Xenharmonic Wiki

7edo: Difference between revisions

← Older edit
VisualWikitext
→Commas: +Prime limit column; chroma > dicot comma
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,215 edits
 
(131 intermediate revisions by 30 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| de =  
| de = 7-EDO
| en = 7edo
| en = 7edo
| es =  
| es = 7 EDO
| ja = 7平均律
| ja = 7平均律
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 7 (prime)
{{ED intro}}
| Step size = 171.429¢
| Fifth = 4\7 = 685.714¢
| Major 2nd = 1\7 = 171¢
| Minor 2nd = 1\7 = 171¢
| Augmented 1sn = 0\7 = 0¢
}}
'''7-edo''' or "Neutral diatonic" divides the 1200-cent [[octave]] into 7 equal parts, making its smallest interval [[cent|171.428¢]], or the seventh root of 2. It is the fourth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].


== Theory ==
== Theory ==
[[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.
=== 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). 
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.
== Intervals ==
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="2" | <!-- empty cell -->
|+ style="font-size: 105%;" | Intervals of 7edo
! prime 2
|-
! prime 3
! rowspan="2" | [[Degree]]
! prime 5
! rowspan="2" | [[Cent]]s
! prime 7
! rowspan="2" | [[Interval region]]
! prime 11
! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])
! prime 13
! rowspan="2" | Audio
|-
|-
! rowspan="2" | error
! [[3-limit]]
! absolute (¢)
! [[5-limit]]
| 0.0
! [[7-limit]]
| -16.2
! Other
| -43.5
| +59.7
| -37.0
| +16.6
|-
|-
! [[Relative error|relative]] (%)
| 0
| 0
| -9
| 0
| -25
| Unison (prime)
| +35
| [[1/1]] (just)
| -22
|
| +10
|
|
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 171.429
| Submajor second
|
| [[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]]
|-
|-
! colspan="2" | [[nearest edomapping]]
| 7
| 4
| 2
| 2
| 6
| 342.857
| Neutral third
|
|
| [[128/105]] (+0.048)
| [[39/32]] (+0.374)<br>[[16/13]] (-16.6)<br>[[11/9]] (-4.551)
| [[File:piano_2_7edo.mp3]]
|-
| 3
| 3
| 514.286
| Fourth
| [[4/3]] (+16.241)
| [[27/20]] (-5.265)
|
| [[35/26]] (-0.326)
| [[File:0-514,29 fourth (7-EDO).mp3|frameless]]
|-
| 4
| 685.714
| Fifth
| [[3/2]] (-16.241)
| [[40/27]] (+5.265)
|
| [[52/35]] (+0.326)
| [[File:0-685,71 fifth (7-EDO).mp3|frameless]]
|-
| 5
| 5
| 857.143
| Neutral sixth
|
|
| [[105/64]] (-0.048)
| [[18/11]] (+4.551)<br>[[13/8]] (+16.6)<br>[[64/39]] (-0.374)
| [[File:0-857,14 sixth (7-EDO).mp3|frameless]]
|-
| 6
| 1028.571
| Supraminor seventh
|
| [[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]]
|-
|-
! colspan="2" | [[fifthspan]]
| 7
| 0
| 1200
| +1
| Octave
| -3
| [[2/1]] (just)
| -2
|  
| -1
|  
| +3
|  
| [[File:0-1200 octave.mp3|frameless]]
|}
|}


Equal-heptatonic scales are used in non-western music in African cultures as well as an integral part of early Thai and early Chinese music. It has been speculated in "Indian music:history and structure", that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones).
== 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]].


7-tet can be thought of as result of stacking seven 11/9s on top of each other, and then tempering to remove the comma (2^-2 3^-14 11^7). As a temperament, William Lynch gives it the name "Neutron[7]" just as the whole tone scale of 12 ET is known as "Hexe[6]".
Because the Pythagorean apotome ([[2187/2048]]) is [[tempered out]], sharps (♯) and flats (♭) are redundant in 7edo. Therefore, 7edo can be notated on a five-line staff without accidentals.  Alternatively, a seven-line stave can be used, with each horizontal line representing one pitch level. There is no distinction between major or minor, so every interval has the [[interval quality]] "perfect" instead.


