Equiheptatonic
An equiheptatonic scale is a scale with 7 approximately equally spaced notes within the octave. A tuning system that generates an equiheptatonic scale may be conceptualized with the 7-form. Below are several examples of equiheptatonic scales.
7edo
7edo is the basic equiheptatonic, where all the steps are tuned to be precisely equal. It features steps of (1200/7) ~= 171.4 cents.
Theory
Edostep interpretations
7edo's edostep has the following interpretations in the 2.3.5 subgroup:
- 9/8 (the diatonic major second)
- 10/9 (the interval separating 9/8 and 5/4)
- 16/15 (the interval separating 5/4 and 4/3)
JI approximation
7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and all intervals are perfect.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -16.2 | -43.5 | +59.7 | -37.0 | +16.6 | +66.5 | +45.3 | +57.4 | -1.0 | +55.0 |
| Relative (%) | 0.0 | -9.5 | -25.3 | +34.9 | -21.6 | +9.7 | +38.8 | +26.5 | +33.5 | -0.6 | +32.1 | |
| Steps
(reduced) |
7
(0) |
11
(4) |
16
(2) |
20
(6) |
24
(3) |
26
(5) |
29
(1) |
30
(2) |
32
(4) |
34
(6) |
35
(7) | |
| Quality | Neutral |
|---|---|
| Cents | 343 |
| Just interpretation | 11/9 |
Diatonic thirds are bolded.
Chords
7edo features, for tertian triadic harmony, only a neutral chord [0 2 4] and the (rather discordant) sus chords [0 1 4] and [0 3 4]. Regardless, due to its triads and due to representing all seven degrees of the diatonic scale, it is the smallest edo where Western functional harmony works.
Scales
7edo is the first edo to distinguish the modes of the pentic scale. However, it is still small enough that it is well-temperable into scales (specifically, those of the 7-form discussed elsewhere in this article). In real world musical cultures which use near-equal 7-note scales, perfect 7edo is almost never used.
Notation
In 7edo, pretty much all reasonable notation schemes collapse to ABCDEFG on A=440Hz. Accidentals are not used.
Whitewood temperament
7edo may be interpreted as Whitewood temperament, which tempers out the Pythagorean chromatic semitone. The most obvious rank-2 extension is to add a free generator corresponding to 7/4, resulting in a system containing multiple copies of 7edo separated by the interval 7/4. This extension is supported by 21edo, which, along with 14edo, supports the omnidiatonic ternary diatonic scale.
Equitetrachordal scale
The equitetrachordal scale is the scale sssLsss where L represents 9/8, and s represents the cube root of 4/3. If s is taken to be 11/10 and s+s to be 6/5, the result is Porcupine temperament, the most common temperament interpretation of the equitetrachordal scale. It is tuned close to just in 29edo.
Just equiheptatonic scale
This is Ptolemy's equable diatonic, the otonal or utonal series 18:20:22:24:27:30:36, which has steps of 10/9, 11/10, and 12/11 in each tetrachord and 9/8 separating them.
Other equiheptatonic scales
Any heptatonic MOS has a range of equiheptatonic tunings. The following is a table of equiheptatonic MOSes with related temperaments, expressed here as temperaments of the 2.3.5 subgroup (though often, other more accurate interpretations are available). Unlike equipentatonic systems, the scale is usually not tuned quite as equally, so extensions to 7 are more common.
| MOS | Pattern | Temperament | EDO |
|---|---|---|---|
| 6L 1s | LLLLLLs | Tetracot | 27 |
| 5L 2s | LLsLLLs | Meantone | 26 |
| 4L 3s | LLsLsLs | Sixix | 25 |
| 3L 4s | LssLsLs | Dicot | 24 |
| 2L 5s | LssLsss | Mavila | 23 |
| 1L 6s | Lssssss | Porcupine | 22 |
Table of equiheptatonic intervals
| # | Name | Tuning range | Just intonation | Equal-tempered | Equitetrachordal | Equable diatonic | Soft antidiatonic | Soft diatonic | Soft mosh |
|---|---|---|---|---|---|---|---|---|---|
| 0 | Unison | 0c | 1/1 | 0c | 0c | 0c | 0c | 0c | 0c |
| 1 | Second | 130-210c | 10/9, 11/10, 12/11, 16/15, 9/8 | 171c | 166c | 182c, 151c | 157-208c | 138-185c | 150-200c |
| 2 | Third | 310-380c | 5/4, 6/5, 11/9 | 343c | 332c | 351c, 316c | 313-365c | 323-369c | 343-350c |
| 3 | Fourth | 480-550c | 11/8, 4/3 | 514c | 498c | 498c | 514-522c | 508-514c | 500-550c |
| 4 | Fifth | 650-720c | 3/2, 16/11 | 686c | 702c | 702c | 678-686c | 686-692c | 650-700c |
| 5 | Sixth | 820-890c | 5/3, 8/5, 18/11 | 857c | 868c | 884c, 849c | 835-887c | 831-877c | 850-857c |
| 6 | Seventh | 990-1070c | 20/11, 9/5, 16/9, 11/6, 15/8 | 1029c | 1034c | 1018c, 1049c | 992-1043c | 1015-1062c | 1000-1050c |
| 7 | Octave | 1200c | 2/1 | 1200c | 1200c | 1200c | 1200c | 1200c | 1200c |
