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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