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'''Ed5''' means '''Division of the Fifth Harmonic (5/1) into n equal parts'''.
'''Ed5''' means '''Division of the Fifth Harmonic (5/1) into n equal parts'''.


=Division of the fifth harmonic into n equal parts=
<span style="font-size: 150%;">Division of the fifth harmonic into n equal parts</span>


The fifth harmonic is particularly wide as far as equivalences go.<span style=""> There are (at absolute most) ~4.8 pentaves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with pentave equivalence, </span>this fact shapes one's musical approach dramatically. Following this, the quintessential example of a pentave based tuning is hyperpyth (see [[17ed5]]). However, perhaps the more common reason to use these scales is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus ([[20ed5]]) which itself is a zeta peak tuning (not "no-fives", full on zeta). Other reasons for taking the nth root of 5 include finding temperaments like orwell, meantone, and thuja. This approach can of course be used indiscriminately.
The fifth harmonic is particularly wide as far as equivalences go. There are (at absolute most) ~4.8 pentaves within the human hearing range; imagine if that were the case with octaves. If one does indeed deal with pentave equivalence, this fact shapes one's musical approach dramatically. Following this, the quintessential example of a pentave based tuning is hyperpyth (see [[17ed5]]). However, perhaps the more common reason to use these scales is in approximation with lower harmonic factors than 5. This approach is highlighted by Hieronymus ([[20ed5]]) which itself is a zeta peak tuning (not "no-fives", full on zeta). Other reasons for taking the nth root of 5 include finding temperaments like orwell, meantone, and thuja. This approach can of course be used indiscriminately.


== Individual pages for ED5s ==
== Individual pages for ED5s ==
Line 104: Line 104:
*[[99ed5]]
*[[99ed5]]
*[[100ed5]] compare [[43edo]]
*[[100ed5]] compare [[43edo]]
==ED5-EDO correspondence==
{| class="wikitable"
|-
! | ED5
! | EDO
! | Comments
|-
| | [[7ed5]]
| | [[3edo]]
| | 7ed5 is 3edo with ~5.9 cent compressed octaves. <br>Equivalently, 3edo is 7ed5 with pentaves stretched by ~13.7 cents. <br>Patent vals match through the 67-limit.
|-
| | [[9ed5]]
| | [[4edo]]
| | Very rough correspondence, but patent vals agree through the 7-limit.
|-
| | [[14ed5]]
| | [[6edo]]
| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
|-
| | [[16ed5]]
| | [[7edo]]
| | Only rough correspondence. Patent vals differ in the 7-limit.
|-
| | [[21ed5]]
| | [[9edo]]
| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 59-limit.
|-
| | [[23ed5]]
| | [[10edo]]
| | 23ed5 is 10edo with ~11 cent stretched octaves. Patent vals match through the 23-limit, with the exception of 11 and 17.
|-
| | [[28ed5]]
| | [[12edo]]
| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 19-limit, with the exception of 13 (as expected).
|-
| | [[30ed5]]
| | [[13edo]]
| | 30ed5 is 13edo with ~7.4 cent stretched octaves. Patent vals match through the 19-limit, with the exception of 3 (which neither represents well).
|-
| | [[35ed5]]
| | [[15edo]]
| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 13-limit.
|-
| | [[37ed5]]
| | [[16edo]]
| | 37ed5 is 16edo with ~4.9 cent stretched octaves. Patent vals match through the 23-limit.
|-
| | [[42ed5]]
| | [[18edo]]
| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals match through the 17-limit, with the exception of 11.
|-
| | [[44ed5]]
| | [[19edo]]
| | 44ed5 is 19edo with ~3.2 cent stretched octaves. Patent vals match through the 13-limit.
|-
| | [[49ed5]]
| | [[21edo]]
| | Same 5.9 cent octave compression as 7ed5~3edo. Patent vals agree through the 17-limit.
|-
| | [[51ed5]]
| | [[22edo]]
| | 51ed5 is 22edo with ~1.9 cent stretched octaves. Patent vals match through the 19-limit.
|-
| | [[56ed5]]
| | [[24edo]]
| | This is a rough correspondence, as the (7n)ed5 ~ (3n)edo sequence begins to break down. Patent vals match through the 13-limit, with the exception of 7.
|-
| | [[58ed5]]
| | [[25edo]]
| | 58ed5 is 25edo with ~1.0 cent stretched octaves. Patent vals match through the 73-limit, with the exception of 13.
|-
| | [[65ed5]]
| | [[28edo]]
| | 65ed5 is 28edo with ~0.26 cent stretched octaves.
|-
| | [[72ed5]]
| | [[31edo]]
| | 72ed5 is 31edo with ~0.34 cent compressed octaves.
|-
| | [[79ed5]]
| | [[34edo]]
| | 79ed5 is 34edo with ~0.83 cent compressed octaves.
|-
| | [[81ed5]]
| | [[35edo]]
| |
|-
| | [[86ed5]]
| | [[37edo]]
| |
|-
| | [[88ed5]]
| | [[38edo]]
| |
|-
| | [[93ed5]]
| | [[40edo]]
| |
|-
| | [[95ed5]]
| | [[41edo]]
| |
|-
| | [[100ed5]]
| | [[43edo]]
| |
|-
| | [[102ed5]]
| | [[44edo]]
| |
|-
| | [[107ed5]]
| | [[46edo]]
| |
|-
| | [[109ed5]]
| | [[47edo]]
| |
|-
| | [[114ed5]]
| | [[49edo]]
| |
|-
| | [[116ed5]]
| | [[50edo]]
| |
|-
| | [[121ed5]]
| | [[52edo]]
| |
|-
| | [[123ed5]]
| | [[53edo]]
| |
|-
| | [[128ed5]]
| | [[55edo]]
| |
|-
| | [[130ed5]]
| | [[56edo]]
| |
|-
| | [[137ed5]]
| | [[59edo]]
| |
|-
| | [[144ed5]]
| | [[62edo]]
| |
|-
| | [[146ed5]]
| | [[63edo]]
| |
|-
| | [[151ed5]]
| | [[65edo]]
| |
|-
| | [[153ed5]]
| | [[66edo]]
| |
|-
| | [[158ed5]]
| | [[68edo]]
| |
|-
| | [[160ed5]]
| | [[69edo]]
| |
|-
| | [[165ed5]]
| | [[71edo]]
| |
|-
| | [[167ed5]]
| | [[72edo]]
| |
|-
| | [[172ed5]]
| | [[74edo]]
| |
|-
| | [[174ed5]]
| | [[75edo]]
| |
|-
| | [[179ed5]]
| | [[77edo]]
| |
|-
| | [[181ed5]]
| | [[78edo]]
| |
|-
| | [[186ed5]]
| | [[80edo]]
| |
|-
| | [[188ed5]]
| | [[81edo]]
| |
|-
| | [[193ed5]]
| | [[83edo]]
| |
|-
| | [[195ed5]]
| | [[84edo]]
| |
|-
| | [[202ed5]]
| | [[87edo]]
| |
|-
| | [[209ed5]]
| | [[90edo]]
| |
|-
| | [[211ed5]]
| | [[91edo]]
| |
|-
| | [[216ed5]]
| | [[93edo]]
| |
|-
| | [[218ed5]]
| | [[94edo]]
| |
|-
| | [[223ed5]]
| | [[96edo]]
| |
|-
| | [[225ed5]]
| | [[97edo]]
| |
|-
| | [[230ed5]]
| | [[99edo]]
| |
|}


==See also==
==See also==
Retrieved from "https://en.xen.wiki/w/ED5"