high-prime scales

There have been *lots* of scales with
primes much higher than 19 described
in the tuning literature over the centuries.
"Just" a few examples, all very important
in the history of music theory:

THEORIST PRIME LIMIT
Ptolemy 31
Dowland 211
Boethius 499

These and lots more are investigated
in my book. Another idea I discuss is my
belief that 13 can be (and has been)
viewed as a kind of limit in harmonic
usage, with higher primes generally being
used as alternate choices for the lower
primes nearby - for example, 17 and 19
are both usually used instead of, and
not together with, 9 or each other; 15 is
generally used instead of, and not together
with, 7.

- Monzo
http://www.ixpres.com/interval/monzo/homepage.html

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Mr. Monzo
I could not agree more that that Ptolemy scale using the 31 is essential to
light those chambers inside the Pyramids.

Joseph L Monzo wrote:

> From: Joseph L Monzo <monz@juno.com>
>
> There have been *lots* of scales with
> primes much higher than 19 described
> in the tuning literature over the centuries.
> "Just" a few examples, all very important
> in the history of music theory:
>
> THEORIST PRIME LIMIT
> Ptolemy 31
> Dowland 211
> Boethius 499
>
> These and lots more are investigated
> in my book. Another idea I discuss is my
> belief that 13 can be (and has been)
> viewed as a kind of limit in harmonic
> usage, with higher primes generally being
> used as alternate choices for the lower
> primes nearby - for example, 17 and 19
> are both usually used instead of, and
> not together with, 9 or each other; 15 is
> generally used instead of, and not together
> with, 7.
>
> - Monzo
> http://www.ixpres.com/interval/monzo/homepage.html
>
> ___________________________________________________________________
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Joseph L Monzo <monz@juno.com> wrote:

>There have been *lots* of scales with
>primes much higher than 19 described
>in the tuning literature over the centuries.
>"Just" a few examples, all very important
>in the history of music theory:
>
>THEORIST PRIME LIMIT
>Ptolemy 31
>Dowland 211
>Boethius 499
>
>These and lots more are investigated
>in my book.

Thanks Joe. Do you really think that their musical uniqueness (assuming they have any, pardon my ignorance) *depends* on these large primes. Or might it be that it was merely convenient to express the scales using these numbers. Maybe logarithms had't been invented?

>Another idea I discuss is my
>belief that 13 can be (and has been)
>viewed as a kind of limit in harmonic
>usage, with higher primes generally being
>used as alternate choices for the lower
>primes nearby - for example, 17 and 19
>are both usually used instead of, and
>not together with, 9 or each other; 15 is
>generally used instead of, and not together
>with, 7.

Ah, yes. This certainly accords with my beliefs. So in a sense, this says to me, for example, that 9/8 can usually be substituted for 17/15 or 19/17 (about 9 cents away). But I am willing to leave the limit at 19 (pending Kraig Grady's response re 23) and say for example that 11/10 can be substituted for 21/19 or 23/21 (about 8c away). Note that around middle C such a deviation corresponds to about a 1 Hz beat, which one would not expect to notice unless the chord were sustained for more than a second (and there was no deliberate vibrato etc. etc.).

So I think that it is more meaningful in most contexts to refer to a 21/19 as "11/10 +8c" or "10/9 -9c".

In another thread there was discussion of whether something was a 45/32. Now this may be significant in its relationship with other notes of the scale but when played with the 1/1 it is surely more significant that this is 7/5 +7.7c.

Maybe there's an ideological/political issue here. If I may be permitted a caricature: "A true JI-ist must never talk in cents or admit that anything is 'a usable approximation' of anything else. JI is ratios, zero beats, and that's that!"

I believe that such a position would fly in the face of the facts.

I understand there is a kind of "uncertainty principle" built into Fourier (or Laplace or wavelet) analysis and presumably also into whatever the ear+brain does. You can't give a definite frequency to any wave that you only sample for a finite duration. The shorter it lasts, the greater the uncertainty in its frequency. delta_f = 1/delta_t. So if a chord only lasts for a second it's impossible to say whether it had zero beats. It might have had a beat slower than one per second and you wouldn't (couldn't?) notice it.

