5edo

5edo compared to other equipentatonic scales. Relative to just intonation, it narrows 7/4, 7/6, and 4/3, and widens 12/7, 8/7, and 3/2.

5edo, or 5 equal divisions of the octave (sometimes called 5-TET or 5-tone equal temperament), is the equal tuning featuring steps of (1200/5) = 240 cents exactly, 5 of which stack to the perfect octave 2/1. It is the basic equipentatonic, where all five steps are tuned to be exactly equal.

General theory

Derivation

5edo is derived by equalizing an equipentatonic scale, which is interpreted by Leriendil as one of the two primary basic forms of scale produced by human cultures, alongside equiheptatonic (or Pythagorean heptatonic) scales; this makes 5edo rather natural as a simplified model for such scales.

JI approximation

5edo is most obviously a 2.3.7 system (and this property carries to 5-form systems as a whole). although its 3 is audibly sharper than a just 3/2. There is no diatonic scale in 5edo; there are too few notes (although for theoretical purposes, 5edo can be taken as having a MOS diatonic where the small step is tuned to 0 cents). However, 5edo interestingly is simple enough, and at least vaguely accurate enough to just intonation, that essentially any combination of notes taken from it sounds consonant to some extent.

Some interpretations treat 5edo as a 2.3.5.7 (5-limit or full 7-limit) system, in which the 5/4 interval, detuned by almost 100 cents, is equated with 4/3, equalizing the triad 3:4:5. Due to the inaccuracy of this temperament, this is mostly only a theoretical curiosity.

In higher limits, 37/32 is the next prime represented reasonably accurately, although as for composite intervals 17/13 is rather accurately represented; so is 55/32.

Approximation of prime harmonics in 5edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	+18.0	+93.7	-8.8	-71.3	+119.5	-105.0	-57.5	+91.7	-69.6	+55.0	-11.3
	Relative (%)	0.0	+7.5	+39.0	-3.7	-29.7	+49.8	-43.7	-24.0	+38.2	-29.0	+22.9	-4.7
Steps (reduced)		5 (0)	8 (3)	12 (2)	14 (4)	17 (2)	19 (4)	20 (0)	21 (1)	23 (3)	24 (4)	25 (5)	26 (1)

Edostep interpretations

5edo's edostep has the following interpretations in the 2.3.7 subgroup:

• 7/6
• 8/7
• 9/8

Intervals and notation

5edo may be notated with any diatonic notation, as long as you keep in mind that E and F are the same note, and B and C are also the same note; the note sequence is usually C-D-E-G-A like a major pentatonic scale, and it is preferable to avoid accidentals. (However, it may be useful to some to treat the third step as E and F simultaneously; Kite calls this "eef".)

Edostep	Cents	Intervals	Notation		Name
		2.3.7	Diatonic	Eef
0	0		D		Unison
1	240	9/8, 8/7, 7/6	E, F	ꘙ	Major second
2	480	21/16, 4/3	G		Perfect fourth
3	720	3/2, 32/21	A		Perfect fifth
4	960	16/9, 7/4, 12/7	B, C	C	Minor seventh
5	1200	2/1	D		Octave

Solfege functions according to standard diatonic solfege: do, re, mi, so, la.

Tempering properties

Tempered commas

Important commas tempered out by the 2.3.7 subgroup interpretation of 5edo include:

• 49/48 (S7), equating 7/6 with 8/7 and 9/7 with 21/16
• 64/63 (S8), equating 8/7 with 9/8 and 21/16 with 4/3

Arithmetic progressions

Equivalently, 5et in the 2.3.7 subgroup may be defined by equalizing 6:7:8:9. In the inaccurate 5-limit, this extends further to 5:6:7:8:9:10.

Notable structural chains

5edo, due to its size, only has two edo-distinct rank-2 temperaments: Slendric (or alternatively Semaphore) and Archy. Semaphore or Slendric is generated by the 240c 8/7 interval, two of which reach 4/3 (semaphore) and three of which reach 3/2 (slendric). Archy is 5edo's version of the standard chain of fifths, where two 4/3s reach 7/4. 5edo is on the sharper end of the tuning range of slendric and archy (which are optimally tuned around 232c and 710c respectively) and on the flatter end of the semaphore tuning range (which is usually around 245c).

Compositional theory

Tertian structure

As the smallest edo in which diatonic interval categories make much sense, 5edo only has thirds by the broadest possible definition of the interval region (which places thirds in the middle third of 3/2) or by the definition based on the degenerate MOS diatonic scale, where the primary major and minor third are 480c and 240c.

Diatonic interval qualities are, however, not particularly useful when composing in 5edo.

Thirds in 5edo

Quality	Minor	Major
Cents	240	480
Just interpretation	7/6 (-27¢), 8/7 (+9c), 9/8 (+36c)	9/7 (+45¢), 4/3 (-18c)

Diatonic thirds are bolded, though note that all thirds shown here appear in the MOS diatonic scale.

