7edo: Difference between revisions

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[[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]

7edo is the basic example of an [[equiheptatonic]] scale, and in terms of tunings with perfect fifths, is essentially the next size up from [[5edo]]. The 7-form is notable as a common structure for many [[5-limit]] systems, including all seven modes of the [[5L 2s|diatonic]] scale—Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian; in 7edo itself, the two sizes of interval in any heptatonic MOS scale are equated, resulting in a [[interval quality|neutral]] feel. All triads are neutral (except if you use suspended triads, which are particularly harsh in 7edo due to the narrowed major second), so functional harmony is almost entirely based on the positions of the chords in the 7edo scale.  

The second (171.429{{c}}) works well as a basic step for melodic progression. The step from seventh to octave is too large as a leading tone - possibly lending itself to a "sevenplus" scale similar to [[elevenplus]].

In terms of just intonation, the 3/2 is flat but usable, but we don't find particularly accurate intervals in pure harmonics outside the 3-limit, which suggests a more melodic approach to just intonation; intervals approximated by each of 7edo's steps include 10/9 for 1 step, 11/9 for 2 steps, 4/3 for 3 steps, and their octave complements. Interestingly, this renders an 8:9:10:11:12 pentad equidistant, from which it can be derived that 7edo supports [[meantone]] (equating the major seconds 10/9 and 9/8) and [[porcupine]] (splitting 4/3 into three equal submajor seconds which simultaneously represent 12/11, 11/10, and 10/9), and is the unique system to do so.  

Due to 7edo's inaccurately tuned [[5/4]] [[major third]] (which is flat by over 40 cents), it supports several exotemperaments in the 5-limit, such as [[dicot]] (which splits the fifth into two equal [[neutral third]]<nowiki/>s, simultaneously representing 5/4 and the [[minor third]] [[6/5]]) and [[mavila]] (which flattens the fifth so that the diatonic "major third" actually approximates 6/5); 6/5 is a slightly more reasonable interpretation of 7edo's third than 5/4, leading to an overall slightly [[minor]] sound.

In higher limits, this third takes on a new role: as a neutral third, it is a decent approximation of the 13th subharmonic, and as such 7edo can be seen as a 2.3.13 temperament. This third is a near perfect approximation of the interval [[39/32]]; the equation of 16/13 and 39/32 is called [[512/507|harmoneutral]] temperament. In general, the inclusion of 13 allows the pentad discussed earlier to be continued to an 8:9:10:11:12:13 hexad, although the specific interval 13/12 is inaccurate due to the errors adding up in the same direction.

7edo represents a 7-step closed [[circle of fifths]], tempering out the Pythagorean chromatic semitone. However, it can also be seen as a circle of neutral thirds, which can be interpreted as 11/9; this is called [[neutron]] temperament.

=== Prime harmonics ===

{{Harmonics in equal|7}}

=== In non-Western traditions ===

[[Equiheptatonic]] scales close to 7edo are used in non-western music in some [[African]] cultures<ref>[https://www.britannica.com/art/African-music ''African music'', Encyclopedia Britannica.]</ref> as well as an integral part of early [[Chinese]] music<ref>Robotham, Donald Keith and Gerhard Kubik.</ref>. Also [[Georgian]] music seems to be based on near-equal 7-step scales.

It has been speculated in ''Indian music: history and structure''<ref>Nambiyathiri, Tarjani. ''[https://archive.org/details/indianmusichistoryandstructureemmietenijenhuisbrill Indian Music History And Structure Emmie Te Nijenhuis Brill]''</ref> that the [[Indian]] three-sruti interval of 165 cents is very similar to one 171-cent step of 7edo.

In [[equiheptatonic]] systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to [[adaptive just intonation]] but with equal tuning instead).   

A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this.

 

 

 

 

 

 

 

 

 

 

 

|+ <span style="font-size: 105%;">Solfege of 7edo</span>

! Standard [[solfege]]<br />(movable do)

 

 

 

 

 

 

 

 

 

 

 

 

 

; [[zpi|15zpi]]

 

; [[11edt]]

 

 

 

7edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].

 

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