Dwarf: Difference between revisions
No edit summary Tags: Reverted Visual edit |
Re-install some changes that are valuable (2/2) |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
A '''dwarf''' is a [[period]]ic [[scale]] obtained by sequentially mapping odd [[harmonic]]s (1, 3, 5, 7, …) using a [[regular temperament]]. A dwarf is a kind of [[detempering|detempered scale]]. The name ''dwarf'' refers to the fact that you are choosing for each degree the smallest [[Tenney height]]. Dwarf scales often produce results which are rich harmonically. They | A '''dwarf''' is a [[period]]ic [[scale]] obtained by sequentially mapping odd [[harmonic]]s (1, 3, 5, 7, …) using a [[regular temperament]]. A dwarf is a kind of [[detempering|detempered scale]]. The name ''dwarf'' refers to the fact that you are choosing for each degree the smallest [[Tenney height]]. Dwarf scales often produce results which are rich harmonically. They are [[otonal]], with intervals expressible as [[octave reduction|octave-reduced]] forms of [[harmonic]]s, and may be flipped to provide [[utonal]] versions of the same scales. | ||
== Construction == | == Construction == | ||
For an equal temperament ''n''- | For an [[equal temperament]] ''n''-et characterized by a [[val]] ''V'' = {{val| ''n'' … }}, suppose the coordinates of the val, reduced {{w|modulo operation|modulo}} ''n'', are distinct. An example would be {{val| 12 19 28 34 }}; reduced mod 12 this is {{val| 0 7 4 10 }} and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers within the [[subgroup]] of the equal temperament in order of increasing size, 1, 3, 5, 7, … and map them by the val ''V'', octave-reducing the result. If this number (from 0 to (''n'' - 1)) has not appeared before, add the odd positive integer to a set. When ''n'' values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(''V''), the dwarf scale resulting from the val ''V''. | ||
== Example == | |||
{{See also| Scalesmith }} | |||
Of particular interest are dwarf scales resulting from equal temperament vals which are [[epimorphic]] for the val ''V'', but even vals far removed from an equal temperament will produce a scale. | |||
Let us construct a JI dwarf by {{val| 12 19 28 34 }} of [[12edo|12et]]. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is | |||
: 135/128, 9/8, 75/64, 5/4, 21/16, 45/32, 3/2, 27/16, 7/4, 15/8, 2/1 | |||
And that is exactly [[Dwarf12 7]], the dwarf of 12et in the 7-limit. | |||
For another example, consider {{val| 7 11 16 }} of 7et in the 5-limit. Here 3/2 is mapped to 4\7, 5/4 to 2\7, 9/8 to 1\7, 15/8 to 6\7, 25/16 to 4\7 (discarded), 27/16 to 5\7, and 45/32 to 3\7, so the scale is: | |||
: 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1 | |||
In fact, this is the Lydian mode of [[Zarlino]], which exposes the fact that all the intervals of Zarlino Lydian can be expressed as octave-reduced harmonics. | |||
== Generalization == | |||
=== Symmetrical dwarf === | |||
For a symmetrical scale, we may consider for each odd harmonic both the octave reduction and octave complement, with equal priority. In the case of a tie, like in an even edo, either may be chosen. Unlike standard dwarves, which are strictly otonal, these are more balanced. For example, the symmetrical dwarf of 7et in the 5-limit is | |||
: 9/8, 5/4, 4/3, 3/2, 8/5, 16/9, 2/1 | |||
== See also == | == See also == | ||
Latest revision as of 10:15, 12 November 2025
A dwarf is a periodic scale obtained by sequentially mapping odd harmonics (1, 3, 5, 7, …) using a regular temperament. A dwarf is a kind of detempered scale. The name dwarf refers to the fact that you are choosing for each degree the smallest Tenney height. Dwarf scales often produce results which are rich harmonically. They are otonal, with intervals expressible as octave-reduced forms of harmonics, and may be flipped to provide utonal versions of the same scales.
Construction
For an equal temperament n-et characterized by a val V = ⟨n …], suppose the coordinates of the val, reduced modulo n, are distinct. An example would be ⟨12 19 28 34]; reduced mod 12 this is ⟨0 7 4 10] and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers within the subgroup of the equal temperament in order of increasing size, 1, 3, 5, 7, … and map them by the val V, octave-reducing the result. If this number (from 0 to (n - 1)) has not appeared before, add the odd positive integer to a set. When n values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(V), the dwarf scale resulting from the val V.
Example
Of particular interest are dwarf scales resulting from equal temperament vals which are epimorphic for the val V, but even vals far removed from an equal temperament will produce a scale.
Let us construct a JI dwarf by ⟨12 19 28 34] of 12et. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is
- 135/128, 9/8, 75/64, 5/4, 21/16, 45/32, 3/2, 27/16, 7/4, 15/8, 2/1
And that is exactly Dwarf12 7, the dwarf of 12et in the 7-limit.
For another example, consider ⟨7 11 16] of 7et in the 5-limit. Here 3/2 is mapped to 4\7, 5/4 to 2\7, 9/8 to 1\7, 15/8 to 6\7, 25/16 to 4\7 (discarded), 27/16 to 5\7, and 45/32 to 3\7, so the scale is:
- 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1
In fact, this is the Lydian mode of Zarlino, which exposes the fact that all the intervals of Zarlino Lydian can be expressed as octave-reduced harmonics.
Generalization
Symmetrical dwarf
For a symmetrical scale, we may consider for each odd harmonic both the octave reduction and octave complement, with equal priority. In the case of a tie, like in an even edo, either may be chosen. Unlike standard dwarves, which are strictly otonal, these are more balanced. For example, the symmetrical dwarf of 7et in the 5-limit is
- 9/8, 5/4, 4/3, 3/2, 8/5, 16/9, 2/1