5edo: Difference between revisions
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| Fifth = 3\5 = 720¢ | | Fifth = 3\5 = 720¢ | ||
| Major 2nd = 1\5 = 240¢ | | Major 2nd = 1\5 = 240¢ | ||
| | | Semitones = 1\5 : 0\5 | ||
}} | }} | ||
'''5 equal divisions of the octave''' (or ''' | '''5 equal divisions of the octave''' (or '''5EDO''') is the [[tuning system]] derived by dividing the [[octave]] into 5 equal steps of 240 [[cent]]s each, or the fifth root of two. 5EDO is the third [[prime EDO]], after [[2edo|2EDO]] and [[3edo|3EDO]]. Most importantly, 5EDO is the smallest [[EDO]] containing xenharmonic intervals — 1EDO, 2EDO, 3EDO, and 4EDO are all subsets of [[12edo|12EDO]]. | ||
== Theory == | == Theory == | ||
| Line 48: | Line 47: | ||
| -24 | | -24 | ||
|- | |- | ||
! colspan="2" | [[nearest | ! colspan="2" | [[Patent val|nearest EDO-mapping]] | ||
| 5 | | 5 | ||
| 3 | | 3 | ||
| Line 69: | Line 68: | ||
|} | |} | ||
If | If 5EDO is regarded as a temperament, which is to say as 5-TET, then the most salient fact is that 16/15 is tempered out. This means in 5-TET 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. | 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. | ||
Despite its lack of accuracy, | 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-odd-limit|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|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|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]]. | ||
=== Differences between distributionally-even scales and smaller | === Differences between distributionally-even scales and smaller EDOs === | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| Line 82: | Line 81: | ||
!s-Nedo | !s-Nedo | ||
|- | |- | ||
|2 | | 2 | ||
|120¢ | | 120¢ | ||
| -120¢ | | -120¢ | ||
|- | |- | ||
|3 | | 3 | ||
|80¢ | | 80¢ | ||
| -160¢ | | -160¢ | ||
|- | |- | ||
|4 | | 4 | ||
|180¢ | | 180¢ | ||
| -60¢ | | -60¢ | ||
|} | |} | ||
== Intervals == | == Intervals == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! Steps | ! Steps | ||
! [[Cent]]s | ! [[Cent]]s | ||
! Closest diatonic <br> interval name | ! Closest diatonic <br>interval name | ||
! The "neighborhood" of just intervals | ! The "neighborhood" of just intervals | ||
|- | |- | ||
| Line 111: | Line 109: | ||
| 240 | | 240 | ||
| second, third | | second, third | ||
| +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]] | | +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]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 480 | | 480 | ||
| fourth | | fourth | ||
| +9.219¢ from narrow fourth [[21/16]] <br> -0.686¢ from smaller fourth [[33/25]] <br> -18.045¢ from just fourth [[4/3]] | | +9.219¢ from narrow fourth [[21/16]] <br>-0.686¢ from smaller fourth [[33/25]] <br>-18.045¢ from just fourth [[4/3]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 720 | | 720 | ||
| fifth | | fifth | ||
| +18.045¢ from just fifth [[3/2]] <br> +0.686¢ from bigger fifth [[50/33]] <br> -9.219¢ from wide fifth [[32/21]] | | +18.045¢ from just fifth [[3/2]] <br>+0.686¢ from bigger fifth [[50/33]] <br>-9.219¢ from wide fifth [[32/21]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 960 | | 960 | ||
| sixth, seventh | | sixth, seventh | ||
| 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]] | | 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]] | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 139: | Line 137: | ||
== Notation == | == Notation == | ||
* via Reinhard's cents notation | * via Reinhard's cents notation | ||
* naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C | * naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C | ||
* a four-line hybrid treble/bass staff. | * 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 | [[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, | * 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 | |||
* 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 === | === Cycles, Divisions === | ||
5 is a prime number so 5EDO contains no sub-EDOs. Only simple cycles: | |||
5 is a prime number so | |||
* Cycle of seconds: 0-1-2-3-4-0 | * Cycle of seconds: 0-1-2-3-4-0 | ||
| Line 164: | Line 159: | ||
=== 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. | |||
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 | |||
