Re: [tuning] Re: Part 3: Definition Question for Paul Erlich: "True" Interval

Jacky Ligon wrote,

> This leads to the following question: How far beyond the simple
integer ratios does this band of classification extend in light of the
reality of what the ear/neurological processing perceives? From what
I've been able to deduce from the Harmonic Entropy threads, the width
of this is different for each interval, with the gravitational pull of
the 3/2 being the most strong. So am I correct to assume that there is
a sort of gradient scale of tonal gravity for each interval in terms
of where it falls in integer size? If true, then the harmonic series
would provide the order of gravitational pull: 1/1, 2/1, 3/1, 4/1, 5/1
etc...right - wrong?

Hi Jacky,

Though Paul will certainly be the one most apt to address this most
clearly (as it's his baby), I would just like to say that if you look
at something like Helmholtz' intensity of influence chart, you can see
something very much like an octave reduced n*d (numerator times
denominator) ranking; as he gives a 9-odd limit ("partials are given
up to the 9th inclusive" p. 187) example of:

1/1 2/1 3/2 4/3 5/3 5/4 7/4 6/5 7/5 8/5 7/6 9/5 8/7 9/7 9/8

However, if you look at the depth of the troughs, or local minima in
Paul's various harmonic entropy graphs, you can see that they all
differ depending slightly on the input of N, and more so on the type
of series used, i.e., "s"; for instance, note the distinct differences
in trough depth between the Farey and the Mann series in these
examples:

<http://www.ixpres.com/interval/td/erlich/entropy.htm>

If I'm reading this all correctly, then I would think that Paul would
argue that it's the consistent occurrences of minima across the two
variables of N (given that you start with a high enough N in the first
place) and s that are most pertinent, rather than something like a
cleanly ordered trough depth... but I'll digress and leave the cleanup
job to Paul!

> if I map a 13/10 to "F", which I must do if I want to tune it to 454
cents - well I certainly don't hear this interval as a 4/3 - I hear it
as a very wide major 3rd.

Interesting, I actually tend to hear this more as some flavor of a
fourth... maybe an interesting listening experiment would be to take a
slate of some of Paul's various local dyadic maxima and have folks
offer there opinion of what those intervals most resemble -- maybe a
future project for Dr. Pehrson's Tuning Lab?

On a somewhat related note, I remember an old chart from Carl
Seashore's "Psychology of Music" that gives an "order of merit of
interval in the consonance-dissonance series" based on a empirical
listening classification method that combined three criteria: S, P,
and B -- smoothness, purity, and blending (see pp. 131 - 33 for more
details):

C'C''
C'G' (the chart actually says c'g'' here, but I can only imagine that
that must be a typo...)
C'A'
C'E'
C'F'
C'Ab'
C'Eb'
C'Gb'
C'Bb'
C'D'
C'B'
C'Db'

Hmm... Interestingly Seashore summarizes that, "The significance of
the above experiments lies not so much in the determination of
consonance-dissonance as in showing what the contributing factors are.
In general, we may say that consonance-dissonance depends primarily
upon two factors; namely, roughness-smoothness and purity-richness.
The factor of blending covers both of these and does not seem to add
any new element, but merely represents a point of view in the
judgement."

Dan