ED5: Difference between revisions
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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]], pentave) into n equal parts'''. | ||
== Theory == | == Theory == | ||
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 range restriction is a crucial consideration. Pentave equivalence itself may have a basis in Western music seeing as minor chords have an octave of 5 in their root (i.e. 10:12:15). | 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 range restriction is a crucial consideration. Pentave equivalence itself may have a basis in Western music seeing as minor chords have an octave of 5 in their root (i.e. 10:12:15). | ||
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 quintessential example of a pentave based tuning is hyperpyth (see [[17ed5|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|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. | ||
Some equal divisions of the pentave are known by alternate names or have special interest: | Some equal divisions of the pentave are known by alternate names or have special interest: | ||
* [[3ed5]] [[orwell]] generator | * [[3ed5|3ED5]] [[orwell]] generator | ||
* [[4ed5]] [[meantone]] generator | * [[4ed5|4ED5]] [[meantone]] generator | ||
* [[5ed5]] [[2L_7s|thuja]] generator | * [[5ed5|5ED5]] [[2L_7s|thuja]] generator | ||
* [[6ed5]] [[Trienstonic clan #Uncle|uncle]] generator | * [[6ed5|6ED5]] [[Trienstonic clan #Uncle|uncle]] generator | ||
* [[8ed5]] [[mohajira]] generator | * [[8ed5|8ED5]] [[mohajira]] generator | ||
* [[Hyperpyth]] tuning (e.g. [[17ed5]]) | * [[Hyperpyth]] tuning (e.g. [[17ed5|17ED5]]) | ||
* [[20ed5]] Hieronymus Tuning | * [[20ed5|20ED5]] Hieronymus Tuning | ||
* [[25ed5]] (Stockhausen, McLaren) | * [[25ed5|25ED5]] (Stockhausen, McLaren) | ||
== Individual pages for ED5s == | == Individual pages for ED5s == | ||
* [[7ed5]] | * [[7ed5|7ED5]] | ||
* [[10ed5]] | * [[10ed5|10ED5]] | ||
* [[11ed5]] | * [[11ed5|11ED5]] | ||
* [[12ed5]] | * [[12ed5|12ED5]] | ||
* [[13ed5]] | * [[13ed5|13ED5]] | ||
* [[15ed5]] | * [[15ed5|15ED5]] | ||
* [[17ed5]] | * [[17ed5|17ED5]] | ||
* [[19ed5]] | * [[19ed5|19ED5]] | ||
* [[20ed5]] | * [[20ed5|20ED5]] | ||
* [[22ed5]] | * [[22ed5|22ED5]] | ||
* [[24ed5]] | * [[24ed5|24ED5]] | ||
* [[25ed5]] | * [[25ed5|25ED5]] | ||
* [[28ed5]] | * [[27ed5|27ED5]] | ||
* [[33ed5]] | * [[28ed5|28ED5]] | ||
* [[39ed5]] | * [[33ed5|33ED5]] | ||
* [[48ed5]] | * [[39ed5|39ED5]] | ||
* [[56ed5]] | * [[48ed5|48ED5]] | ||
* [[67ed5]] | * [[49ed5|49ED5]] | ||
* [[72ed5]] | * [[56ed5|56ED5]] | ||
* [[95ed5]] | * [[67ed5|67ED5]] | ||
* [[72ed5|72ED5]] | |||
* [[95ed5|95ED5]] | |||
== ED5-EDO correspondence == | == ED5-EDO correspondence == | ||
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| | [[323ed5]] | | | [[323ed5]] | ||
| | [[139edo]] | | | [[139edo]] | ||
| | 323ed5 is 139edo with ~.94 cent compressed compressed octaves. Patent vals match alternating primes through the 11-limit | | | 323ed5 is 139edo with ~.94 cent compressed compressed octaves. Patent vals match alternating primes through the 11-limit. | ||
|- | |- | ||
| | [[325ed5]] | | | [[325ed5]] | ||
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| | [[332ed5]] | | | [[332ed5]] | ||
| | [[143edo]] | | | [[143edo]] | ||
| | 332ed5 is 143edo with ~.13 cent stretched octaves. Patent vals match through the 29-limit with the exception of 7, 17 and 19 | | | 332ed5 is 143edo with ~.13 cent stretched octaves. Patent vals match through the 29-limit with the exception of 7, 17 and 19. | ||
|- | |- | ||
| | [[337ed5]] | | | [[337ed5]] | ||