# 5edo

## Contents

## Theory

**5-edo** divides the 1200-cent octave into 5 equal parts, making its smallest interval exactly 240 cents, or the fifth root of two. 5-edo is the 3rd 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.)

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 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.

## Intervals

degrees | cents | Closest diatonic interval name |
The "neighborhood" of just intervals |
---|---|---|---|

0 | 0 | unison / prime | exactly 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 | exactly 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, 5-edo 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 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":
- 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:

- 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 10-EDO).

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

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

Ratio | Monzo | cents | Color Name | Name | Second Name | Third Name |
---|---|---|---|---|---|---|

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

27/25 | | 0 3 -2 > | 133.238 | Gugu | Large diatonic semit. | ||

16/15 | | 4 -1 -1 > | 111.731 | Gubi | Diatonic semitone | ||

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

2889416/2882415 | | 24 -21 4 > | 4.200 | Sasa-quadyo | Vulture | ||

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

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

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

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

1728/1715 | | 6 3 -1 -3 > | 13.074 | Triru-agu | Orwellisma | Orwell Comma | |

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

19683/19600 | | -4 9 -2 -2 > | 7.316 | Labiruru | Cataharry | ||

5120/5103 | | 10 -6 1 -1 > | 5.758 | Saruyo | Hemifamity | ||

1065875/1063543 | | -26 -1 1 9 > | 3.792 | Latritrizo-ayo | Wadisma | ||

420175/419904 | | -6 -8 2 5 > | 1.117 | Quinzo-ayoyo | Wizma | ||

11/10 | | -1 0 -1 0 1 > | 165.004 | Logu | Large neutral second | ||

99/98 | | -1 2 0 -2 1 > | 17.576 | Loruru | Mothwellsma | ||

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

385/384 | | -7 -1 1 1 1 > | 4.503 | Lozoyo | Keenanisma | ||

441/440 | | -3 2 -1 2 -1 > | 3.930 | Luzozogu | Werckisma | ||

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

14/13 | | 1 0 0 1 0 -1 > | 128.298 | Thuzo | |||

91/90 | | -1 -2 -1 1 0 1 > | 19.130 | Thozogu | Superleap | ||

676/675 | | 2 -3 -2 0 0 2 > | 2.563 | Bithogu | Parizeksma |

## 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-edo scale MIDI:

## Music

- Herman Miller:
*Daybreak on Slendro Mountain*(2000) - Aaron K. Johnson:
*5tet funk*(2004) - Andrew Heathwaite: //Pinta Penta// (2004) play (rendered in 6 alternative pentatonics as well)
- Hans Straub: Asîmchômsaia play
- Brian Wong: Slendronica#1b 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
- "True Island" (album) by Small Scale Revolution (2011)
- Ralph Jarzombek: Micro12

There is much 5-edo (or nearly so) world music, just search for "gyil" or "amadinda" or "slendro".