Yahoo Tuning Groups Ultimate Backup tuning Fokker periodicity blocks from the 3-5-7-harmonic lattice

I took the smallest commas (those under 35 cents -- ranging from 4375:4374
to 50:49) in Fokker's table (which he credits to John Chalmers) and
calculated the periodicity blocks generated by all possible triplets of
commas. The most common sizes of the periodicity blocks are shown in the
following table:

# of notes # of occurrences ET with that # of notes consistent in
7-limit?
22 381 Yes
0 308 *
31 289 Yes
44 282 Yes
12 235 Yes
24 225 No (but 24=2*12)
41 198 Yes
19 183 Yes
48 180 No (but 48=4*12)
62 175 Yes
10 169 Yes
53 168 Yes
66 166 No (but 66=3*22)
38 159 No (but 38=2*19)
36 156 Yes
72 149 Yes
46 145 Yes
34 144 No
106 141 No (but 106=2*53)
15 136 Yes
93 135 Yes
17 135 No
82 130 Yes
68 123 Yes
30 123 No (but 30=2*15)
88 120 Yes
7 118 No
58 117 Yes
60 109 Yes
92 107 No (but 92=2*46)
27 107 Yes
63 101 Yes
65 100 No
5 96 Yes
80 94 Yes
49 88 Yes
14 88 No
84 87 Yes
56 87 Yes
96 85 No (but 96=8*12)
110 83 No (but 110=5*22)
50 82 Yes
20 78 No
26 77 Yes
29 76 Yes
3 76 No
etc. (total 12341 from 43 commas; largest scale had 481 notes)

* 0 notes indicates that the triplet of commas contained a linear dependence
and thus did not define a periodicity block.

Reference: Fokker, A.D. "Unison vectors and periodicity blocks in the
three-dimensional (3-5-7) harmonic lattice of notes." Proceedings of the
Koninkliike Nederlandse Akademie van Wetenschappen. Series B, Physical
sciences / Vol. 72 (1969), pp. 135-168. Amsterdam : North-Holland Pub. Co.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

I repeated the experiment but this time with no commas larger than the
septimal comma (64:63 = 27.26 cents).

# of notes # of occurrences 7-limit consistency of ET
31 276 Yes
22 205 Yes
0 198 *
12 181 Yes
62 164 Yes
53 149 Yes
24 145 No (but 24=2*12)
44 127 Yes
41 124 Yes
93 118 Yes
48 108 No (but 48=4*12)
19 107 Yes
72 103 Yes
106 98 No (but 106=2*53)
10 97 Yes
36 88 Yes
46 87 Yes
17 85 No
34 81 No
15 80 Yes
66 78 Yes
27 77 Yes
68 74 Yes
5 68 Yes
etc. (total 7140 from 36 commas, largest scale had 481 notes)

With no commas larger than the syntonic comma (81:80=21.51 cents):

# of notes # of occurrences 7-limit consistency of ET
31 172 Yes
53 112 Yes
22 110 Yes
0 105 *
62 87 Yes
106 86 No (but 106=2*53)
12 63 Yes
19 62 Yes
44 61 Yes
93 51 Yes
46 48 Yes
41 47 Yes
68 46 Yes
77 44 Yes
10 42 Yes
24 41 No (but 24=2*12)
48 38 No (but 48=4*12)
17 37 No
27 36 Yes
72 35 Yes
36 35 Yes
58 34 Yes
159 33 Yes
65 33 No
15 33 Yes
80 32 Yes
38 32 No (but 38=2*19)
118 31 Yes
89 31 Yes
66 31 No (but 66=3*22)
5 31 Yes
171 29 Yes
154 28 No (but 154=7*22)
82 28 Yes
92 26 No (but 92=2*46)
49 26 Yes
99 25 Yes
34 25 No
etc. (total 3276 from 28 commas, largest scale had 472 notes)

With no commas larger than 126:125 = 13.79 cents:

# of notes # of occurrences 7-limit consistency of ET
31 70 Yes
53 65 Yes
22 50 Yes
0 40 *
106 39 No (but 106=53*2)
171 28 Yes
118 28 Yes
19 27 Yes
99 24 Yes
77 24 Yes
41 23 Yes
62 22 Yes
159 19 Yes
72 17 Yes
130 16 Yes
89 16 Yes
44 16 Yes
27 16 Yes
140 15 Yes
108 13 Yes
15 13 Yes
12 13 Yes
49 11 Yes
46 10 Yes
etc. (total 1140 from 20 commas, largest scale had 472 notes)

With no commas larger than Fokker's fave u.v., 225:224 = 7.71 cents
(complete listing):

