Marvelous dwarves: Difference between revisions
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A marvelous dwarf is a scale with the following attributes: | |||
(1) It is a [[Marvel_family|marvel]] tempering of a 5-limit [[Dwarves|dwarf]]. | |||
(2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads. | (2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads. | ||
(3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads. | (3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads. | ||
(4) It has more 5-limit triads than pentads. | (4) It has more 5-limit triads than pentads. | ||
(5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224. | (5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224. | ||
If every condition but the third--the covering condition--is | If every condition but the third--the covering condition--is | ||
satisfied, I'm calling it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage we've discussed before as genus(15^4), is particularly striking from the point of view of the quantity of pentads it supplies. | satisfied, I'm calling it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage we've discussed before as genus(15^4), is particularly striking from the point of view of the quantity of pentads it supplies. | ||
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<11 17 26| 1 6-6 11 semimarvelous | <11 17 26| 1 6-6 11 semimarvelous | ||
<12 19 28| 2 6-6 6 marvelous | <12 19 28| 2 6-6 6 marvelous | ||
<13 20 30| 1 7-6 13 semimarvelous | <13 20 30| 1 7-6 13 semimarvelous | ||
<14 22 33| 2 7-6 7 semimarvelous | <14 22 33| 2 7-6 7 semimarvelous | ||
<15 24 35| 3 8-8 5 marvelous | <15 24 35| 3 8-8 5 marvelous | ||
<16 25 37| 2 7-6 8 semimarvelous | <16 25 37| 2 7-6 8 semimarvelous | ||
<17 27 40| 4 10-9 4.25 semimarvelous | <17 27 40| 4 10-9 4.25 semimarvelous | ||
<18 29 42| 4 10-10 4.5 marvelous | <18 29 42| 4 10-10 4.5 marvelous | ||
<19 30 44| 5 12-11 3.8 marvelous | <19 30 44| 5 12-11 3.8 marvelous | ||
<20 32 47| 6 12-12 3.333 marvelous | <20 32 47| 6 12-12 3.333 marvelous | ||
<21 33 49| 5 12-12 4.2 marvelous | <21 33 49| 5 12-12 4.2 marvelous | ||
<22 35 51| 6 14-13 3.667 semimarvelous | <22 35 51| 6 14-13 3.667 semimarvelous | ||
<25 40 58| 9 16-16 2.778 marvelous | |||
Revision as of 00:00, 17 July 2018
A marvelous dwarf is a scale with the following attributes:
(1) It is a marvel tempering of a 5-limit dwarf.
(2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads.
(3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads.
(4) It has more 5-limit triads than pentads.
(5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224.
If every condition but the third--the covering condition--is
satisfied, I'm calling it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage we've discussed before as genus(15^4), is particularly striking from the point of view of the quantity of pentads it supplies.
There is a marvelous or semimarvelous dwarf for each scale size from 11 to 22, and then the 25 note scale. So far as I know this is the complete list.
Here is a brief description; the numbers are pentad number, numbers of major-minor triads, and size/pentad ratio.
<11 17 26| 1 6-6 11 semimarvelous
<12 19 28| 2 6-6 6 marvelous
<13 20 30| 1 7-6 13 semimarvelous
<14 22 33| 2 7-6 7 semimarvelous
<15 24 35| 3 8-8 5 marvelous
<16 25 37| 2 7-6 8 semimarvelous
<17 27 40| 4 10-9 4.25 semimarvelous
<18 29 42| 4 10-10 4.5 marvelous
<19 30 44| 5 12-11 3.8 marvelous
<20 32 47| 6 12-12 3.333 marvelous
<21 33 49| 5 12-12 4.2 marvelous
<22 35 51| 6 14-13 3.667 semimarvelous
<25 40 58| 9 16-16 2.778 marvelous