Typically, 7-edo exists as the tuning for pentatonic scales in traditional thai music with the other two pitches acting as auxiliary tones. However, it can be used as an interesting diatonic scale choice as well in tunings such as 14 EDO or 21 EDO.
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 7edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 171.429
| '''Perfect second (P2)'''
| '''E'''
|-
| 2
| 342.857
| '''Perfect third (P3)'''
| '''F'''
|-
| 3
| 514.286
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 4
| 685.714
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 5
| 857.143
| '''Perfect sixth (P6)'''
| '''B'''
|-
| 6
| 1028.571
| '''Perfect seventh (P7)'''
| '''C'''
|-
| 7
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}
 
In 7edo:
* [[Ups and downs notation]] is identical to circle-of-fifths notation;
* Mixed and pure [[sagittal notation]] are identical to circle-of-fifths notation.


The seventh of 7-edo 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|harmonic entropy]] than [[Harmonic seventh|7/4]], a much simpler overtone seventh.
===Sagittal 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]].


Similarly, in equi-heptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. One of the most impressive areas in 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 culture 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 (386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system. For the notation of such music, a seven-line stave is most appropriate, with each horizontal line representing one pitch level.
<imagemap>
File:7-EDO_Sagittal.svg
desc none
rect 80 0 246 50 [[Sagittal_notation]]
rect 246 0 406 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 246 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:7-EDO_Sagittal.svg]]
</imagemap>


([http://www.britannica.com/EBchecked/topic/719112/African-music "African music." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 05 Jul. 2009])
Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.


A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.
=== Alternative notations ===
[[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.


== Intervals ==
== Solfege ==
7edo can be notated on a five-line staff without accidentals. There is no distinction between Major or Minor; each pitch class is unique.
{| class="wikitable center-all"
{| class="wikitable"
|+ <span style="font-size: 105%;">Solfege of 7edo</span>
|-
|-
! Interval
! [[Degree]]
! Cents
! [[Cents]]
! interval name
! Standard [[solfege]]<br />(movable do)
! The "neighborhood" of just intervals
! [[Uniform solfege]]<br />(1 vowel)
|-
|-
| 0
| 0
|0.000
| 0
| unison / prime
| Do
| exactly [[1/1]]
| Da
|-
|-
| 1
| 1
| 171.429
| 171.429
| second
| Re
| 6.424¢ from Ptolemy (neutral) second [[11/10]] <br /> 3.215¢ from second [[54/49]] <br> -1.006¢ from the 29th subharmonic [[32/29]] <br /> -10.975¢ from major second (small whole tone) [[10/9]]
| Ra
|-
|-
| 2
| 2
| 342.857
| 342.857
| third
| Mi
| 0.374¢ from neutral third [[39/32]]
| Ma
-4.55¢ from neutral third [[11/9]]
|-
|-
| 3
| 3
| 514.286
| 514.286
| fourth
| Fa
| 16.241¢ from just fourth [[4/3]] (498.045¢) <br /> -5.265¢ from wide fourth [[27/20]]
| Fa
|-
|-
| 4
| 4
| 685.714
| 685.714
| fifth
| So
| 5.265 ¢ from narrow fifth [[40/27]] <br /> -16.241¢ from just fifth [[3/2]] (701.955¢)
| Sa
|-
|-
| 5
| 5
| 857.143
| 857.143
| sixth
| La
| 4.551¢ from neutral sixth [[18/11]]
| La
-0.374¢ from neutral sixth [[64/39]]
|-
|-
| 6
| 6
| 1028.571
| 1028.571
| seventh
| Ti
| 10.975¢ from (Didymus) minor seventh [[9/5]] <br /> -6.424¢ from neutral seventh [[20/11]] <br /> -1.006¢ from the 29th harmonic [[29/16]] <br /> -3.215¢ from seventh [[49/27]]
| Ta
|-
|-
| 7
| 7
| 1200
| 1200
| octave
| Do
| exactly [[2/1]]
| Da
|}
|}


[[File:7ed2-001.svg|alt=alt : Your browser has no SVG support.]]
== Approximation to JI ==
 
[[File:7ed2-001.svg]]
[[:File:7ed2-001.svg|7ed2-001.svg]]
 
== Observations ==
Related in a lateral way to traditional Thai music. Subset of [[14edo|14 EDO]] and [[21edo|21 EDO]].
 
There is a neutral feel between whole tone scale and major/minor diatonic scale. The second (171.429 c) works well as a basic step for melodic progression.
 
The step from seventh to octave is too large for the leading tone.


== Notation ==
== Regular temperament properties ==
[[William Lynch]] proposes using numbers 1 through 7 as the nominals of 7 ET with sharp signs being possible to expand to [[14edo|14 EDO]] or even [[21edo|21 EDO]].
=== Uniform maps ===
{{Uniform map|edo=7}}


== Commas ==
=== Commas ===
7 EDO tempers out the following commas. This assumes val {{val| 7 11 16 20 24 26 }}.  
7et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 7 11 16 20 24 26 }}.  


