Mathematical explanation of consonance? - Simple!

Charles Lucy wrote,

>> Yes, Robin. It seems to be determined by the number of steps between
the
>> pitches on the "spiral" of fourths and fifths.

Gary Morrison wrote,

> (Predictable...)

> Well Mr. Lucy, I am at a lack to think of *ANY* concept of
consonance in
>which a harmonic major second (two steps up the circle of fifths),
would be
>more consonant than a harmonic major third (four steps up the circle).

Of course Gary is right, especially in the context of tempered fifths,
which applies to 12-tET, LucyTuning, and meantones. In addition, in just
intonation, the "major third" formed by eight fourths is more
acoustically consonant than that formed by four fifths, since the former
is only 2 cents off a just 5:4 while the latter is 21.5 cents off.
However, musical context would still have to determine whether the
acoustical consonance could translate into musical consonance.

An obvious problem with Lucy's definition is that you can get
arbitrarily close to any interval if you are allowed an unlimited number
of fifths (whether these are just fifths, Lucy fifths, or any other
irrational fraction of an octave). So there will be very large numbers
of fifths which result in very consonant intervals. For example, 666
just fifths lead to a frequency ratio of 1.50007, or 702.03 cents. Even
musical context would not help to distinguish between this and 1 just
fifth, which is a frequency ratio of 1.5 or 701.96 cents. Lucy's own
tempered fifth is 695.49 cents, while 52 just fourths produce an
interval of 698.34 cents, and 89 Lucy fifths produce an interval of
698.87 cents, both clearly more consonant.