ED5: Difference between revisions

Line 3:

== 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).

One way to treat 5/1 as an equivalence is by eliminating the primes 2 and 3. The triad in this paradigm is 5:7:11. This chord can be approximated in a 5.7.11 (or "no-twos-or-threes [[11-limit]]") subgroup [[regular temperament]] by eliminating the comma 859375/823543, equating [[7/5]]<sup>7</sup> with [[11/5]]. Other equivalences that could be used for such "no-two-or-threes" music include [[Ed11/5|equal divisions of 11/5]] and [[Ed11/7|equal divisions of 11/7]].

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.

Much like how [[EDT]]s can be used for "no-twos" harmony, ED5s can be used for "no-twos-or-threes" harmony, but it might be better to use a narrower equivalence interval for this like [[Ed11/5]], especially since the 11/5.7.11 fractional subgroup can be equated to the 5.7.11 subgroup.

Some equal divisions of the pentave are known by alternate names or have special interest:

@@ Line 3: / Line 3: @@
 == 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).
+One way to treat 5/1 as an equivalence is by eliminating the primes 2 and 3. The triad in this paradigm is 5:7:11. This chord can be approximated in a 5.7.11 (or "no-twos-or-threes [[11-limit]]") subgroup [[regular temperament]] by eliminating the comma 859375/823543, equating [[7/5]]<sup>7</sup> with [[11/5]]. Other equivalences that could be used for such "no-two-or-threes" music include [[Ed11/5|equal divisions of 11/5]] and [[Ed11/7|equal divisions of 11/7]].
 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.
-Much like how [[EDT]]s can be used for "no-twos" harmony, ED5s can be used for "no-twos-or-threes" harmony, but it might be better to use a narrower equivalence interval for this like [[Ed11/5]], especially since the 11/5.7.11 fractional subgroup can be equated to the 5.7.11 subgroup.
 Some equal divisions of the pentave are known by alternate names or have special interest: