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Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
periodicity block
A term used by Fokker in his
lattice theory to describe
a whole multi-dimensional region of the lattice that
could substitute for another, by means of
unison vectors.
This concept is allied with that of
finity, which I
developed independently of Fokker. The periodicity
blocks quantify the finity of the system.
Periodicity blocks enclose a certain
number of discrete
categorical
intervals or
pitch-classes,
and the unison vectors are small enough intervals
that pitches within the block can represent
or imply pitches outside of it which have different
prime/odd-factors,
a process which I had named bridging.
[Paul Erlich comments:
This number of pitches can be calculated by
a Matrix determinant,
using the prime-factors of the
ratios
at either ends of the bridges or
unison vectors to fill the Matrix.
[from Joe Monzo, JustMusic: A New Harmony]
See Erlich's
A Gentle Introduction to Fokker
Periodicity Blocks.
also see Kees
van Prooijen's periodicity-block webpage.
This is Gene Ward Smith's formula for finding
what he calls a "notation" for the ratios enclosed
within a JI periodicity-block:
for any non-zero a scale can be defined by calculating
for 0 <= n < d :
step[n] = (u1/v1)^round(h1(2)*n/d) * (u2/v2)^round(h2(2)*n/d)
* ... (ui/vi)^round(hi(2)*n/d) .
Further notes from Paul Erlich:
Yahoo tuning-math group, message 2158:
From: "paulerlich"
--- In tuning-math@y..., "genewardsmith" wrote:
The only published articles on PBs are Fokker's. Inferring strict
definitions from these articles would suggest that a parallelepiped
(or N-dimensional equivalent) are the only accepted shape (thus I
call these _Fokker_ periodicity blocks, or FPBs), and that if there
is an even number of notes, one needs to produce two alternative
versions so that symmetry about 1/1 is maintainted.
[note from Monz: Paul shows how to construct the parallelepiped
type of periodicity-block in his
A gentle introduction
to Fokker periodicity blocks, part 2, and then shows that
periodicity-blocks need not be parallelepipeds in his
A Gentle
Introduction to Fokker Periodicity Blocks:
an excursion.]
Onelist
Tuning Digest 465
... all
JI
scales
involve a sort-of-arbitrary decision of where to stop.
What I've shown is, this decision is not completely arbitrary, it almost
always seems to conform to a periodicity-block construction.
Onelist
Tuning Digest 340
Let me try graphing the 27-tone periodicity block that I posted on Thu
9/23/99 5:26 PM.
This one is amazingly rich in 7-limit
triads and
tetrads, so much so that
it's straining my meager ASCII symbology to show all the connections.
Onelist
Tuning Digest 340
These three basic
scales,
identified in my paper as the melodic bases for
3-, 5-, and 7-limit
harmony, respectively, have
JI representations that come
out as Fokker periodicity blocks when the typical
"chromatic" interval
implied by the scales is used as a
unison vector.
Interestingly, all these periodicity blocks are of the "most natural" type
catalogued by Kees van Prooijen (he skipped the 3-limit ones but they're
trivial -- apparantly not enough so to satisfy Carl [Lumma]?)
In the 3-limit
pentatonic
case, modulating the scale by a single
ratio of 3
simply moves one note by 256:243. Using this as the unison vector (only one
is needed since
octave-equivalent
3-limit space is one-dimensional), we get
the following periodicity block:
Interestingly, as I was writing this, Carl posted something about 1D
periodicity blocks, to which this may relate.
In the 5-limit diatonic case, 81:80 is already assumed as a unison vector,
and the chromatic interval by which one note moves when modulating by a
ratio of 3 is 25:24. Using these as the unison vectors, the resulting
periodicity block is:
In the 7-limit
decatonic
case, 64:63 and 50:49 are already assumed as unison
vectors. If you are unfamiliar with decatonic scales, see
my paper.
When modulating by a
3-limit ratio, two notes move by a 48:49. Using these as the unison vectors,
the resulting periodicity block is:
I don't think the
pentachordal
decatonic scales can be thought of as
periodicity blocks, but I could be wrong . . .
Updated: 2001.12.25
That's correct, but Fokker generally set the overall
prime-limit (either 5
or 7) beforehand and all unison vectors connect pitches (or intervals)
within this limit.
Paul has pointed out that my concept of 'bridges' refers
specifically to the kinds of prime-factors and not
the numbers of prime-factors, which is what Fokker
described with the periodicity block concept.]
where M is the matrix composed of a set of i
rational
vectors {u1/v1, u2/v2,... ui/vi} in which
u1/v1 is a
step-vector and {u2/v2 ... ui/vi} are
commatic or
chromatic
unison-vector
generators of the kernel,
and where {h1, h2, ...hi} is the top row of M^-1, and
where round() is the function which rounds to the
nearest integer,
Date: Tue Dec 25, 2001 4:12 am
Subject: Re: The epimorphic property
> --- In tuning-math@y..., "paulerlich" wrote:
>
> > > Does "shape" entail connectedness, or can it be scattered islands
> > > all over the place?
> >
> > The latter. Especially as preimages of ETs, such constructs would be
> > just fine.
>
> Under that definition, PB <==> epimorphic. Are you sure it is the
> accepted one?
Message: 6
Date: Thu, 30 Dec 1999 04:12:36 -0500
From: "Paul H. Erlich"
Subject: 11-limit, 31 tones, 9 hexads within 2.7c of just
Message: 14
Date: Mon, 4 Oct 1999 18:40:06 -0400
From: "Paul H. Erlich"
Subject: RE: Re: Fokker periodicity blocks from the 3-5-7-harmonic lattice
The matrix of unison vectors is
2 -3 1
3 -1 -3
-4 3 -2
and the coordinates are
-3 2 -2 329
-2 1 -2 645
-2 1 -1 414
-2 2 -3 62
-1 0 -2 960
-1 0 -1 729
-1 1 -3 378
-1 1 -2 147
0 -1 -1 1045
0 -1 0 814
0 0 -3 694
0 0 -2 462
0 0 -1 231
0 0 0 0
0 1 -4 111
1 -2 0 1129
1 -1 -3 1009
1 -1 -2 778
1 -1 -1 547
1 0 -3 195
2 -2 -2 1094
2 -2 -1 862
2 -1 -3 511
2 -1 -2 280
3 -3 -1 1178
3 -2 -2 596
4 -3 -2 911
I think I'll go back to the usual orientation for this one:
5
/:\
/ : \
/ 7 \
/,' `.\
1---------3
And we're off:
329
\`.
\ 62
\ : 414
\: ,' `.
645-------147
\`. ,'/:\ 0
\ 378 / : \ ,'/
\ : /.729-------231 /
\:/,'**\`.\ ,'/:\/
960-------462 / :/\
`. : ,' : / 814 \
694--\:/,'195`.\
\ 1045-/:\--547
\ `/ : \' : `.
\ / 778-\-----280
\ /.' \`.\@@,'/:\
1009------511 / : \
\ : / 862 \
\:/,' `.\
1094------596
:\
: \
1178\
`.\
911
**111 goes here, but there was not enough room to write it.
@@1129 goes here, but there was not enough room to write it.
Message: 16
Date: Mon, 4 Oct 1999 22:00:20 -0400
From: "Paul H. Erlich"
Subject: JI pentatonic, "diatonic", and decatonic scales as periodicity bl ocks
ratios for major
1/1-------3/2-------9/8------27/16-----81/64
ratios for minor
32/27-----16/9-------4/3-------1/1-------3/2
ratios for major
5/3-------5/4------15/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
4/3-------1/1-------3/2-------9/8
or
10/9-------5/3-------5/4------15/8
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
4/3-------1/1-------3/2
or
100/81-----50/27
\ / \
\ / \
\ / \
\ / \
40/27-----10/9-------5/3
\ / \
\ / \
\ / \
\ / \
4/3--------1/1
ratios for minor
1/1-------3/2-------9/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
8/5-------6/5-------9/5------27/20
or
4/3-------1/1-------3/2-------9/8
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
8/5-------6/5-------9/5
or
40/27-----10/9
\ / \
\ / \
\ / \
\ / \
16/9-------4/3-------1/1
\ / \
\ / \
\ / \
\ / \
8/5--------6/5
ratios for symmetrical major
5/4------15/8 7/4------21/16
,'/:\`. ,'/:\`. ,'/:\`. ,'/:\`.
10/7-/-:-\15/14/-:-\45/28 or 1/1-/-:-\-3/2-/-:-\-9/8
: / 7/4------21/16\ : : /49/40----147/80\ :
:/,' `.\:/,' `.\: :/,' `.\:/,' `.\:
1/1-------3/2-------9/8 7/5------21/20-----63/40
or
16/9-------4/3-------1/1 80/63-----40/21-----10/7
:\`. ,'/:\`. ,'/: :\`. ,'/:\`. ,'/:
: \32/21/-:-\-8/7 / : : 160/147-:-\80/49/ :
56/45-----28/15------7/5 or 16/9-------4/3-------1/1
`.\:/,' `.\:/,' `.\:/,' `.\:/,'
16/15------8/5 32/21------8/7
or
5/4 7/4
.'/:\`. .'/:\`.
40/21-----10/7-/-:-\15/14 4/3-------1/1-/---\-3/2
:\`. ,'/: / 7/4 \ : :\`. ,'/: /49/40\ :
: \80/49/ :/,' `.\: or : \ 8/7 / :/.' `.\:
4/3-------1/1-------3/2 28/15------7/5------21/20
`.\:/,' `.\:/,'
8/7 8/5
ratios for symmetrical minor
4/3-------1/1 40/21-----10/7
,'/:\`. ,'/:\`. ,'/:\`. ,'/:\`.
32/21/-:-\-8/7-/-:-\12/7 or 160/147/-:-\80/49/-:-\60/49
: /28/15------7/5 \ : : / 4/3-----/-1/1 \ :
:/,' `.\:/,' `.\: :/,' `.\:/,' `.\:
16/15------8/5-------6/5 32/21------8/7------12/7
or
40/21-----10/7------15/14 4/3-------1/1-------3/2
:\`. ,'/:\`. ,'/: :\`. ,'/:\`. ,'/:
: \80/49/-:-\60/49/ : : \ 8/7-/-:-\12/7 / :
4/3-------1/1-------3/2 or 28/15------7/5------21/20
`.\:/,' `.\:/,' `.\:/,' `.\:/,'
8/7------12/7 8/5-------6/5
or
21/16 15/8
.'/:\`. .'/:\`.
1/1-------3/2-/-:-\-9/8 10/7------15/14/---\45/28
:\`. ,'/: 147/80\ : :\`. ,'/: /21/16\ :
: \12/7 / :/,' `.\: or : \60/49/ :/.' `.\:
7/5------21/20-----63/40 1/1-------3/2-------9/8
`.\:/,' `.\:/,'
6/5 12/7
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