Just a note to those who are

interested in my work:

I have not yet read anything by

A. D. Fokker - most of his work

is still in Dutch, but some has been

translated into English and I'm chomping

at the bit to get my hands on it. I've

found it very difficult to find any.

Are some of his books available?

(perhaps from Corpus Microtonale?)

But I've realized from what I do know

of his work (mainly from Mandelbaum's

thesis) that it's probably closer than

anyone else's to what I'm doing.

Fokker (as far as I know) was the first

to draw ratios on a 3-dimensional lattice,

and apparently he also formalized what I've

called "finity" and "bridging". If anyone else

out there is more familiar with his theories,

I'd appreciate a knowledgeable response.

- Monzo

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

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I sent this to Joe Monzo before I read his post. I will now share it

with everyone (excuse the copyright infringement against Fokker):

> -----Original Message-----

> From: Paul H. Erlich

> Sent: Tuesday, February 23, 1999 7:39 PM

> To: 'Joseph L Monzo'

> Subject: RE: thanks for all the hospitality

>

> A Fokker article just fel into my lap! I didn't even look for it! It's

> "Selections from the Harmonic Lattice of Perfect Fifths and Major

> Thirds Containing 12, 19, 22, 31, 41, or 53 Notes." This is how it

> starts -- this sounds just like your finity concept:

>

> **********************************************************************

_Summary and Conclusion_

> The infinite multitude of notes is investigated, which are

> generated by reiterated traspositions of a central note by perfect

> fifths (3/2) and perfect major thirds (5/4). The notes are represented

> in a rectangular lattice. The numbers of factors 3 and 5 involved in

> their definition are coordinates in the plane of the lattice. Around

> the central note (0,0) the notes are thus defined by pairs of numbers

> (p,q), where both p and q may run from - to + infinity.

>

> In practice it will be convenient to use letters, from a to g

> inclusive, with additional sharps and flats. In order to discriminate

> between members of a commatic pair, commatic strokes, sloping up or

> down, may preced the letters. For reasons of symmetry the central note

> will be chosen to be D.

>

> The pitches of some notes are so close one to another, that they

> may be mistaken to be unison. Again, too, on purpose they may be taken

> to be unisons, in order to reduce the infinite multitude of notes to

> tractable smaller numbers. Such vanishing intervals are represented by

> vectors in the lattice from the centre to the postulated unisonious

> note. We call them _unison vectors_.

>

> Any unison vector introduces a periodicity into the lattice. Two

> unison vectors postulate a two-fold periodicity, producing _periodic

> meshes_. These are parallelograms having for sides the two unison

> vectors. They include a _number of notes_, a varying number according

> to the choice of the unison vectors. Such numbers have served to

> constitute equal temperaments, with equal intervals between

> consecutive members of one periodicity mesh.

>

> The common equal temperament takes a selection of twelve

> semitones. Nineteen tritotones have been advocated by Wesley Woolhouse

> (1835) and in modern times by Joseph Yasser (1932). The Indian

> musicians discriminate twenty-two sruti's. Christiaan Huygens (1691)

> computed a system of thirty-one dieses. This is supported by the

> present writer. Forty-one supracomma's have been put forward by Paul

> Von Janko (1901), and Nicolaus Mercator, as early as 1608, extended

> the selection to fifty-three comma's.

> **********************************************************************

>

--------------------------------

That's what I sent to Joe. Now I'll skip to almost the end of the

article.

**********************************************************************

_Geometry in the lattice_

Geometry shows that a vector (a,b) and a second vector (p,q)

define a parallelogram with an are (aq-bp). This is written in the

algebraical form

|a b|

| | = aq-bp

|p q|

Therefore, in the lattice, two unison vectors will define an

area, i.e., a number of lattice area units. Each unit carrying one note,

that area will give the number of the notes in the selection.

The selections examined in the preceding pages are the

following.

|0 -3| |4 -1|

12 = | | = | |

|4 -1| |8 1|

|4 -1| |4 -1|

19 = | | = | |

|-1 5| |-5 6|

|4 2| |4 2|

22 = | | = | |

|-1 5| |3 7|

|4 -1| |1 -8|

31 = | | = | |

|3 7| |4 -1|

|8 1| |8 1|

41 = | | = | |

|-1 5| |7 6|

|8 1| |8 1|

53 = | | = | |

|-5 6| |3 7|

**********************************************************************

When Joe was over at my house this weekend I already knew about this

aspect of Fokker's theory (from Graham Breed and/or Paul Hahn) so we

tried this with all kinds of 5-limit commas and got 34 in addition to

all the values above.

