Reply to Joe Monzo

On Tue, 9 Mar 1999, Joseph L Monzo wrote:
> [Hahn:]
>> A 15/8 alone vs. a 7/4 alone?
>> You better believe the 7/4 is more
>> consonant!

Just for the record, that was Paul Erlich, not me, though I agree with
him.

> [Erlich:]
>> Plus, as Paul Hahn pointed out, the
>> "correct" lengths of the lines would
>> not be equal, but proportional to the
>> log of the odd limit.

I dunno if I would go so far as to say "correct" or "incorrect"--rather
that for certain purposes, an equidistant lattice is best, and for
others, a weighted/proportional one works better.

On Tue, 9 Mar 1999, Joseph L Monzo wrote:
> So it sounds like what you're saying is that
> the equidistant lines in the lattices we post
> here are just the result of ASCII limitations
> (again).

Not in my case. My "diameter" metric is a simple integer, and using a
proportional lattice for it is entirely unnecessary.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

>To use an overworked example, 225/224 would appear on
>the odd-lattice exactly the same as on the prime-lattice.
>(except for different angles, vector-lengths, etc.)
>A composite odd like 9 doesn't even enter the picture.

Not true -- as Paul Hahn recently discussed in regard to this particular
example. But your point is that 225/224 is a potential unison vector, or
"bridge", regardless of whether you use an odd-lattice or a prime-lattice.
That's quite true.

>So speaking of mutually exlcusive, this kind of
>discrepancy is why I was thinking, a few TDs ago,
>that our uses of the word 'wormhole' might be
>mutually exclusive.

Yes, as I said, your use of the word "bridge" and my use of the word
"wormhole" refer to two completely different things, though the terminology
might in fact be better suited if we switched.

One again, what I called a "wormhole" (because it's a "warp" in the
Euclidean fabric of the lattice) refers to the fact that certain
composite-based ratios (such as 9:5) are in fact simpler than distances on a
prime-based lattice would indicate. At least if you're talking about
octave-equivalent lattices. 9:5 is clearly not going to be used as a unison
vector, so it is not a "bridge" in your sense.

>I'm interested in studying the historical conceptions of
>various shapes and sizes of periodicity blocks in music
>all over the world. I believe that this 'history of finity
>in tuning' (which, I now realize, is what my book(s?) attempts
>to be) can enrich our knowledge of many other aspects of our
>lives and histories, especially ancient religious beliefs,
>possibly even extending to modern scientific theories about
>the universe.

The Hindu system of 22 srutis, in its common JI form is essentially a Fokker
periodicity block, but not one of the good ones I'm looking for. Since this
is a 5-limit system, we need two unison vectors. The first one is the
diaschisma, ratio 2048/2025 or 3^-4 * 5^-2, as we've discussed before,
there's evidence that the same pitch (sruti #2) functioned as 135/128 and as
16/15 (or in ma-grama, as 45/32 and as 64/45). That's about 19.6 cents. The
other unison vector is large: the difference between sruti #1 in the two
gramas, 3^-9 * 5 or 68.7 cents. So the Fokker matrix is

-5 3
-9 1

The inverse of this matrix is

1/22 -3/22
9/22 -5/22

If we transform the set of lattice points with this matrix, we can define a
periodicity block within any 1 X 1 square of the transformed lattice. The
usual approach is to use the "unit square" from the origin to (1,1), but of
course we are free to translate this square wherever we want since the
result will still tile the plane with the unison vectors as generators. So
let us choose the square where the position along each dimension is greater
than -1/2 and less than or equal to 1/2, so the origin will serve as a
central note rather than a corner. Taking the resulting 22 points

-5/11 1/11
-9/22 2/11
-4/11 3/11
-7/22 4/11
-3/11 5/11
-5/22 -5/11
-2/11 -4/11
-3/22 -3/11
-1/11 -2/11
-1/22 -1/11
0 0
1/22 1/11
1/11 2/11
3/22 3/11
2/11 4/11
5/22 5/11
3/11 -5/11
7/22 -4/11
4/11 -3/11
9/22 -2/11
5/11 -1/11
1/2 0

and transforming them back to the lattice (using the original Fokker
matrix), we get

-1 -1 112
0 -1 814
1 -1 316
2 -1 1018
3 -1 520
-5 0 90
-4 0 792
-3 0 294
-2 0 996
-1 0 498
0 0 0
1 0 702
2 0 204
3 0 906
4 0 408
5 0 1110
-3 1 680
-2 1 182
-1 1 884
0 1 386
1 1 1088
2 1 590

in lattice form:

680---182---884---386---1088--590
/ \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \
90---792---294---996---498----0----702---204---906---408---1110
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
112---814---316---1018--520

Now of course all the properties of the block are preserved if we transpose
one (or more) note(s) by one of the unison vectors used to create the block.
This corresponds to distorting the edges of the block in parallel as I've
discussed before. Let's take (-3,1) (680.5 cents) and lower it by the unison
vector (-9,1) (68.7 cents) to obtain the note (6,0) (612 cents):

182---884---386---1088--590
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
90---792---294---996---498----0----702---204---906---408---1110--612
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
112---814---316---1018--520

This is the scale of srutis according to most reputable sources, such as S.
Ramanathan, Mathieu (see his _Harmonic Experience), etc. etc.

Clearly the scales distinguish notes a syntonic comma (22 cents) apart,
while one of the unison vectors is over three times larger. So this isn't
one of the blocks I'm looking for.