Important chords: | Important chords: | ||
| Line 176: | Line 170: | ||
=== 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 | |||
=== 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 == | == Commas == | ||
5EDO [[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" | ||
| Line 367: | Line 358: | ||
== Ear Training == | == Ear Training == | ||
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- | 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: | ||
* http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid | * http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid | ||
== Music == | == Music == | ||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
!Title | !Title | ||
| Line 482: | Line 471: | ||
* [[Brian McLaren]]: various and sundry | * [[Brian McLaren]]: various and sundry | ||
* [[Paul Rubenstein]]: various, with electric guitars in 10- and | * [[Paul Rubenstein]]: various, with electric guitars in 10- and 15EDO | ||
There is also much 5edo-like world music, just search for "gyil" or "amadinda" or "slendro". | There is also much 5edo-like world music, just search for "gyil" or "amadinda" or "slendro". | ||
| Line 489: | Line 478: | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:5-tone scales]] | [[Category:5-tone scales]] | ||
[[Category:7-limit]] | |||
[[Category:9-limit]] | [[Category:9-limit]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
| Line 494: | Line 484: | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||
Revision as of 13:34, 13 October 2021
| ← 4edo | 5edo | 6edo → |
(convergent)
5 equal divisions of the octave (or 5EDO) is the tuning system derived by dividing the octave into 5 equal steps of 240 cents each, or the fifth root of two. 5EDO is the third prime EDO, after 2EDO and 3EDO. Most importantly, 5EDO is the smallest EDO containing xenharmonic intervals — 1EDO, 2EDO, 3EDO, and 4EDO are all subsets of 12EDO.
Theory
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | ||
|---|---|---|---|---|---|---|---|---|---|
| error | absolute (¢) | 0.0 | +18.0 | +93.7 | -8.8 | -71.3 | +119.5 | -105.0 | -57.5 |
| relative (%) | 0 | +8 | +39 | -4 | -30 | +50 | -44 | -24 | |
| nearest EDO-mapping | 5 | 3 | 2 | 4 | 2 | 4 | 0 | 1 | |
| fifthspan | 0 | +1 | -1 | -2 | -1 | -2 | 0 | +2 | |
If 5EDO is regarded as a temperament, which is to say as 5-TET, then the most salient fact is that 16/15 is tempered out. This means in 5-TET the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit father temperament.
Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain 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.
Despite its lack of accuracy, 5EDO is the second zeta integral EDO, after 2EDO. It also is the smallest equal division representing the 9-limit 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 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.
Differences between distributionally-even scales and smaller EDOs
| N | L-Nedo | s-Nedo |
|---|---|---|
| 2 | 120¢ | -120¢ |
| 3 | 80¢ | -160¢ |
| 4 | 180¢ | -60¢ |
Intervals
| Steps | Cents | Closest diatonic interval name |
The "neighborhood" of just intervals |
|---|---|---|---|
| 0 | 0 | unison / prime | 1/1 |
| 1 | 240 | second, third | +8.826¢ from septimal second 8/7 -4.969¢ from diminished third 144/125 -13.076¢ from augmented second 125/108 -26.871¢ from septimal minor third 7/6 |
| 2 | 480 | fourth | +9.219¢ from narrow fourth 21/16 -0.686¢ from smaller fourth 33/25 -18.045¢ from just fourth 4/3 |
| 3 | 720 | fifth | +18.045¢ from just fifth 3/2 +0.686¢ from bigger fifth 50/33 -9.219¢ from wide fifth 32/21 |
| 4 | 960 | sixth, seventh | 26.871¢ from septimal major sixth 12/7 13.076¢ from diminished seventh 216/125 4.969¢ from augmented sixth 125/72 -8.826¢ from septimal seventh 7/4 |
| 5 | 1200 | octave | 2/1 |
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
Related scales
- By its cardinality, 5EDO is related to other pentatonic scales, and it is especially close in sound to many Indonesian slendros.
- 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. 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).
Important chords:
- 0+1+3
- 0+2+3
- 0+1+3+4
- 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.
Commas
5EDO tempers out the following commas. This assumes the val ⟨5 8 12 14 17 19].