# of notes # of occurrences 7-limit consistency of ET
171 28 Yes
22 28 Yes
130 16 Yes
118 16 Yes
99 16 Yes
31 16 Yes
0 14 *
248 8 Yes
140 8 Yes
77 8 Yes
53 8 Yes
41 8 Yes
19 8 Yes
279 3 Yes
270 3 Yes
217 3 Yes
152 3 Yes
149 3 Yes
108 3 Yes
94 3 Yes
72 3 Yes
63 3 Yes
12 3 Yes
125 1 Yes
62 1 Yes
50 1 Yes
46 1 Yes
44 1 Yes
34 1 No
32 1 No
10 1 Yes

It's surprising how well 22-tone holds up even in the context of these very
small commas.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

No one had any comments on these results I posted two weeks ago, but I have
some questions/comments:

1. John Chalmers, Fokker attributed his tables of commas to you. Do you
happen to remember (I know it was 32 years ago) what portion of the harmonic
lattice was searched to find these commas?

2. I suspect the answer to (1) is some sort of symmetrical region in the
Euler-Fokker (rectangular) lattice. I think we should instead use a
symmetrical region in the triangular lattice, so that, for example, major
sevenths are less direct than minor thirds.

3. It doesn't make much sense to define a periodicity block with three
unison vectors if the block contains intervals smaller than one of the
unison vectors. I'm sure many of the inconsistent scales I was getting would
not appear had this been enforced.

4. Although (2) and (3) may change this, it appears that 22-tone systems
have a somewhat privileged status with respect to Fokker's idea.
Particularly surprising is that using the 12 unison vectors no larger than
225:224, 22-tone and 171-tone are the most commonly appearing systems by a
wide margin. To what extent can we interpret this to mean that 22 notes is a
natural amount for 7-limit harmony? (This is not a new idea -- Ben Johnston
used a 22-tone scale for the 7-limit section of his String Quartet #4.)

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

I've still received no comments about my ideas here, so let me explain why I
think this is important.

A just intonation system will often, for practical reasons, have to have a
finite number of notes. Erv Wilson, Kraig Grady, and of course Fokker have
given many examples of logical ways of "snipping" the lattice. After
snipping, one may or may not wish to use temperament to connect the ends
back together. But rather than look for particulars, there are some very
general questions that the Fokker formalism should allow us to answer.

The sensible place to "snip" the lattice is right before a new note is added
that is so close to an old note, that it can function like the old note (a
wide or narrow tolerance for deviations could be implied here, depending on
the context). In the context of 7-limit harmony, three different types of
"snips", applied consistenly, lead to a finite set of JI pitches. A very
wide variety of JI scales exist, but given three intervals that define your
"snips", all the possibilities will have the same number of notes. That
number is what the Fokker determinant gives you. Although this number
usually described in terms of the volume of a parallelopiped, the edges can
be deformed into almost any shape one wishes (as long as in each of the
three sets of four parallel edges, the four remain translationally
congruent) and one gets a JI scale with the same number of notes.

Now in my earlier posts I described the results of trying out every possible
triplet of "snips" out of a large set, but some restrictions should be
placed on this process. Most evidently, it wouldn't make sense if the JI
scale contained an interval smaller than any of the intervals defining a
"snip". In more mathematical terms, the periodicity block defined by three
unison vectors should not contain within it any smaller (in cents) unison
vectors, built upon any of the eight vertices. Any periodicity block will be
a "constant structures" scale. But this is a more stringent criterion. I
read about it in Mandelbaum and someone derived the sizes of 5-limit scales
that satisfy it. Does anyone have this reference handy?

My question to the programmers on this list (Manuel, Paul H, . . .): Can we
devise a simple algorithm to find all the solutions here? If we start with
the simplest (defined by distance on the triangular lattice) unison vectors,
and then replace the largest one with the next simplest, and repeat the
process, I think we make some omissions and include some we don't want. For
example, if we use odd limit instead of the Hahn-Erlich triangular metric,
and Fokker's list of unison vectors under 35 cents, the results of doing
that are

|-2 0 -1| |-2 0 -1|
| 0 2 -2| = 10; | 0 2 -2| = 22
|-3 0 -4| | 3 -1 3|

|-2 0 -1|
| 3 -1 -3| = 5
|-3 0 4|

| 4 -1 0|
| 3 -1 -3| = 5
|-3 0 -4|

| 2 -3 1| |-4 3 -2| | 1 -3 -2|
| 3 -1 -3| = 58; | 3 -1 -3| = 53; | 3 -1 -3| = 53
|-3 0 -4| |-3 0 -4| |-3 0 -4|

| 2 -3 1| |-4 3 -2| | 1 -3 -2|
| 3 -1 -3| = 27; | 3 -1 -3| = 53; | 3 -1 -3| = 31
|-4 3 -2| | 1 -3 -2| | 2 -3 1|

