Yahoo Tuning Groups Ultimate Backup tuning-math Marvelous dwarves

A marvelous dwarf is my name for 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. I've uploaded Scala files to the files section:

/tuning-math/files/marvdwarf/

Here is a brief description; the numbers are pentad number, numbers of
major-minor triads, and size/pentad ratio.

Marvelous dwarves

🔗Gene Ward Smith <gwsmith@svpal.org>