7edo: Difference between revisions

| de = 7-EDO

| ja = 7平均律

[[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.  

The seventh of 7edo is almost exactly the 29th harmonic (29/16), which can have a very agreeable sound with harmonic timbres. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher harmonic entropy than 7/4, a much simpler overtone seventh.  

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.

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).

One region of Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern [[Angolan]] area. This music is [[heptatonic]] and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds ([[5/4]], 386{{c}}), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.

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

It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>

=== Subsets and supersets ===

7edo is the 4th [[prime edo]], after [[5edo]] and before [[11edo]]. It does not contain any nontrivial subset edos, though it contains [[7ed4]]. Multiples such as [[14edo]], [[21edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 7edo, while improving on other intervals.

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Intervals of 7edo

! rowspan="2" | [[Interval region]]

! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])

| 0

| 0

| Unison (prime)

| [[1/1]] (just)

| 1

| 171.429

| [[10/9]] (-10.975)

| [[54/49]] (+3.215)

| [[11/10]] (+6.424)<br>[[32/29]] (-1.006)

| [[File:0-171,43 second (7-EDO).mp3|frameless]]

| 2

| 342.857

| [[128/105]] (+0.048)

| [[39/32]] (+0.374)<br>[[16/13]] (-16.6)<br>[[11/9]] (-4.551)

| 3

| 514.286

| [[4/3]] (+16.241)

| [[27/20]] (-5.265)

| [[35/26]] (-0.326)

| 4

| 685.714

| [[3/2]] (-16.241)

| [[40/27]] (+5.265)

| [[52/35]] (+0.326)

| [[File:0-685,71 fifth (7-EDO).mp3|frameless]]

| 5

| 857.143

| [[105/64]] (-0.048)

| [[18/11]] (+4.551)<br>[[13/8]] (+16.6)<br>[[64/39]] (-0.374)

| 6

| 1028.571

| [[9/5]] (+10.975)

| [[49/27]] (-3.215)

| [[29/16]] (-1.006)<br>[[20/11]] (-6.424)

| [[File:0-1028,57 seventh (7-EDO).mp3|frameless]]

| 7

| 1200

| [[2/1]] (just)

== Notation ==

The usual [[Musical notation|notation system]] for 7edo is the [[chain-of-fifths notation]], which is directly derived from the standard notation used in [[12edo]].

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Notation of 7edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Chain-of-fifths notation]]

|-

|-

| '''D'''

|-

| 1

| 171.429

|-

| 2

| 342.857

| '''Perfect third (P3)'''

| '''F'''

|-

| 3

| 514.286

| '''G'''

|-

| 4

| 685.714

|-

| 5

| 857.143

|-

| 1028.571

| '''C'''

|-

| 7

| '''D'''

|}

* [[Ups and downs notation]] is identical to circle-of-fifths notation;

* Mixed and pure [[sagittal notation]] are identical to circle-of-fifths notation.

 

This notation is a subset of the notations for EDOs [[14edo#Sagittal notation|14]], [[21edo#Sagittal notation|21]], [[28edo#Sagittal notation|28]], [[35edo#Sagittal notation|35]], and [[42edo#Second-best fifth notation|42b]].

 

File:7-EDO_Sagittal.svg

rect 246 0 406 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

default [[File:7-EDO_Sagittal.svg]]

 

 

[[William Lynch]] proposes using numbers 1 through 7 as the nominals of 7edo with sharp signs being possible to expand to 14edo or even 21edo.

 

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

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

! [[Uniform solfege]]<br />(1 vowel)

| 171.429

| 1028.571

 

 

 

=== Commas ===

 

{| class="commatable wikitable center-1 center-2 right-4 center-5"

! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

| [[1600000/1594323|(14 digits)]]

| {{monzo| 9 -13 5 }}

|-

| <abbr title="1119744/1071875">(14 digits)</abbr>

| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>

| [[243/242]]

| [[385/384]]

| 4.50

| 3.03

| [[52/49]]

| {{monzo| 2 0 0 -2 0 1 }}

| 102.87

| [[27/26]]

|-

| {{monzo| -6 0 1 0 0 1 }}

 

7edo is the first edo in which [[regular temperament theory]] starts to make sense as a way of subdividing the steps into [[mos scale]]s, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another.

 

 

 

 

* Step size: 171.429{{c}}, octave size: 1200.0{{c}}

Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It doesn't approximate 7.

 

 

; [[18ed6]]

 

; [[WE|7et, 2.3.5.11.13 WE]]

 

; [[zpi|15zpi]]

 

; [[11edt]]

 

 

{{Main| 7edo/Music }}

 

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

 

<references group="note" />

 

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