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7edo: Difference between revisions

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zeta is not the reason 7edo is close to JI
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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>
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>
=== Octave stretch ===
[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.
The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo's approximation of JI in a similar way.


=== Subsets and supersets ===
=== Subsets and supersets ===
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! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! rowspan="2" | [[Interval region]]
! colspan="5" | Approximated [[JI]] intervals ([[error]] in [[`¢]])
! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])
! rowspan="2" | Audio
! rowspan="2" | Audio
|-
|-
! [[3-limit]]
! [[3-limit]]
!2.3.13
! [[5-limit]]
! [[5-limit]]
! [[7-limit]]
! [[7-limit]]
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| Unison (prime)
| Unison (prime)
| [[1/1]] (just)
| [[1/1]] (just)
|
|
|
|  
|  
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| 171.429
| 171.429
| Submajor second
| Submajor second
|
|
|
| [[10/9]] (-10.975)
| [[10/9]] (-10.975)
| [[54/49]] (+3.215)
| [[54/49]] (+3.215)
| [[11/10]] (+6.424)<br />[[32/29]] (-1.006)
| [[11/10]] (+6.424)<br>[[32/29]] (-1.006)
| [[File:0-171,43 second (7-EDO).mp3|frameless]]
| [[File:0-171,43 second (7-EDO).mp3|frameless]]
|-
|-
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| Neutral third
| Neutral third
|  
|  
|[[39/32]] (+0.374)
[[16/13]] (-16.6)
|
|
| [[128/105]] (+0.048)
| [[128/105]] (+0.048)
| <br />[[11/9]] (-4.551)
| [[39/32]] (+0.374)<br>[[16/13]] (-16.6)<br>[[11/9]] (-4.551)
| [[File:piano_2_7edo.mp3]]
| [[File:piano_2_7edo.mp3]]
|-
|-
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| Fourth
| Fourth
| [[4/3]] (+16.241)
| [[4/3]] (+16.241)
|
| [[27/20]] (-5.265)
| [[27/20]] (-5.265)
|  
|  
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| Fifth
| Fifth
| [[3/2]] (-16.241)
| [[3/2]] (-16.241)
|
| [[40/27]] (+5.265)
| [[40/27]] (+5.265)
|  
|  
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| Neutral sixth
| Neutral sixth
|  
|  
|[[13/8]]
|  
(+16.6)
[[64/39]] (-0.374)
|
| [[105/64]] (-0.048)
| [[105/64]] (-0.048)
| [[18/11]] (+4.551)<br />
| [[18/11]] (+4.551)<br>[[13/8]] (+16.6)<br>[[64/39]] (-0.374)
| [[File:0-857,14 sixth (7-EDO).mp3|frameless]]
| [[File:0-857,14 sixth (7-EDO).mp3|frameless]]
|-
|-
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| Supraminor seventh
| Supraminor seventh
|  
|  
|
| [[9/5]] (+10.975)
| [[9/5]] (+10.975)
| [[49/27]] (-3.215)
| [[49/27]] (-3.215)
| [[29/16]] (-1.006)<br />[[20/11]] (-6.424)
| [[29/16]] (-1.006)<br>[[20/11]] (-6.424)
|[[File:0-1028,57 seventh (7-EDO).mp3|frameless]]
| [[File:0-1028,57 seventh (7-EDO).mp3|frameless]]
|-
|-
| 7
| 7
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| Octave
| Octave
| [[2/1]] (just)
| [[2/1]] (just)
|
|  
|
|  
|  
|  
|  
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== Approximation to JI ==
== Approximation to JI ==
[[File:7ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:7ed2-001.svg]]
 
[[:File:7ed2-001.svg|7ed2-001.svg]]
 
=== Zeta peak index ===
{{ZPI
| zpi = 15
| steps = 6.95668765658792
| step size = 172.495885863671
| tempered height = 4.166936
| pure height = 3.940993
| integral = 1.162332
| gap = 14.234171
| octave = 1207.47120104570
| consistent = 6
| distinct = 5
}}


== Regular temperament properties ==
== Regular temperament properties ==
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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.
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 ==
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<references />
<references />


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