By the way, this process of transposing notes by unison vectors can in
general affect the analysis of whether the block is of the 'good' type or
not, as can the translation of the unit square mentioned earlier. Also one
could consider the diagonals of the block as representing unisons as well as
the edges. So my quest is complicated beyond belief.

>Maybe it's
>the case that my mind refuses to accept it as anything other
>than JI, simply because I know that 64:75 is a 5-limit ratio
>and I know how it fits into the 5-limit lattice, and my
>foreknowledge affects my perception.

You better believe it! It's certainly not something you can _hear_.

>> As Dave Beardsley pointed out, an 8:13 (or even an 8:15?)
>> might be justly intoned, but you'd hardly call them
>> consonances.

>I would. I almost always treat both of these intervals
>as consonances in my JI pieces, and find that as chord
>members they help to produce beautiful chords that I
>think of as 'consonant'.

There they are parts of very stable otonal chords, but as dyads?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> http://www.egroups.com/message/tuning/15944
>
> [Dave Keenan:]
> >>
> >> As Dave Beardsley pointed out, an 8:13 (or even an 8:15?)
> >> might be justly intoned, but you'd hardly call them
> >> consonances.
>
> > [me, monz:]
> >
> > I would. I almost always treat both of these intervals
> > as consonances in my JI pieces, and find that as chord
> > members they help to produce beautiful chords that I
> > think of as 'consonant'.
>
> There they are parts of very stable otonal chords, but as dyads?

Good of you to point that out, Paul, because I suppose Dave K.
probably meant *as dyads*. I certainly was thinking in terms
of otonal chords, or just in more general terms such as
melodically, etc.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> > http://www.egroups.com/message/tuning/15944
> >
> > [Dave Keenan:]
> > >>
> > >> As Dave Beardsley pointed out, an 8:13 (or even an 8:15?)
> > >> might be justly intoned, but you'd hardly call them
> > >> consonances.
> >
> > > [me, monz:]
> > >
> > > I would. I almost always treat both of these intervals
> > > as consonances in my JI pieces, and find that as chord
> > > members they help to produce beautiful chords that I
> > > think of as 'consonant'.
> >
> > There they are parts of very stable otonal chords, but as dyads?
>
>
> Good of you to point that out, Paul, because I suppose Dave K.
> probably meant *as dyads*. I certainly was thinking in terms
> of otonal chords,

I did mean as dyads or I would have said otherwise. But when they
appear in chords, how could you tell if they were consonant, or merely
dissonances in consonant chords? i.e. how can you apportion consonance
among the components of a chord, except by listening to them
separately? But I perceive an otonal chord as more consonant than the
sum of its parts.

But one can certainly tell if a pitch is justly intoned in the context
of a chord, by varying it slightly either way and listening for beats.

> or just in more general terms such as
> melodically, etc.

I assume that ocurrence of "just" above means "simply" since it isn't
capitalised and is otherwise ambiguous. I really don't think there is
such a thing as melodic consonance. I _might_ be able to stretch the
concept of justness to pure melody (in which case the claimed ratios
need to be very simple and don't need any great accuracy), but melodic
consonance?

Regards,
-- Dave Keenan

--- In tuning@egroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/16006

> I did mean as dyads or I would have said otherwise. But when they
> appear in chords, how could you tell if they were consonant, or
merely dissonances in consonant chords? i.e. how can you apportion
consonance among the components of a chord, except by listening to
them separately? But I perceive an otonal chord as more consonant
than the sum of its parts.
>

I believe, Paul correct me if I'm wrong, that this was shown nicely
in Erlich's recent Harmonic Entropy experiment, where the consonant
rankings of separate diads SUMMED for the tetrads he considered
ranked UTONAL chords pretty nicely according to
concordance/discordance, but failed pretty miserably in ranking the
OTONAL chords, where there was a different kind of combined energy
going on...

________ ___ __ __
Joseph Pehrson

Dave Keenan wrote,

>I did mean as dyads or I would have said otherwise. But when they
>appear in chords, how could you tell if they were consonant, or merely
>dissonances in consonant chords? i.e. how can you apportion consonance
>among the components of a chord, except by listening to them
>separately? But I perceive an otonal chord as more consonant than the
>sum of its parts.

>But one can certainly tell if a pitch is justly intoned in the context
>of a chord, by varying it slightly either way and listening for beats.

Dave, it appears you answered your own question here! I don't doubt that if
you had a big fat otonal 8:9:10:11:12:13:14:15 chord, you could hear the
"beats" from detuning 13 or 15 -- but these would be beats involving
combination tones and/or the virtual pitch, rather than overtones. Remember
our tetrad listening experiments and all the "beats" that were heard?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave, it appears you answered your own question here! I don't doubt
that if
> you had a big fat otonal 8:9:10:11:12:13:14:15 chord, you could hear
the
> "beats" from detuning 13 or 15 -- but these would be beats involving
> combination tones and/or the virtual pitch, rather than overtones.
Remember
> our tetrad listening experiments and all the "beats" that were
heard?

Yes. But Paul, how does that tell you how to apportion consonance
among the intervals. You are simultaneously damaging many intervals by
detuning one note. In the case of the 15 at least, the damage to 10:15
= 2:3 may be more significant than 8:15 or the others.

Dave Keenan wrote,

>Yes. But Paul, how does that tell you how to apportion consonance
>among the intervals.

You can't -- the effects are synergistic (nonlinear).