{| class="wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]s
! [[Ratio]]<ref group="note">Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! [[Color notation/Temperament names|Color name]]
! Name(s)
! Name(s)
|-
|-
| 3
| 3
| [[2187/2048]]
| [[2187/2048]]
| {{Monzo| -11 7 }}
| {{monzo| -11 7 }}
| 113.69
| 113.69
| Lawa
| Lawa
| Apotome, Pythagorean chromatic semitone
| Whitewood comma, apotome, Pythagorean chroma
|-
|-
| 5
| 5
| [[135/128]]
| [[135/128]]
| {{Monzo| -7 3 1 }}
| {{monzo| -7 3 1 }}
| 92.18
| 92.18
| Layobi
| Layobi
| Major chroma, major limma, pelogic comma
| Mavila comma, major chroma
|-
|-
| 5
| 5
| [[25/24]]
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| {{monzo| -3 -1 2 }}
| 70.67
| 70.67
| Yoyo
| Yoyo
| Classic chromatic semitone, dicot comma
| Dicot comma, classic chroma
|-
|-
| 5
| 5
| [[250/243]]
| [[250/243]]
| {{Monzo| 1 -5 3 }}
| {{monzo| 1 -5 3 }}
| 49.17
| 49.17
| Triyo
| Triyo
| Maximal diesis, porcupine comma
| Porcupine comma, maximal diesis
|-
|-
| 5
| 5
| [[20000/19683]]
| [[20000/19683]]
| {{Monzo| 5 -9 4 }}
| {{monzo| 5 -9 4 }}
| 27.66
| 27.66
| Saquadyo
| Saquadyo
| Minimal diesis, tetracot comma
| Tetracot comma, minimal diesis
|-
|-
| 5
| 5
| [[81/80]]
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
| Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| 5
| 5
| [[1600000/1594323]]
| [[1600000/1594323|(14 digits)]]
| {{Monzo| 9 -13 5 }}
| {{monzo| 9 -13 5 }}
| 6.15
| 6.15
| Saquinyo
| Saquinyo
| Amity comma
| [[Amity comma]]
|-
| 7
| <abbr title="1119744/1071875">(14 digits)</abbr>
| {{monzo| 9 7 -5 -3 }}
| 75.64
| Triru-aquingu
| [[Superpine comma]]
|-
|-
| 7
| 7
| [[36/35]]
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal quartertone
| Mint comma, septimal quartertone
|-
|-
| 7
| 7
| [[525/512]]
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| {{monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
Line 216: Line 350:
| 7
| 7
| [[64/63]]
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ru
Line 223: Line 357:
| 7
| 7
| [[875/864]]
| [[875/864]]
| {{Monzo| -5 -3 3 1 }}
| {{monzo| -5 -3 3 1 }}
| 21.90
| 21.90
| Zotriyo
| Zotriyo
Line 230: Line 364:
| 7
| 7
| [[5120/5103]]
| [[5120/5103]]
| {{Monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyo
| Hemifamity
| Hemifamity comma
|-
|-
| 7
| 7
| [[6144/6125]]
| [[6144/6125]]
| {{Monzo| 11 1 -3 -2 }}
| {{monzo| 11 1 -3 -2 }}
| 5.36
| 5.36
| Sarurutriyo
| Sarurutriyo
| Porwell
| Porwell comma
|-
|-
| 7
| 7
| [[4375/4374]]
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| {{monzo| -1 -7 4 1 }}
| 0.40
| 0.40
| Zoquadyo
| Zoquadyo
Line 250: Line 384:
|-
|-
| 7
| 7
| …
| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>
| {{Monzo| 47 -7 -7 -7 }}
| {{monzo| 47 -7 -7 -7 }}
| 0.34
| 0.34
| Trisa-rugu
| Trisa-rugu
| Akjaysma, 5\7 octave comma
| [[Akjaysma]]
|-
| 11
| [[33/32]]
| {{monzo| -5 1 0 0 1}}
| 53.27
| Ilo
| Io comma, undecimal quartertone
|-
|-
| 11
| 11
| [[100/99]]
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyo
Line 265: Line 406:
| 11
| 11
| [[121/120]]
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
Line 272: Line 413:
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lurugugu
| Lurugugu
Line 279: Line 420:
| 11
| 11
| [[65536/65219]]
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| 8.39
| Satrilu-aruru
| Satrilu-aruru
Line 286: Line 427:
| 11
| 11
| [[243/242]]
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| {{monzo| -1 5 0 0 -2 }}
| 7.14
| 7.14
| Lulu
| Lulu
Line 293: Line 434:
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
Line 300: Line 441:
| 11
| 11
| [[4000/3993]]
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Triluyo
| Triluyo
| Wizardharry
| Wizardharry comma
|-
| 13
| [[14641/13312]]
| {{monzo| -10 0 0 0 4 -1 }}
| 164.74
|
|
|-
| 13
| [[52/49]]
| {{monzo| 2 0 0 -2 0 1 }}
| 102.87
| thoruru unison
| Hammerisma
|-
| 13
| [[27/26]]
| {{monzo| -1 3 0 0 0 -1 }}
| 65.33
| Thu
| Small tridecimal third tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
|
| Wilsorma
|}
|}
== Temperaments ==
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.
1/7 can be considered the intersection of sharp [[porcupine]] and flat [[tetracot]] temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat [[Mohajira]] or critically sharp [[amity]], and creates mosses of 322 and 2221.
3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.
== Octave stretch ==
What follows is a comparison of stretched-octave 7edo tunings.
; 7edo
* 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.
{{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}}
{{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}}
; [[WE|7et, 2.3.11.13 WE]]
* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
{{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
{{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}
; [[18ed6]]
* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. The tuning 18ed6 does this.
{{Harmonics in equal|18|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}
{{Harmonics in equal|18|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}
; [[WE|7et, 2.3.5.11.13 WE]]
* Step size: 172.390{{c}}, octave size: 1206.7{{c}}
Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
{{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
{{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}
; [[zpi|15zpi]]
* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.
{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}
{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}
; [[11edt]]
* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.
{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}
{{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}
== Instruments ==
* [[Lumatone mapping for 7edo]]


