Hydra: Difference between revisions

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'''Hydra''' is a 9-note subset of [[17edo|17edo]], having step pattern 3 3 1 1 2 3 2 1 1.

Hydra was originally envisioned as the ~~[https://en.wikipedia.org/wiki/Union_~~(~~set_theory~~) union] of the [[5L 2s|diatonic]] [[~~MOS~~]] and the [[Mohajira|rast]] [[~~MODMOS~~]] in [[17edo]]. Since those two scales have five out of seven notes in common, this results in an enneatonic scale. In addition to diatonic and rast, hydra also contains a number of other very useful scales as subsets (such as bayati/[[~~Screamapillar|~~screamapillar]]), and therefore can do everything those scales can, and more.

Hydra was originally envisioned as the {{w|Union (set theory)|union}} of the [[5L 2s|diatonic]] [[mos]] and the [[Mohajira|rast]] [[modmos]] in [[17edo]]. Since those two scales have five out of seven notes in common, this results in an enneatonic scale. In addition to diatonic and rast, hydra also contains a number of other very useful scales as subsets (such as bayati/[[screamapillar]]), and therefore can do everything those scales can, and more.

As an enneatonic scale, hydra allows for a great deal more variety than heptatonic scales do, but with a relatively manageable number of pitches, and still fits within the "~~[https://en.wikipedia.org/wiki/The_Magical_Number_Seven~~,~~_Plus_or_Minus_Two~~ seven plus or minus two]" range.

As an enneatonic scale, hydra allows for a great deal more variety than heptatonic scales do, but with a relatively manageable number of pitches, and still fits within the "{{w|The Magical Number Seven, Plus or Minus Two|seven plus or minus two}}" range.

Perhaps the most immediately obvious feature of hydra that distinguishes it from more familiar scales is that it is not ~~an [[MOS]]~~, or even close. In fact, the large and small steps tend to cluster together, resulting in a highly uneven scale. This unevenness can be analyzed using ~~[https://en.wikipedia.org/wiki/Moment_~~(mathematics) moments], where the zeroth moment is simply the cardinality of the scale, the first moment is the tendency to form clusters separated by one period, the second moment the tendency to form clusters separated by 1/2 period, etc. As it turns out, hydra minimizes the first moment, but maximizes the second, so its pitches tend to fall near the notes of an overlaid [[~~2edo|~~2edo]].

Perhaps the most immediately obvious feature of hydra that distinguishes it from more familiar scales is that it is not a mos, or even close. In fact, the large and small steps tend to cluster together, resulting in a highly uneven scale. This unevenness can be analyzed using {{w|Moment (mathematics)|moments}}, where the zeroth moment is simply the cardinality of the scale, the first moment is the tendency to form clusters separated by one period, the second moment the tendency to form clusters separated by 1/2 period, etc. As it turns out, hydra minimizes the first moment, but maximizes the second, so its pitches tend to fall near the notes of an overlaid [[2edo]].

This unevenness is not necessarily a bad thing, since it adds more dynamic and contrast to the melody and causes the scale to have a more defined tonality. In fact, it may well be the case that the best-sounding large (i. e., with more than 7 pitches) scales are those that have a highly uneven, rather than maximally even or ~~MOS~~-like structures.

This unevenness is not necessarily a bad thing, since it adds more dynamic and contrast to the melody and causes the scale to have a more defined tonality. In fact, it may well be the case that the best-sounding large (i.e. with more than 7 pitches) scales are those that have a highly uneven, rather than maximally even or mos-like structures.

The name hydra is a reference to the nine-headed beast of Greek mythology. Given that this scale was derived by merging together several heptatonic scales into one, the name of a multi-headed beast is particularly fitting.

There are analogues of hydra in other edos, too, obtained by fusing that edo's version of diatonic with its version of the rast ~~MODMOS~~. The [[~~31edo|~~31edo]] version (5 5 3 1 4 5 4 1 3) could be called Nāga; its smallest interval is a diesis, which makes it blatantly microtonal in a way that hydra is not. The 24edo version (4 4 2 1 3 4 3 1 2) could be called ~~[https://en.wikipedia.org/wiki/~~Xiangliu Xiangliu]. Both of these scales have four different step sizes, whereas hydra only has three.

There are analogues of hydra in other edos, too, obtained by fusing that edo's version of diatonic with its version of the rast modmos. The [[31edo]] version (5 5 3 1 4 5 4 1 3) could be called Nāga; its smallest interval is a diesis, which makes it blatantly microtonal in a way that hydra is not. The 24edo version (4 4 2 1 3 4 3 1 2) could be called {{w|Xiangliu|Xiangliu}}. Both of these scales have four different step sizes, whereas hydra only has three.