The two bars surrounding a square matrix of numbers is called the

determinant. The general formula for computing the determinant in N

dimensions is complicated, but it is still equivalent to computing the

area of the N-dimensional parallelopiped formed by N vectors. Fokker

wrote a paper about the three-dimensional (7-limit) case, but I haven't

seen it yet. I would assume looks at sets of 3 7-limit unison vectors,

calculates their determinants, and gets numbers which correspond to ETs

with good 7-limit approximations, such as 31 and 72.

Joseph L Monzo wrote:

> Fokker (as far as I know) was the first

> to draw ratios on a 3-dimensional lattice,

> and apparently he also formalized what I've

> called "finity" and "bridging".

I could be wrong but I remember Euler having some 3D lattices. The stuff I

have of Adrian's is all 2d though!

-- Kraig Grady

North American Embassy of Anaphoria Island

www.anaphoria.com

The two essential items of musical Fokkeriana are:

Fokker, Adriaan Danie"l, _Selected Musical Compositions (1948-1972)_,

edited by Rudolph Rasch, Urecht, the Diapason Press, 1987. Includes

complete bibliography of Fokker's musical and non-musical writings and a

detailed summary of Fokker's musical theories as well as the scores for 31

compositions. It is a beautiful volume and a bargain.

Fokker, A.D., _New music with 31 notes_, trans. by Leigh Gerdine. Bonn:

Verlag f�r systematische Musikwissenschaft, 1975. 96 pp. (Orpheus - Series

of Monographs on Basic Questions in Music, Volume 5).

On Wed, 24 Feb 1999, Paul H. Erlich wrote:

> Fokker

> wrote a paper about the three-dimensional (7-limit) case, but I haven't

> seen it yet. I would assume looks at sets of 3 7-limit unison vectors,

> calculates their determinants, and gets numbers which correspond to ETs

> with good 7-limit approximations, such as 31 and 72.

Actually, the 7-limit paper examines the same ETs as does the 5-limit

one; it just derives 3D periodicity blocks for them instead of 2D. Once

you get the basic idea, there's not that much new there.

I believe I mentioned some time ago that both of these papers are held

by the library where I work, and can be obtained from us through

InterLibrary Loan.

I have been using Fokker's unison vector/determinant method for some

time. At this point I am not so much interested in using them to

generate ETs (I'm pretty much committed to 31TET) as for "generalized

diatonic" scales embedded within them. For example, the simplest set of

unison vectors for 12TET at the 5-limit form the matrix:

| 4 -1 | = 12

| 0 3 |

By substituting (-1 2) for (0 3) (implying that the 25/24, chromatic

semitone, "vanishes"--IOW, the 6/5 and the 5/4 are the same generic

interval) you get

| 4 -1 | = 7

|-1 2 |

implying heptatonic scales, the simplest of which is the diatonic set.

But you can embed the diatonic set in other scales which use (4 -1) as a

unison vector such as 19TET [(4 -1), (-1 5)] or 31TET [(4 -1), (-1 8)].

Using (1 1), the 16/15, instead of (-1 2) gives you pentatonicism.

(-ness?)

In the 7-limit, my 9-out-of-31 scales work like this:

| 2 2 -1 | | 2 2 -1 |

| 4 -1 0 | = 31 ==> |-1 0 2 | = 9

| 1 0 3 | | 2 -1 -1 |

(Some of the signs may be wrong on those determinants. I'm used to just

thinking of the vanishing intervals/unison vectors.)

Paul Erlich's 10-out-of-22 scales work like this:

| 2 2 -1 | | 2 2 -1 |

| 0 -2 2 | = 22 ==> | 0 -2 2 | = 10

| 3 -1 -3 | |-1 2 0 |

--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. <*>

Mark Nowitzky wrote,

>When I talked to David Rosenboom (Dean of Calarts School of Music)

about an

>alternate tuning keyboard of mine back in 1995 (four years ago!), he

>mentioned Adriaan Fokker. I've been looking for stuff by this Fokker

guy

>ever since. I haven't found anything in English yet either!

Fokker's book _New Music with 31 Notes_ has been translated to English.

Rasch wrote a book in English with Fokker's theories and music in it.

Check the microtonal bibliography

(ftp://ella.mills.edu/ccm/tuning/papers/bib.html) for more.

I recall also an early 1/1 article by Rasch following a talk he gave

at JIN in San Fran in the mid-eighties.

> Fokker, Adriaan Danie"l, _Selected Musical Compositions (1948-1972)_,

> edited by Rudolph Rasch, Urecht, the Diapason Press, 1987. Includes

> complete bibliography of Fokker's musical and non-musical writings and a

> detailed summary of Fokker's musical theories as well as the scores for 31

> compositions. It is a beautiful volume and a bargain.

>

> Fokker, A.D., _New music with 31 notes_, trans. by Leigh Gerdine. Bonn:

> Verlag f�r systematische Musikwissenschaft, 1975. 96 pp. (Orpheus - Series

> of Monographs on Basic Questions in Music, Volume 5).