Marvelous dwarves

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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