
Line 335: Line 435:
| 19.130
| 19.130
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 344: Line 444:
| Island comma, parizeksma
| Island comma, parizeksma
|}
|}
<references/>


== Ear Training ==
== 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:
5edo ear-training exercises by Alex Ness available here:
* https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web
* 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-edo scale MIDI:
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]


* http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid
== Instruments ==
* [[Lumatone mapping for 5edo]]


== Music ==
== Music ==
{{Main|Music in 5edo}}
{{Catrel|5edo tracks}}


* [http://www.io.com/%7Ehmiller/ Herman Miller]: ''[http://micro.soonlabel.com/herman_miller/Daybreak.mp3 Daybreak on Slendro Mountain]'' (2000)
There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".  
* [[Aaron Krister 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<li>Paul Rubenstein: various, with electric guitars in 10- and 15-edo</li><li>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.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele True Island]" (album) by Small Scale Revolution (2011)
* [[Ralph Jarzombek]]: [http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3 Micro12]
* [https://soundcloud.com/uz1kt3k/prelude-in-5et Prelude In 5ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/invention-in-5et Invention In 5ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (first and third parts in 5-edo)
*[https://youtu.be/Xh2EUwg34pk "Winter Forest"] by NullPointerException Music (from his 1-EDO to 31-EDO album "Edolian")


There is much 5-edo (or nearly so) world music, just search for "gyil" or "amadinda" or "slendro".
== See also ==
* [[Alpha, beta, and gamma family of equal divisions]]


[[Category:5-tone]]
== Notes ==
[[Category:5edo| ]] <!-- main article -->
<references group="note" />
[[Category:9-limit]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Theory]]
[[Category:Zeta]]


[[Category:todo:unify_precision]]
[[Category:3-limit record edos|#]] <!-- 1-digit number -->
[[Category:5-tone scales]]
Retrieved from "https://en.xen.wiki/w/5edo"