Technically, 5edo is a mint temperament, tempering together 6/5 and 7/6 to produce 2 qualities from the 4 main 7-limit thirds, although its 6/5 is inaccurate enough as to be basically useless.

Scales

5edo is simple enough that the entire edo is usually used as a scale (although composer Hideya Amano avoids this by using subsets such as [0 2 3 5] in order to establish a clear tonal center). The construction of scales from 5edo is usually not by taking subsets but by detuning the notes to reach one of the equipentatonic scales discussed in Equipentatonic. In fact, in real world musical cultures that use equipentatonic scales, they are almost never perfect 5edo.

Every possible octave-equivalent subset of 5edo is a MOS; 5edo is the largest edo with this property. There are seven such scales, up to rotation: 5edo, 1edo, 4-1, 3-2, 3-1-1, 2-2-1, and 2-1-1-1. Each scale is generated by at least one of 5edo's temperaments:

Scale	Chart	Notes	Temperament
4-1	├┴───┤ 1 4		Slendric
3-2	├──┴─┤ 3 2		Archy
3-1-1	├┴┴──┤ 1 1 3		Slendric
2-2-1	├─┴┴─┤ 2 1 2	Used by Amano	Archy
2-1-1-1	├┴┴┴─┤ 1 1 1 2	Used by Amano	Slendric, Archy

(Archy generates Lsss by coincidence; this is not a general property of archy.)

Harmony

Essentially any combination of notes in 5edo is a chord of some kind. Up to inversion and modulation, there are a total of 5 chords in 5edo, not counting a single note or the whole scale. Predictably, triadic harmony is rare in 5edo, and most harmonic systems in this tuning rely on drones or modulation.

The full set of chords or scales including modes is as follows:

Chord	Chart	Info
0 1	├┴───┤ 1 4
0 2	├─┴──┤ 2 3	Perfect fourth trine
0 3	├──┴─┤ 3 2	Perfect fifth trine
0 4	├───┴┤ 4 1
0 1 2	├┴┴──┤ 1 1 3	Chthonic triad, see Chthonic harmony
0 1 3	├┴─┴─┤ 1 2 2	"Minor triad", see Extraclassical tonality
0 1 4	├┴──┴┤ 1 3 1
0 2 3	├─┴┴─┤ 2 1 2	"Major triad", see Extraclassical tonality
0 2 4	├─┴─┴┤ 2 2 1
0 3 4	├──┴┴┤ 3 1 1	Representation of the chord 4:6:7(:8).
0 1 2 3	├┴┴┴─┤ 1 1 1 2	"Major/minor triad", see Extraclassical tonality. Also the slendric tetrad.
0 1 2 4	├┴┴─┴┤ 1 1 2 1
0 1 3 4	├┴─┴┴┤ 1 2 1 1	"Minor seventh" chord
0 2 3 4	├─┴┴┴┤ 2 1 1 1	"Dominant seventh" chord

However, it is often better to avoid relying on chord progressions, especially as most useful chrods are inversions of each other and the step of the edo is found as the step in many chords, making voice leading difficult.

SS

It's best in my opinion to use a monophonic drone and modulate between different keys, that or use a perfect fifth drone in harmony.

Instruments

On a Lumatone

As a diatonic edo, 5edo can be cleanly represented on a Lumatone:

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On a guitar

If a guitar is tuned according to its standard tuning in 5edo, it becomes isomorphic, because the diatonic major third is also the perfect fourth. It's not necessary to re-fret the guitar for most notes (meaning that a guitar tuned like this actually plays in a subset of 60edo).

Supersets and subsets

Owing to 5edo's accuracy in 2.3.7 but failure to represent 5, a perfectly tuned 5edo can be taken as the 2.3.7 structure in a rank-2 temperament, with 5 (or any other prime) added as a generator. This is called blackwood temperament. Blackwood tempers out the diatonic semitone, meaning that the 3-limit diatonic MOS scale is degenerate, collapsing to a 5-note scale. However, intervals of 5 are distinct, and may be used to generate the Zarlino diatonic.

Supersets of 5edo that support blackwood include 10edo, 15edo, 20edo, 25edo, and 30edo. All of these share 5edo's 2.3.7 subgroup, and its slendric, semaphore, and archy temperaments.

Comparisons to other tuning systems

5edo's temperaments are shared by other edos separated by 5:

• 16edo, 21edo, 26edo, 31edo, 36edo, 41edo, and 46edo all support Slendric.
• 9edo, 14edo, 19edo, 24edo, 29edo, and 34edo all support Semaphore.
• 12edo, 17edo, 22edo, 27edo, 32edo, and 37edo all support Archy.
• As previously mentioned, the multiples of 5edo itself support Blackwood.