n+d

[Keenan]
>Nah. Why? His whole point was that the dissonance was proportional to d
>WHEN THE FRACTION IS IN LOWEST TERMS. This is also true for n+d.
>
>What we are suggesting here is that the dissonance is related to the period
>(inverse of frequency) of the virtual fundamental (VF). When the ratio for
>an interval is in lowest terms, the VF corresponds to a 1.

Right, this makes sense.

[Keenan]
>If instead we fix the average frequency of the two notes at 450Hz. The
>notes of a 3:2 will be 360Hz and 540Hz and the VF will be 180Hz (period
>5.56ms), while the notes of a 5:4 will be 400Hz and 500Hz and the VF will
>be 100Hz (period 10ms). Periods are proportional to n+d or (n+d)/2.

This makes sense too. I wonder what Paul E's derrivation is. BTW, my idea
behind n*d is to measure the period of the VF in terms of the periods of
the fundamentals of the notes in the dyad, rather than the clock time.

[Keenan]
>If we want to talk about the realtive dissonance of intervals (or chords
>for that metter) without considering frequency, then the only sensible
>thing to do is to compare them all with the same average frequency, hence
>n+d or (n+d)/2 is the right approximation for dyads, not d.

Sticking to dyads just for a moment, I'm not sure how considering the fixed
frequency to be the average of n and d improves anything over assuming
either n or d to be fixed. That is, if we need to specify frequency, don't
we need to specify frequency?

[Erlich]
>>How so? One dimension for every odd factor. Or as you say, the (odd
>>limit minus one) over two.
>
>Yes, but the odd limit would have to be specified in advance.

When not? Harmonic limit seems such a fundamental part of how we hear
music, and of what music in turn does to oblige.

[Erlich]
>>Only up to the point you've declared consonant! The problem with your
>>algorithm is it has no way to measure intervals outside of those you
>>declare consonant.
>
>Well, if you are allowed to declare an odd limit for consonance, it
>_can_ measure intervals that are composites of consonant intervals
>(using the lattice)

But not always by the shortest route, as Paul Hahn showed.

[Erlich]
>>Hence (n+d)/2. Is this an incorrect generalization for dyads?
>
>I'm still not following you.

Uh...

[Erlich]
>Well, if you weight by prime or odd limit like you said, then the 2 will
>never come into the weighting of any intervals

2 is prime.

C.