Equiheptatonic: Difference between revisions

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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.
[[File:Qetqbnx.png|thumb|The derivation of [[equipentatonic]] and equiheptatonic scales by adding melodic steps to [1/1 4/3 3/2 2/1].]]
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'''. (Inthar prefers the term '''homoioheptatonic''' (Greek ''homoios'' 'similar') to avoid using the term ''equi-'' which may imply perfectly equal steps to some readers.)


== 7edo ==
The basic equiheptatonic is [[7edo]], where all the steps are tuned to be precisely equal. It features steps of (1200/7) ~= 171.4 cents. Below are several other examples of equipentatonic scales.
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.
{{Harmonics in ED|7|31}}
{| class="wikitable"
|+Thirds in 7edo
!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 (which Cole Parker claims are the most common cross-culturally), 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 [[Diatonic|omnidiatonic]] ternary diatonic scale.


== Equitetrachordal scale ==
== Equitetrachordal scale ==
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!MOS
!MOS
!Pattern
!Pattern
!Temperament
!Temperament [2.3.5]
!Temperament [2.3.11]
!Temperament [2.3.13]
!EDO
!EDO
|-
|-
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|LLLLLLs
|LLLLLLs
|Tetracot
|Tetracot
|7 & 27
|''find 2.3.13 interpretation cuz sintel temper is not cooperating''
|27
|27
|-
|-
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|LLsLLLs
|LLsLLLs
|Meantone
|Meantone
|Alpharabian flattone
|Deeptone
|26
|26
|-
|-
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|LLsLsLs
|LLsLsLs
|Sixix
|Sixix
| colspan="2" |7 & 25
|25
|25
|-
|-
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|LssLsLs
|LssLsLs
|Dicot
|Dicot
|Rastmic
|Intertridecimal
|24
|24
|-
|-
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|LssLsss
|LssLsss
|Mavila
|Mavila
|7 & 23
|2 & 7 (sets 13/8 to the major sixth)
|23
|23
|-
|-
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|Lssssss
|Lssssss
|Porcupine
|Porcupine
|Alpharabian porcupine
|7 & 22
|22
|22
|}
|}

Latest revision as of 22:06, 15 April 2026

The derivation of equipentatonic and equiheptatonic scales by adding melodic steps to [1/1 4/3 3/2 2/1].

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. (Inthar prefers the term homoioheptatonic (Greek homoios 'similar') to avoid using the term equi- which may imply perfectly equal steps to some readers.)

The basic equiheptatonic is 7edo, where all the steps are tuned to be precisely equal. It features steps of (1200/7) ~= 171.4 cents. Below are several other examples of equipentatonic scales.

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 [2.3.5] Temperament [2.3.11] Temperament [2.3.13] EDO
6L 1s LLLLLLs Tetracot 7 & 27 find 2.3.13 interpretation cuz sintel temper is not cooperating 27
5L 2s LLsLLLs Meantone Alpharabian flattone Deeptone 26
4L 3s LLsLsLs Sixix 7 & 25 25
3L 4s LssLsLs Dicot Rastmic Intertridecimal 24
2L 5s LssLsss Mavila 7 & 23 2 & 7 (sets 13/8 to the major sixth) 23
1L 6s Lssssss Porcupine Alpharabian porcupine 7 & 22 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
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