5edo: Difference between revisions
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{{interwiki | |||
| de = 5-EDO | |||
| en = 5edo | |||
| es = 5 EDO | |||
| ja = 5平均律 | |||
| ro = 5DEO | |||
}} | |||
{{Infobox ET}} | |||
{{ED intro}} | |||
5edo is notable for being the smallest [[edo]] containing xenharmonic intervals—1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]]. | |||
== 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. | |||
5 is | |||
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. | |||
==Harmony== | 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" | |||
|+ 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. | |||
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" | |||
|+ 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) | |||
|} | |||
== Approximation to JI == | |||
=== Selected 7-limit intervals === | |||
[[File:5ed2-001.svg]] | |||
== Observations == | |||
=== Related scales === | |||
* By its cardinality, 5edo 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. | |||
* 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. | |||
=== 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 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 | |||
=== 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. | |||
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: | Important chords: | ||
1+2+ | * 0+1+3 | ||
1+3+4 | * 0+2+3 | ||
* 0+1+3+4 | |||
1 | * 0+2+3+4 | ||
=== 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. | |||
=== 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. | |||
== Regular temperament properties == | |||
=== Uniform maps === | |||
{{Uniform map|edo=5}} | |||
=== 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<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 == | |||
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] | |||
== Instruments == | |||
* [[Lumatone mapping for 5edo]] | |||
== Music == | |||
{{Main|Music in 5edo}} | |||
{{Catrel|5edo tracks}} | |||
[[ | There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]". | ||
== See also == | |||
* [[Alpha, beta, and gamma family of equal divisions]] | |||
== Notes == | |||
<references group="note" /> | |||
[[Category:3-limit record edos|#]] <!-- 1-digit number --> | |||
[[Category:5-tone scales]] | |||