Re: [tuning math] Re: everyone concerned

Hello, there, everyone, and how one analyzes 3:4:6 depends a great
deal upon what period or style of Western European or other theory one
is focusing on.

To me, 3:4:6 is a variation on the more conclusive 2:3:4, with the
latter arrangement (outer 2:4 octave, 2:3 fifth below, 3:4 fourth
above) as a complete stable sonority in 13th-14th century music of
Western Europe.

As one theorist around 1300 explains, in 2:3:4 the intervals follow a
"natural series" of numbers 2-3-4, with the 2:3 fifth "preceding" the
more complex 3:4 fourth.

Interestingly, this passage was written about three centuries before
the discovery of the harmonic series, or of the way a vibrating string
in effect subdivides itself to produce partials such as the 2:1 octave
and 3:1 twelfth (fifth plus octave).

In the era around 1300, a theorist named Johannes de Grocheio
describes a complete 2:3:4 sonority as manifesting _trina harmoniae
perfectio_, the "threefold perfection of harmony," since it includes
all three simple stable consonances: the 2:1 octave, 3:2 fifth, and
4:3 fourth.

This phrase leads to the modern English term "trine" for a complete
2:3:4 sonority; in Gothic and neo-medieval styles, it defines the
standard of full stable harmony, and the ideal resolution for unstable
sonorities.

Theorists around 1300 also note that 2:3:4 is smoother than 3:4:6,
with the 3:4 fourth below the 4:6 or 2:3 fourth. Either is regarded by
Jacobus of Liege (c. 1325) as perfectly concordant, but with the first
form generally preferable.

This preference, however, may vary: around 1030, Guido d'Arezzo in his
_Micrologus_ prefers the 3:4 fourth to the 2:3 fifth as a basic
interval, and advocates the placement of the fourth below the fifth in
a technique of three-voice singing which involves a series of 3:4:6
sonorities.

If we want a name for the 3:4:6 reflecting later stylistic trends
around the 13th century, we could call it a "converse trine," since
the intervals are said to be arranged "conversely" (_e converso_) from
the more conclusive 2:3:4 sonority with the fifth below the fourth.

A standard Gothic Pythagorean tuning, generally assumed in medieval
European theory (with a few interesting variations), makes both the
2:3:4 and its "conversity" of 3:4:6 pure.

In neo-medieval music, however, the fifths are often tuned somewhat
wider than pure, and the fourths somewhat narrower, with about 2 cents
of tempering standard.

It should be noted that in the 18th-19th century styles discussed by
Jon Szanto, a complete 2:3:4 or 3:4:6 trine of medieval or
neo-medieval styles would be considered some kind of incomplete
sonority, for example a portion of a 4:5:6 triad. Similarly, in the
21st-century decatonic harmony of Paul Erlich based on a complete
4:5:6:7 tetrad, the complete 4:5:6 triad of 18th-century music would
be considered an incomplete portion of a tetrad.

Most appreciatively,

Margo Schulter
mschulter@value.net