For example, Jazz appears to me to use 12-tET as if it were approximating ratios of 7 (among others). 12-tET is very poor at this (up to 31 cent errors) but long sustained chords are notably absent from Jazz, so it "works". I'm sure people can train themselves to recognise more complex ratios, but there will still be some limit on the complexity, and I see no way around requiring a longer time to do the recognition.

Regards,

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote [TD 54]:

> Maybe there's an ideological/political issue
> here. If I may be permitted a caricature:
> "A true JI-ist must never talk in cents or
> admit that anything is 'a usable approximation'
> of anything else. JI is ratios, zero beats, and
> that's that!"

When I first got indoctrinated by Partch's book,
I used to feel this way, but read on . . .

> I believe that such a position would fly in
> the face of the facts.

> I understand there is a kind of "uncertainty
> principle" built into Fourier (or Laplace or
> wavelet) analysis and presumably also
> into whatever the ear+brain does. You
> can't give a definite frequency to any
> wave that you only sample for a finite
> duration. The shorter it lasts, the greater
> the uncertainty in its frequency.

This last sentence is true, but as a matter
of fact, no matter how long the sample,
I have now come to believe that there is
indeed an "uncertainty principle" at work,
simply due to the limitations of our auditory
and cognitive apparatus (= "ear+brain").
This is why the "xenharmonic bridges" exist
in the first place.

In the course of doing my research, I've
come to appreciate the usefulness and
appropriateness of ETs much more than
in the past, but I still think more in terms
of JI myself. That fact is that no matter
how complex and accurate you wish to
make your JI model (i.e., lattice diagram),
there are perceptual and conceptual
limits built into our physiology that make
it necessary to reduce the complexity
of the model in order for it to more accurately
represent what is really going on when
we listen to and anyalyze musical intervals,
scales, harmonies, etc. This is indeed
an "uncertainty principle", exactly related
to the one defined in quantum mechanics.

As I've said, I'm not all that familiar with
Fokker's work, but he apparently formalized
this idea already. It also plays an important
part in Erv Wilson's ideas.

In terms of my concepts of "finity" and
"bridging", I'd like to recap a discussion I
had with Paul Erlich regarding Fokker
Determinants (Paul can give a better
mathematical description of what these
are - I'll just give the results of our discussion):

Given a set of independent xenharmonic
bridges, the Fokker Determinant is the operation
which specifies the number of distinct elements
which constitute the finity of the system.

Each bridge reduces the dimensions of
the conceptual model (i.e., lattice diagram)
by one, and the final bridge closes the
theoretically infinite system of the model,
to make it finite.

As examples:

- to reduce an infinite 3-limit JI system to
a finite one, you need one bridge. This
is most commonly the Pythagorean Comma,
leaving a closed 12-EQ system. Another
commonly used bridge is the Mercator
Comma, leaving a closed 53-EQ system.

- to reduce an infinite 5-limit system to a
finite one, you need 2 bridges. The first
one bridges the syntonic comma, reducing
the 5-limit system to a 3-limit one, and the
second bridge acts as the ones described
above

- similarly, a 7-limit system requires 3 bridges
to become a closed system, an 11-limit requires
4, and so on.

An important point is that the resulting closed
systems do not have to be equal temperaments.
They can also be finite JI systems.

The determinant of a matrix (or lattice diagram)
is the area of the parallelogram bounded by the
bridging vectors making up the matrix. Therefore
the determinant is the number of lattice points in
the finite system.

- Monzo (with help from Erlich)
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There was a post that I had wanted to
respond to, but forgotten. This is
more THEORY, so skip it if you're not
interested. (OK?! you've been warned)

[Monzo, TD 64:]
> There have been *lots* of scales with
> primes much higher than 19 described
> in the tuning literature over the centuries.
> "Just" a few examples, all very important
> in the history of music theory:
>
> THEORIST PRIME LIMIT
> Ptolemy 31
> Dowland 211
> Boethius 499
>
> These and lots more are investigated
> in my book.

[Keenan:]
> Thanks Joe. Do you really think that
> their musical uniqueness (assuming
> they have any, pardon my ignorance)
> *depends* on these large primes. Or
> might it be that it was merely convenient
> to express the scales using these numbers.
> Maybe logarithms had't been invented?

Indeed, logarithms were not invented
until (at the earliest) the 1600s.
I believe Euler was one of the first
to use them.