== Music ==
== Music ==
{{Main| 7edo/Music }}
{{Catrel|7edo tracks}}


* [https://soundcloud.com/overtoneshock/death-teasing-monolith-7-edo-premiere Death Giving Monolith] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] (dulcimer and voice)
== Ear training ==
* [https://xenharmonicgod.bandcamp.com/track/jingle-bells-7-edo Jingle Bells cover!] (recorded by Stephen Weigel)
7edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].
* [http://sethares.engr.wisc.edu/mp3s/PagansRevenge.mp3 Pagan's Revenge] by [[Bill Sethares]] (synthetic gamelan)
* [http://www.akjmusic.com/audio/i_dream_of_tibet.mp3 I dream of Tibet] {{dead link}} by [http://www.akjmusic.com/works.html Aaron K. Johnson] (electronic swirlies) {{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Walker/Seven%20Equal%20Trio.mp3 Seven Equal Trio] by [http://robertinventor.tripod.com/tunes/tunes.htm#7_equal_trio Robert Walker] ((synth) violin, viola, glockenspiel)
* [http://micro.soonlabel.com/gene_ward_smith/Others/Hunt/7ET.mp3 Two-part Invention in 7TET] by [http://www.h-pi.com/musicFiles.html Aaron Hunt]
* [http://www.uvnitr.cz/flaoyg/flao_yg/pavouci.html Pavouci], [http://www.uvnitr.cz/flaoyg/flao_yg/kelt.html Kelt] by Milan Guštar
* [http://www.seraph.it/dep/det/7edo%20dance.mp3 7edo Dance] ''by [[Carlo_Serafini|Carlo Serafini]]''
* ''[https://www.youtube.com/watch?v=IQ4hY41TvAo&feature=youtu.be Nightfire (video)]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/ac7b7a4c36a28d28f624fc2066b98f3c-236.html blog entry])
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/06%20-%206.%207%20octave.mp3 Comets Over Flatland 6] by [[Randy_Winchester|Randy Winchester]]
* [http://media.soundcloud.com/stream/IUcgFYhtu3Rk?stream_token=4QIvd Sävelmä] {{dead link}} by [http://soundcloud.com/juhani-nuorvala/s-velm-long-version Sävelmä (long version)] by [[Juhani Nuorvala]]
* [https://soundcloud.com/santiagocosentino/rock-in-7edo Rock in 7edo by Santiago Cosentino] {{dead link}}
* [https://soundcloud.com/user-544568549/zhaleyka-7-edo Zhaleyka] by [[Dmitry Bazhenov]]


== Ear Training ==
== Notes ==
7 EDO ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].
<references group="note" />


[[Category:7-tone]]
== References ==
[[Category:7edo]]
<references />
[[Category:Equal divisions of the octave]]
[[Category:Golden]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Zeta]]


[[Category:todo:unify_precision]]
[[Category:3-limit record edos|#]] <!-- 1-digit number -->
[[Category:7-tone scales]]
Retrieved from "https://en.xen.wiki/w/7edo"