Marvelous dwarves
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Original Wikitext content:
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. (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
Original HTML content:
<html><head><title>Marvelous dwarves</title></head><body>A marvelous dwarf is a scale with the following attributes:<br /> <br /> (1) It is a <a class="wiki_link" href="/Marvel%20family">marvel</a> tempering of a 5-limit <a class="wiki_link" href="/Dwarves">dwarf</a>.<br /> (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.<br /> (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.<br /> (4) It has more 5-limit triads than pentads.<br /> (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.<br /> <br /> If every condition but the third--the covering condition--is<br /> 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.<br /> <br /> 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. <br /> <br /> Here is a brief description; the numbers are pentad number, numbers of major-minor triads, and size/pentad ratio.<br /> <br /> <11 17 26| 1 6-6 11 semimarvelous<br /> <12 19 28| 2 6-6 6 marvelous<br /> <13 20 30| 1 7-6 13 semimarvelous<br /> <14 22 33| 2 7-6 7 semimarvelous<br /> <15 24 35| 3 8-8 5 marvelous<br /> <16 25 37| 2 7-6 8 semimarvelous<br /> <17 27 40| 4 10-9 4.25 semimarvelous<br /> <18 29 42| 4 10-10 4.5 marvelous<br /> <19 30 44| 5 12-11 3.8 marvelous<br /> <20 32 47| 6 12-12 3.333 marvelous<br /> <21 33 49| 5 12-12 4.2 marvelous<br /> <22 35 51| 6 14-13 3.667 semimarvelous<br /> <25 40 58| 9 16-16 2.778 marvelous</body></html>