These numbers were definitely chosen
because they facilitated calculations
(believe or not!) or had some other
aesthetic _raison d'etre_.

As an example, I'll explain the 19 used
by Boethius in his chromatic genus, and
the 499 prescribed for his enharmonic -
which is the highest prime I have ever seen
specified in the theoretical literature
(and I've searched thru quite a bit).

Both of them arose as a result of calculating
an *arithmetic* mean of a monochord string
length - so no, there was nothing special
about 499. He was just trying to achieve
as closely as possible a geometrically equal
division.

I'll explain briefly, using only one
tetrachord, and I'll just call the notes
1, 2, 3, 4 (and their modifications in
letter-names), from highest to lowest.

It's a bit complicated, because some
notes were moveable and some were fixed
(that's how ancient Greek theory worked).

Technically, 1 and 4 (the bounding notes
of the tetrachord) were fixed, and the
two inner notes (2 and 3) were moveable,
depending on the genera, which was either
Diatonic, Chromatic, or Enharmonic.

The difference was based on characteristic
intervals withing the tetrachord,
illustrated below in fractional (i.e.,
logarithmic) parts of a tone.

Without logarithmic measurement, ancient
rational tunings only approximate these
intervals, but Boethius came remarkably
close to them. Thus, his Enharmonic
2nd note is exactly the same as his
Diatonic and Chromatic 3rd note (in many
other theorists it was different).

Here's a rough diagram of the intervals,
actually quite similar to the one in the
manuscripts of his book, but with the
monochord bridges labelled simply with
numbers and letters (rather than the
long Greek names).

The numbers are equivalent to the Greek
use of names, and the letters correspond
to our music notes, descending in pitch
from E to B. The ^ sign means a
quarter-tone sharp:

DIATONIC CHROMATIC ENHARMONIC
| |
1-E------- |1-E------- |1-E---------
| |
| |
| |
tone | tone |
| + |
| semitone |
| |
2-D------- | | 2 tones
| |
| |
| |
tone |2-C#------ |
| |
| semitone |
| |
3-C------- |3-C--------|2-C--------
| | 1/4 tone
semitone | semitone |3-B^-------
| | 1/4 tone
4-B------- |4-B--------|4-B--------

Boethius's measurements were done in
string lengths, so remember that the
numbers increase as we go down in pitch,
so 1-E will be the smallest number, and
4-B will be the largest.

The first thing to know is that his
Diatonic genus was the "standard"
Pythagorean

Tone + Tone + Semitone = "4th"
9/8 9/8 256/243 4/3
= 3^2 3^2 3^-5 3^-1

Or in matrix addition:

| 2 | tone
| 2 | tone
|-5 | semitone
-------
-1 "4th"

In the equations, notes 2 and 3 move around,
so they have to be qualified by their genus
(dia./chr./enh.).

So starting with the Pythagorean Diatonic
as a basis, to arrive at the proper placing
of the Chromatic 2nd note (chr.2-C#), he
simply divided the distance, from 1-E to the
Diatonic 2nd note (dia.2-D), in half, then
added that amount to dia.2-D, to get chr.2-C#.

dia.2-D + (dia.2-D - 1-E) = chr.2-C#
-------------
2

It just so happens that this calculation
produces ratios of 19.

Similarly, to arrive at the proper placing
of the Enharmonic 3rd note (enh.3-B^), he simply
divided the distance, from 4-B (the lower
boundary note) to the Enharmonic 2nd
note (enh.2-C), in half, then added that amount
to enh.2-C to get enh.3-B^.

enh.2-C + (4-B - enh.2-C) = enh.3-B^
-------------
2

It just so happens that this calculation
produces ratios of 499!

All you need is the value of one variable,
and you can do the calculations yourself
to find first the diatonic ratios, then
these high primes.

Try 1-E = 3072, and remember, 4-B is fixed
at a 4/3 *below* that (a longer piece of
string, a bigger number). Find the 499!

(Of course, this is Boethius's *theory*.
I stated here not long ago that his book
gives what I think is compelling evidence
of a 5-limit diatonic, in practice!)

And yes, I have improvised in this
enharmonic! Not one of my favorites, tho.

- Monzo
http://www.ixpres.com/interval/monzo/homepage.html
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