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

| 1 -3 -2|
| 1 0 3| = 31
| 2 2 -1|

| 1 -3 -2|
|-1 -2 4| = 31
| 2 2 -1|

| 1 -3 -2|
|-1 -2 4| = 0
| 0 -5 2|

| 1 -3 -2|
|-1 -2 4| = 99
|-7 4 1|

| 8 1 0|
|-1 -2 4| = 171
|-7 4 1|

🔗Joe Monzo <monz@xxxx.xxxx>

> [Paul Erlich, TD 316.3]
>
> I've still received no comments about my ideas here, so let
> me explain why I think this is important.

Paul, thanks for further elaborating your previous post on
this subject. I'm extremely interested in it, but just
haven't been able to spend the time I need to on it to
fully understand the questions you're asking. Please
continue to add more of your own thoughts. Can we get
Paul Hahn, Kami Rousseau, or Daniel Wolf to add anything?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Thanks for your interest Joe. In Mandelbaum's dissertation there is
mentioned a theorist (or several) who worked with the concept of "generating
intervals" (=unison vectors) and "constructing intervals" (=step sizes). One
theorist was said to prove that the only 5-limit JI scales where all
generating intervals were smaller than all constructing intervals had
cardinality 5, 7, 12, 19, 22, 31, 41, 53, 72 . . . (I may have this list
wrong). So basically what I'm asking is, in the context of Fokker
periodicity blocks, can we determine something similar for 7-limit JI
scales? Also, it is of interest to know how often each cardinality comes up
. . . for example, within the range of reasonable unison vectors, it may
seem that 12 is usually the "natural" cardinality for thinking about
(composing in, notating) 5-limit JI . . . can we assess the situation for
7-limit JI, and decide among 22, 27, 31, etc.?

This is an idea that's been floating around my head for about two years now.
While the process of evaluating ETs in terms of how well they approximate JI
occupied my thought for many years, leading to the notion of consistency and
the first two figures in my paper, the newer idea is evaluting JI systems in
terms of how well they approximate ETs. The reason the two questions are
different is that in the first, the only JI intervals one cares about are
the first-order ones, the consonances, the ones up to a certain odd limit.
In the second, since one is constructing a JI system, one has these
higher-order intervals (the unison vectors and constructing intervals) to
worry about.

Fokker periodicity blocks are the natural result of deciding on a minimal
set of unison vectors. But if a smaller interval occurs within the block
than one of the unison vectors, the construction makes little intuitive
sense. If a smaller interval does not occur, then what you have is a clear
delineation of interval classes as in an ET. So what one needs is a
geometrical calculation to assess whether a unison vector is contained
anywhere within the periodicity block formed by three other unison vectors,
at least one of which is larger than the first one. For if and only if it
is, the block has a "generating interval" that is larger than a
"constructing interval", making it inadmissable by the above criterion.

What I'm hoping for (and Paul Hahn is probably best equipped among us to
help) is an algorithm that will start with a small portion of the lattice
and rather large unison vectors, and work its way out to smaller unison
vectors and a larger portion of the lattice, evaluating all possibilities,
without skipping any. Or perhaps we can look to the work of the original
theorist who looked at this in the 5-limit and think of some shortcuts that
can be applied to the 7-limit. Either way, the results would go along way
toward characterizing the seemingly endless possibilities in reasonable,
finite JI systems.

In other words, Joe, this is all about finity! (Once again, a Partch
tonality diamond is most certainly not an example of finity in this context,
while Partch's 43-tone system could be seen as a variation on an 11-limit,
41-tone periodicity block with two auxillary notes.)

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

> [Paul Erlich, TD 319.10]
>
> <snip>
> In other words, Joe, this is all about finity!

Yes, I realized that already.
That's exactly why I'm so interested in it.

Thanks for elaborating still further. I've been pondering
for most of this year how to portray both JIs and ETs on the
same lattice, and this discussion gives me a little more to
chew on.

Considering the absence so far of any comments from Paul Hahn,
can you go into any more detail about how the algorithm would
work? - maybe I can add something, if the math doesn't get too
complicated.

