Octaves not exactly 2^N

While reading through the backlog of messages that built up while I was
overseas for 3 weeks, I saw one about tunings with octaves not being
exactly powers of 2. In most cases of variation it seems that values
greater (rather than less) than 2 are found in practice. The reasons
may relate to overtones being not exact harmonics of the fundamental. I
would like to raise a theoretical reason for an irregular octave to see
whether anyone has practical experience of such a thing.

With just tunings, the whole idea is to have as many simple ratios
between frequencies as possible. For a range of notes this is achieved
by having frequencies which are some fundamental times 2^A*3^B*5^C*7*D
with A, B, C and D all integers with ranges of perhaps 8, 3, 1 and 0 for
a typical western scale, or maybe 12, 5, 2 and 1 for more interesting
possibilities.

[Note: the use of 12^2 does not mean that a ratio of >4096 in frequency
must be used, as the powers of the other factors may be an offsetting
effect. e.g. in the JI scale of 24 27 30 32 36 40 45 48, there is a
variation of 5 powers of 2 although only one octave is spanned.]

It so happens that when 8 octaves are spanned, a greater degree of
strong relationships exist when a ratio of 252 is used than when 256 is
used. It may be argued that 8 octaves are not normally spanned, however
if the implied tuning of chord fundamentals and rhythm equivalents are
included at the low end, and the overtones of notes at the high end,
then the situation will be quite normal to span over 8 octaves.

So how do I measure this "greater degree of strong relationships"?
I use the the number of ways in which a number can be factorised, and
whereas 64 and 128 can be factorised in more unique ways than 63 and
126, the reverse applies for 256 and 252.

It is possible to make some special chords that tend to highlight this
"need" for a reduced octave. Notes with relative frequencies of say
4:5:6:7:8:14:21:42:63 demonstrate the disparity between the 4 & 8 and
the 63. This is a somewhat forced example to put the whole argument
into a single chord. You can fiddle around from there to get the idea
if you want.

I am interested in whether anyone else has thought about such a note as
this reduced tonic, or felt a need for it in their music? My reasons
are purely theoretical.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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On Tue, 17 Jun 1997, Paul Erlich wrote:
> I wonder if there are any 7-limit analogues to these chords (i.e., chords in
> which each interval is within the 7-limit but the chord as a whole is in a
> higher limit). I think the answer is no. Anyone care to come up with a
> counterexample?

Now that I understand the question, it's easy to find a counterexample, or
prove a slightly stronger version of the statement.

To demonstrate this I will use ratio space axes. As I still don't know of a
standard notation, I'll have to explain and use mine.

As an example, here's a chord:

(0 0 0 1) (0 0 0 0) (0 1 1 1)
(0 -1 0 1) (0 1 0 0) (0 0 1 1)
(0 0 -1 1)H (0 0 0 1)H - (0 0 1 0)H (0 -1 -1 0)H + (0 1 0 1)H
(0 0 0 0) (0 0 0 1) (0 1 1 0)

In each matrix, each row is a note and each column corresponds to a prime
factor. The matrix on the right is the otonal version. In ratio space
terms, this is defined so that the lowest number in each column is zero.
This chord is then revealed to be 15:21:35:105.

This is a 105-limit otonality, but a 7-limit utonality (the middle
representation). The intervals are all 7-limit or below. You can tell this
because each interval involves adding a factor in one column and subtracting
one from another. The intervals are, in fact, 5/7, 5/3, 3, 7/3, 5 and 7.

The following assumes octave equivalence -- i.e. you ignore the first
column. It also assumes there are only four columns in total (true for
all 7-limit intervals).

If you take the lower of the otonal and utonal limits, though, this can't
be done. In a 7-limit interval, one column can go up by one, and another
column can go down by one. If any note in the otonal representation has a
number in any column that is not 0 or 1, there must be a >7-limit interval.
If any note has more than one 1 in it, and another has more than one 0 in it,
the interval between these notes must be >7-limit. Hence a chord with only
must be a subset of either the chord above or its otonal analog (1:3:5:7).

This only is true because 3*3>7. It doesn't apply, therefore, to higher
prime limits where the otonal matrix needn't be binary.

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Thanks, Paul, for thanking me for my proof from before. It's
nice to know my work's being appreciated. It's possible that
this message will arrive before the other one I'm sending
today. Read that one first, or this one will make even less
sense!

The chords under discussion are: 12:15:18:20 and 14:18:21:24.
In otonal, non-reduced matrix form, these are:

(2 0 1 0) (3 1 0 0)
(1 2 0 0) (0 1 0 1)
(0 1 1 0)H and (1 2 0 0)H
(2 1 0 0) (1 0 0 1)

They occupy the ROHSs (2 2 1 0)H and (3 2 0 1)H respectively.
The latter chord, which is the larger odd-limit chord, also
occupies the ROHS corresponding to the larger LCM. It doesn't
matter what the LCMs are: just note that 2*7<5. Neither is
a complete filling of the LCM, though, as neither contains
the root. This is, incidentally, a criterion for the maximum
limit of the intervals being less than the limit of the
chord.

The next step is to look at the octave-reduced otonal
versions of the chords. They are:

(-3 1 1 0) (-4 1 0 1)
(-3 2 0 0) (-3 2 0 0)
(-2 0 1 0)H and (-2 0 0 1)H
(-1 1 0 0) (-1 1 0 0)

The two chords look very similar in this form. Are they
complete? The first is clearly a subset of the (x 2 1)H
otonality, or the Euler genus 3.3.5. Here it is complete:

(-5 2 1)
(-3 1 1)
(-3 2 0)
(-2 0 1)H
(-1 1 0)
( 0 0 0)

The chord being considered is a section of this, and
therefore a complete octave reduced (2 2 1)H chord, although
not the utonal or otonal one.

The octave reduced genus 3.3.7 is:

(-5 2 0 1)
(-4 1 0 1)
(-3 2 0 0)
(-2 0 0 1)H
(-1 1 0 0)
( 0 0 0 0)

It can now be seen that the second chord is the complete
octave reduced (3 2 0 1)H chord above the root (-1 1 0 0)H.

I don't know the significance of all this. I worked out
most of it as I was writing. It does look interesting,
though. I think the reason that both chords are above the
root (-1 1 0 0)H, or 3/2, is something to do with the
compositeness of 9.

Graham

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I've got this almost completely sorted out now, and I don't even
need matrices to explain it. If I'm repeating anyone here, it's
because I thought a different approach would be helpful. I'm not
behind on the list or anything like that.

All the numbers we'll be meeting today will be odd, so I won't say
so every time. Numbers with more than one prime factor have a
special significance, so I'll call them "rhubarb numbers" for want
of knowing a better term.

First choose a number m. Either n or m, or both, must be
composite. mProviding neither m nor m are rhubarb, a saturated n-limit chord
can be constructed form all the divisors of n*m except n*m and 1.
This chord will be it's own utonal analog.

Examples are the two 9-limit saturated chords already found.
Also, the 11-limit s-chord 33:11:9:3, the 25-limit s-chord
75:45:25:15:9:5:3 and the 35-limit s-chord
105:63:45:35:21:15:9:7:5:3.

When mj is rhubarb, we have a problem. The intervals n*i/j and
n*j/i will be beyond the n-limit. The same problem occurs when
nj is rhubarb unless i>m and j>m, so that i*mThis problem complicates matters, but is not insuperable as I
shall demonstrate with the example n, m

This gives us the chord 35:21:15:7:5:3. The intervals 35/3 and
21/5 are >15-limit. Therefore [21 or 5] and [35 or 3] must be
removed. If 35 and 21 are removed, we have a 15-limit otonality.
Removing 5 and 3 gives its utonal analog. The choice is then
removing [21 and 3] or [35 and 5]. The resulting chords 15-limit
s-chords 35:15:7:5 and 21:15:7:3 are then otonal/utonal analogs.

I otonal/utonal pairs will arise whenever the rhubarb prooblem
occurs. I haven't looked at any cases where n and m are both
rhubarb. The first of these is n5, m3. Doubly rhubarb
numbers can safely be ignored until someone starts writing 105-
limit music.

That should be all you need to know.

Graham

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