[[Category:17edo]]

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 '''Hydra''' is a 9-note subset of [[17edo|17edo]], having step pattern 3 3 1 1 2 3 2 1 1.
-Hydra was originally envisioned as the [https://en.wikipedia.org/wiki/Union_(set_theory) union] of the [[5L 2s|diatonic]] [[MOS]] and the [[Mohajira|rast]] [[MODMOS]] in [[17edo]]. Since those two scales have five out of seven notes in common, this results in an enneatonic scale. In addition to diatonic and rast, hydra also contains a number of other very useful scales as subsets (such as bayati/[[Screamapillar|screamapillar]]), and therefore can do everything those scales can, and more.
+Hydra was originally envisioned as the {{w|Union (set theory)|union}} of the [[5L 2s|diatonic]] [[mos]] and the [[Mohajira|rast]] [[modmos]] in [[17edo]]. Since those two scales have five out of seven notes in common, this results in an enneatonic scale. In addition to diatonic and rast, hydra also contains a number of other very useful scales as subsets (such as bayati/[[screamapillar]]), and therefore can do everything those scales can, and more.
-As an enneatonic scale, hydra allows for a great deal more variety than heptatonic scales do, but with a relatively manageable number of pitches, and still fits within the "[https://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two seven plus or minus two]" range.
+As an enneatonic scale, hydra allows for a great deal more variety than heptatonic scales do, but with a relatively manageable number of pitches, and still fits within the "{{w|The Magical Number Seven, Plus or Minus Two|seven plus or minus two}}" range.
-Perhaps the most immediately obvious feature of hydra that distinguishes it from more familiar scales is that it is not an [[MOS]], or even close. In fact, the large and small steps tend to cluster together, resulting in a highly uneven scale. This unevenness can be analyzed using [https://en.wikipedia.org/wiki/Moment_(mathematics) moments], where the zeroth moment is simply the cardinality of the scale, the first moment is the tendency to form clusters separated by one period, the second moment the tendency to form clusters separated by 1/2 period, etc. As it turns out, hydra minimizes the first moment, but maximizes the second, so its pitches tend to fall near the notes of an overlaid [[2edo|2edo]].
+Perhaps the most immediately obvious feature of hydra that distinguishes it from more familiar scales is that it is not a mos, or even close. In fact, the large and small steps tend to cluster together, resulting in a highly uneven scale. This unevenness can be analyzed using {{w|Moment (mathematics)|moments}}, where the zeroth moment is simply the cardinality of the scale, the first moment is the tendency to form clusters separated by one period, the second moment the tendency to form clusters separated by 1/2 period, etc. As it turns out, hydra minimizes the first moment, but maximizes the second, so its pitches tend to fall near the notes of an overlaid [[2edo]].
-This unevenness is not necessarily a bad thing, since it adds more dynamic and contrast to the melody and causes the scale to have a more defined tonality. In fact, it may well be the case that the best-sounding large (i. e., with more than 7 pitches) scales are those that have a highly uneven, rather than maximally even or MOS-like structures.
+This unevenness is not necessarily a bad thing, since it adds more dynamic and contrast to the melody and causes the scale to have a more defined tonality. In fact, it may well be the case that the best-sounding large (i.e. with more than 7 pitches) scales are those that have a highly uneven, rather than maximally even or mos-like structures.
 The name hydra is a reference to the nine-headed beast of Greek mythology. Given that this scale was derived by merging together several heptatonic scales into one, the name of a multi-headed beast is particularly fitting.
-There are analogues of hydra in other edos, too, obtained by fusing that edo's version of diatonic with its version of the rast MODMOS. The [[31edo|31edo]] version (5 5 3 1 4 5 4 1 3) could be called Nāga; its smallest interval is a diesis, which makes it blatantly microtonal in a way that hydra is not. The 24edo version (4 4 2 1 3 4 3 1 2) could be called [https://en.wikipedia.org/wiki/Xiangliu Xiangliu]. Both of these scales have four different step sizes, whereas hydra only has three.
+There are analogues of hydra in other edos, too, obtained by fusing that edo's version of diatonic with its version of the rast modmos. The [[31edo]] version (5 5 3 1 4 5 4 1 3) could be called Nāga; its smallest interval is a diesis, which makes it blatantly microtonal in a way that hydra is not. The 24edo version (4 4 2 1 3 4 3 1 2) could be called {{w|Xiangliu|Xiangliu}}. Both of these scales have four different step sizes, whereas hydra only has three.
 [[Category:17edo]]