Re: [harmonic_entropy] note-naming grid

> ----- Original Message -----
> From: D.Stearns <STEARNS@...>
> To: <harmonic_entropy@yahoogroups.com>
> Sent: Tuesday, June 12, 2001 10:47 PM
> Subject: [harmonic_entropy] note-naming grid
>

> Hey Paul,
>
> Something I've been interested in every now and again is some
> reasonable way to lay a theoretical note-naming grid on top of the
> pitch continuum. As an example of what I'm trying to get at here I'd
> say that you could call twelve equal where all pitches are rounded to
> the nearest twelfth of an octave a "crude" example of this. So on this
> grid an 11/8 is F# (or Gb or any other enharmonic equivalence of
> twelve), and 7/4 is Bb, etc.
>
> What I'm interested in is a more "sophisticated" note-naming grid
> based on the results of harmonic entropy. Say where the base of valley
> represent an alphabetized note name's center, and the slopes its
> range... something along this line.
>
> Any ideas?

Hi Dan. I found this interesting because, except for the
reference to Paul Erlich's Harmonic Entropy specifcially
(I had my own vague idea of the same concept in mind), it
is very much what I was striving for about 10 years ago,
which felt satisfied when I found prime-factorization.

So that's not really an example of note *naming*, since it's
numbers and basic math. But using a vector notation for
the exponents of the prime series makes it at the same time
both simple and sophisticated.

Your two examples are:

2 3 5 7 11

| -3 0 0 0 1| = 11/8 = Gb / F# (or one might even use F)
| -2 0 0 1 0| = 7/4 = Bb

which in my 72-EDO would be F^ / F#v / Gbv and Bb< , and in the
version everyone else is using I think that's F] / F#[ / Gb[ and
Bb< .

I don't know... maybe this doesn't have much relevance as a
response to your question. But the prime-factor notation in
my mind ties into the valleys on the harmonic entropy graph.

I'd appreciate further comments on this.

-monz
http://www.monz.org
"All roads lead to n^0"

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<< is an 11/9 a major or a minor third? >>

The term "third" is a scale degree (III == 2) and may not be well
attributed to an isolated ratio. It depends of context.

I leave others to attribute terms as "major", "minor" or "neutral" but I
say that these terms correspond to a numerical property (having sense in
rational space only).

Considering P(X) the power of 2 in the ratio X, for instance,

P(5/4) = -2
P(9/8) = -3
P(16/15) = 4
P(24/19) = 3

a major interval is simplifying in upper octave until it reach its "pivot"

5/4 -> 5/2 -> 5/1
9/8 -> 9/4 -> 9/2 -> 9/1

while a minor one is simplifying in lower octave until it reach its "pivot"

1/15 <- 2/15 <- 4/15 <- 8/15 <- 16/15
3/19 <- 6/19 <- 12/19 <- 24/19

As you may see, between 5/4 and 24/19 there is only few cents (96/95) but
one is major (-2) an the other minor (3).

11/9 is a "pivot", so neither major nor minor.

Pierre

>Something I've been interested in every now and again is some
>reasonable way to lay a theoretical note-naming grid on top of the
>pitch continuum. As an example of what I'm trying to get at here I'd
>say that you could call twelve equal where all pitches are rounded to
>the nearest twelfth of an octave a "crude" example of this. So on this
>grid an 11/8 is F# (or Gb or any other enharmonic equivalence of
>twelve), and 7/4 is Bb, etc.

>What I'm interested in is a more "sophisticated" note-naming grid
>based on the results of harmonic entropy. Say where the base of valley
>represent an alphabetized note name's center, and the slopes its
>range... something along this line.

>Any ideas?

I don't think this idea will work because harmonic entropy involves
_intervals_ -- incommensurate intervals, if you're looking at the base of
each valley -- while note names involve _pitches_. So in your example, if
5:4 were represented by a 12-tET major third in terms of naming, then you
immediately have a problem in that F# to Bb is 14:11 instead of 5:4 . . .
see what I'm getting at?

The only way I could see this kind of note-naming being useful is in a
system of music where a fixed 1/1 is played as a constant drone -- only then
would the set of local minima of harmonic entropy "mean something" as a
pitch set.

>Hi Joe,

>What I had in mind was something like this: Given any arbitrary
>interval what would be a nice way to go about naming it. So in other
>words I want to ask a model simple questions based on shoehorning any
>given interval into traditional intervallic relationships -- is an
>11/9 a major or a minor third? From there it's on to things like how
>many gradations or shades are sufficiently fine in a theoretical
>sense -- is 11/9 neither a major nor a minor third?

>Does this make any better sense now?

>--Dan Stearns

One reasonable proposal along these lines is based on a 31-tET grid:

http://www.uq.net.au/~zzdkeena/Music/IntervalNaming.htm

Of course, if you're using a non-meantone temperament, this scheme will have
problems.

>Hi Paul,

>I was just assuming the idea of a fixed 1/1. So with that aside, what
>would you see the harmonic entropy graphs possibly spelling along the
>lines of this note-naming grid idea?

Just use the ratios themselves!

>I'll take a look at the 31 link, but my hope was for something that
>would cast larger shadows were you'd expect them -- near lower n*d
>zones -- and tighter bands were you'd expect them. The trick I think
>would be to strike some sort of a balance between too fine and too
>crude. This would probably require some fudging at the abutments of
>the stronger consonance areas.

We kind of went through this a while back . . . remember your "poor man's
harmonic entropy" which turned out to really be a poor man's van Eck model?
Shall we go through that again?

>Did it just get a bit frosty in here or is that just me!

Brrrr . . . sorry for the curt replies . . . just trying to get through
umpteen lists in a short time . . .

Hello Dan,

The terms major and minor are used about modes and about chords. It is
surely hard to share my opinion about the modality notion -- a measure in
the gradation minor/major -- if the terminology of chords is the reference.

The terms major and minor applied to the triad had a sense (linked to
harmonics and subs) as a modality of relation for three elements. However,
the terminology has evolved from the model of third superposition. In
complex chord, major or minor is now only qualifying the first third rather
than characterizing the type of relation of all the elements in the chord.

I don't use the modality notion in the theory itself but in periphery, for
instance in the vectorial representation of the modes with lattices (where
the lines are the steps).

I'm not worry about musical terms. It's a matter of musicians. I'm worry
about coherence in models. I can say only "look at these images"

<http://www.aei.ca/~plamothe/pix/canonic.gif>
<http://www.aei.ca/~plamothe/pix/majmin1.gif>

and if you are interested to dig about the modality notion I could explain
more.

-----

About "pivot".

There exist two current systems representing the interval classes modulo 2.
I give only examples with intervals

8/45
9/4
24/5
5/32

One corresponds to equivalent tone in the first octave, what I name "ton"

64/45
9/8
6/5
5/4

The other corresponds to equivalent interval where power of 2 is zero, what
I name "pivot" (for all other equivalent intervals have a higher complexity)

1/45
9/1
3/5
5/1

Pierre

What the heck is this discussion doing in harmonic_entropy?