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.

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. 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.

[WIP]

Chords

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.

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]). 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 elsewhere on the page. 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.

Blackwood temperament

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, and is supported by, most notably, 15edo. 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.