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 3 | 256/243 | [8 -5⟩ | 90.225 | Sawa | Limma, Pythagorean diatonic semitone |
| 5 | 27/25 | [0 3 -2⟩ | 133.238 | Gugu | Large limma |
| 5 | 16/15 | [4 -1 -1⟩ | 111.731 | Gubi | Classic diatonic semitone |
| 5 | 81/80 | [-4 4 -1⟩ | 21.506 | Gu | Syntonic comma, Didymus comma, meantone comma |
| 5 | (22 digits) | [24 -21 4⟩ | 4.200 | Sasa-quadyo | Vulture |
| 7 | 36/35 | [2 2 -1 -1⟩ | 48.770 | Rugu | Septimal quarter tone |
| 7 | 49/48 | [-4 -1 0 2⟩ | 35.697 | Zozo | Slendro diesis |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.264 | Ru | Septimal comma, Archytas' comma, Leipziger Komma |
| 7 | 245/243 | [0 -5 1 2⟩ | 14.191 | Zozoyo | Sensamagic |
| 7 | 1728/1715 | [6 3 -1 -3⟩ | 13.074 | Triru-agu | Orwellisma, Orwell comma |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.433 | Latrizo | Gamelisma |
| 7 | 19683/19600 | [-4 9 -2 -2⟩ | 7.316 | Labiruru | Cataharry |
| 7 | 5120/5103 | [10 -6 1 -1⟩ | 5.758 | Saruyo | Hemifamity |
| 7 | (18 digits) | [-26 -1 1 9⟩ | 3.792 | Latritrizo-ayo | Wadisma |
| 7 | (12 digits) | [-6 -8 2 5⟩ | 1.117 | Quinzo-ayoyo | Wizma |
| 11 | 11/10 | [-1 0 -1 0 1⟩ | 165.004 | Logu | Large undecimal neutral 2nd |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.576 | Loruru | Mothwellsma |
| 11 | 896/891 | [7 -4 0 1 -1⟩ | 9.688 | Saluzo | Pentacircle |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.503 | Lozoyo | Keenanisma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.930 | Luzozogu | Werckisma |
| 11 | 3025/3024 | [-4 -3 2 -1 2⟩ | 0.572 | Loloruyoyo | Lehmerisma |
| 13 | 14/13 | [1 0 0 1 0 -1⟩ | 128.298 | Thuzo | Tridecimal 2/3-tone, trienthird |
| 13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.130 | Thozogu | Superleap |
| 13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.563 | Bithogu | Island comma, parizeksma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Ear Training
5EDO ear-training exercises by Alex Ness available here:
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:
Music
| Title | Composer | Year | Genre | Additional links |
|---|---|---|---|---|
| Daybreak | Herman Miller | 2000 | Folk (Indonesian) | |
| Pinta Penta in 5tET | Andrew Heathwaite | 2004 | Electronic | Alternative versions (SoundClick) |
| 5tet funk[dead link] | Aaron Krister Johnson | 2004 | Funk | |
| Sleeping Through It All | X. J. Scott | 2004 | Electronic | |
| 5-tet funk | Bill Sethares | 2004 | Funk | |
| Pentacle | Bill Sethares | 2004 | (?) | |
| Asîmchômsaia | Hans Straub | 2005 (?) | Folk (Japanese) | |
| Slendronica#1b | Brian Wong | 2009 (?) | Folk (Indonesian) | |
| Random Explorations (for ukulele) | Cenobyte | 2011 | Folk | |
| True Island : 5 Equal Divisions Of The Octave Ukulele (album) | Small Scale Revolution | 2011 | Folk | |
| Micro12 | Ralph Jarzombek | 2010 (?) | Electronic | |
| Prelude in 5ET | Aaron Andrew Hunt | 2015 | Neobaroque | |
| Invention in 5ET | Aaron Andrew Hunt | 2015 | Neobaroque | |
| Hey, ule! (1st and 3rd sections) | Dmitriy Bazhenov | 2020 | Ambiance | |
| "Winter Forest" (from Edolian) | NullPointerException Music | 2020 | Classical | |
| "Ether" (from STAFFcirc vol. 7) | Vince Kaichan | 2021 | Electronic | Album (Bandcamp) |
- Brian McLaren: various and sundry
- Paul Rubenstein: various, with electric guitars in 10- and 15EDO
There is also much 5edo-like world music, just search for "gyil" or "amadinda" or "slendro".