-monz

🔗Kees van Prooijen <kees@xxxx.xxxx>

On seeing these posts I remembered working along these lines in a previous
century. Today I finely had the time to dig up some results. Indeed I did a
search through the possible intervals keeping the smallest ones, taking the
determinant for finding the number of tones in the set. I didn't know about
the work of Fokker then, but I basically reinvented the principle of the
periodicity blocks.

The problem algorithmically is to search in the right order, indeed to not
have any smaller intervals inside the block. Also I remembered needing
tricks to be able to continue with big numbers without losing precision.

For the 5 limit it was relatively easy, searching in 2 dimensions. The trick
is to only search half the plane, in a triangle where both axes are positive
and in a rectangle where the signs are opposite. Here are the results
http://www.dnai.com/~kees/tuning/s235.html

For the 7 limit case the thing gets tricky. You have to search in four
octants (? 3D quadrants) of the space. Where all axes are positive that's a
4 sided pyramid, in the others, where 2 axes have the same signe, and one
is opposite, you end up with a prisma. It was too much trouble too search
these with the right relative scaling (log3 log5 log7) as I had done in the
5 limit case. So you don't actually get the unison intervals in the right
order and in the worst case you might miss one. Anyway here are these
results.
(I corrected the order for the ones that have their actual interval numbers
calculated)
http://www.dnai.com/~kees/tuning/s2357.html

I wonder if I have the record for smallest intervals in each case with
(94848, -23569)
and
(179, 732, 1066)
or with the largest ETs with 1003611167 and 1105633117 :-)

If I have really much time I will try too clear this up with some
illustrations.

Kees
------------------------------------------
Kees van Prooijen
email: kees@kees.cc
web: http://www.kees.cc

🔗PERLICH@xxxxxxxxxxxxx.xxx

Kees van Prooijen-

Your results, like the odd-limit-based ones I posted earlier, seem incomplete.
The 5-limit results I read about in Mandelbaum included 22-, 31-, and 41-tone
scales (if I remember correctly). I would love to discuss your algorithm and these
issues further.

Joe-

I can't really think of an algorithm that would solve this problem fully. The
short-cut approaches I (and probably Kees) have tried involve increasing the portion
of the lattice one is looking at little by little, and at each stage computing the
periodicity block formed from the three smallest (in cents) unison vectors. This
approach gives neither a subset nor a superset of the solutions to the problem,
which was: Find all the periodicity blocks (up to, say, 100 notes) where there is
no interval between notes of the block that is smaller than any of the unison
vectors.

I'll probably have to ask Ara to get Mandelbaum from the library again. . . .

🔗Carl Lumma <clumma@xxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

> [Kees van Prooijen, TD 321.25]
>
> On seeing these posts I remembered working along these lines
> in a previous century. <snip> I didn't know about the work of
> Fokker then, but I basically reinvented the principle of the
> periodicity blocks. <snip> Here are the results
> http://www.dnai.com/~kees/tuning/s235.html
> http://www.dnai.com/~kees/tuning/s2357.html
>
> <snip>
>
> If I have really much time I will try too clear this up with
> some illustrations.

To Kees:

Please do! I'm really tired right now and don't remember
if Paul posted lattices to describe any of what he discussed,
but I'm very interested in seeing the graphs of both of your
results. Thanks for the info.

To all:

I'm interested in studying the historical conceptions of
various shapes and sizes of periodicity blocks in music
all over the world. I believe that this 'history of finity
in tuning' (which, I now realize, is what my book(s?) attempts
to be) can enrich our knowledge of many other aspects of our
lives and histories, especially ancient religious beliefs,
possibly even extending to modern scientific theories about
the universe.

The most important aspect, it seems to me, is to determine
the proper lattice metric to portray all the different tunings.
There is no clear consensus on this yet, altho I recall Pauls
Erlich and Hahn being the most outspoken in favor of the one(s?)
they like, and doesn't Carl Lumma agree, at least partially?

The only other metrics I'm familiar with are those of Fokker,
Wilson, Chalmers (who has also made some with me), Vogel,
Johnston, Doty, and Terpstra. Did Mandelbaum have a preferred
one of his own? Can anyone (Daniel Wolf perhaps?) post
something about the metrics and angles of Tanake and other
early tonal-lattice designers? I haven't had a chance to look
up Daniel's earlier references and am very impatient to know
about them now.

If this thread is new to you and you don't have a clue what
I'm saying, see Paul Erlich's posts on it from the last couple
of weeks, and my definition of finity and its related links:
http://www.ixpres.com/interval/dict/finity.htm

-monz

🔗Carl Lumma <clumma@xxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

> [Paul Erlich, TD 327.24]
>
> <snip explanation and excellent lattice diagram>
>
> