reply to Joe Monzo

OK, you went back to Euclidean distances, got some weird results in the ordering on your octave-equivalent lattice, and some weirder results in the octave-specific lattice. Meanwhile, you continually make arguments based on the city-block metric. You should look at the city-block metric in the depth that you looked at the Euclidean metric and see if that doesn't serve your purposes better.

With lengths of p rather that log(p), you are indeed giving odd composites a lower complexity than their odd-limits suggest. I am opposed to that because I see no evidence that consonance works this way. In a full musical context, composites will most likely come into play more than higher primes. But that is by means of chains of connections, which the lattice can already display quite nicely. I prefer the lattice to be constructed devoid of any considerations of musical context, since those can always vary. Without context, I see (and more importantly, hear) no reason to consider composite simpler than primes with similar-sized numbers.

>> 3rd axis: steps of 7:4 have length log(7)
>> the angle between the the first and third axes >is determined by
>> the condition that steps of 7:5 have length >log (7)
>> the angle between the second and third axes is >determined by the
>> condition that steps of 7:6 have length log (7)

>This isn't logical to me.� You're using the >*limit* prime

Actually, odd

>of each ratio to determine the vector, whereas >my lattice
>formula *adds the complexity for each prime* >found as a factor.

>So in your formula, it doesn't matter whether 7 >is combined
>with 2, 3, or 5 - they all have a metric based >on 7.

>In my formula, 7:4, 7:6, and 7:5 are all >different, each metric
>longer than the one before it, based on all >their prime factors
>(numbers inside the table are the exponents of >the prime factors
>on the top line):

>�������������� 2� 3� 5� 7
>����� 7:4 = | -2� 0� 0� 1|
>����� 7:6 = | -1 -1� 0� 1|
>����� 7:5 = |� 0� 0 -1� 1|

Joe, we're talking about octave-equivalent lattices, right?

>This reflects well my feeling that 7:4 is less >complex than both
>7:6 and 7:5, that 7:6 is more complex than 7:4 >but less complex
>than 7:5, and 7:5 more complex than both 7:4 and >7:6.

That only seems to be true if you ignore factors of two. In an octave-specific lattice, the ratios of 2 contribute to the complexity of the ratios.

>(Of course,
>in this simple example an integer-limit would do >the same.)

How so? Naively, they all have an integer-limit of 7.

Being a little more clever, the octave-specific ratios that could be octave-equivalent to the ratios in question include

8:7
7:6
7:5
10:7
12:7
7:4
16:7
7:3

etc. Ranking these in order of complexity, just by the size of the numbers (integer limit, denominator, sum, or product), certainly doesn't agree with your prime-based complexity argument. Since the ranking doesn't clearly place the octave-equivalents of 7:4 higher than the octave-equiavalents of 7:5 or of 7:6, I prefer to use the same length for all three.

>This is somewhat contradicted
>by Ptolemy (c. 150 AD) who made accurate measurements of the intervals in
>all types of scales *actually in use*

Accurate measurements? Actually, I think the original Pentium chip came out
in 150 AD, so his measurements may have been afflicted by that chip's
division bug.

Come on, Joe, what kind of accurate measurements could Ptolemy have made?

P.S. Didn't Johnny Reinhard tell you that diatonic music in Europe derived
not from the Greeks but from another source (related to the Huns)? According
to him, there was a definite discontinuity between the civilizations of
classical antiquity and Christian Europe. Only later did scholars try to tie
the two together, and apparently they were so succesful that many today
suppose that one civilzation grew out of the other.