Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
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:
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.]
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:
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,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"
Date: Tue Dec 25, 2001 4:12 am
Subject: Re: The epimorphic property
--- In tuning-math@y..., "genewardsmith" wrote:
> --- 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?
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
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
... 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
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
Let me try graphing the 27-tone periodicity block that I posted on Thu 9/23/99 5:26 PM.
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.
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
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
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:
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
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:
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
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:
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
I don't think the pentachordal decatonic scales can be thought of as periodicity blocks, but I could be wrong . . .
Updated: 2001.12.25
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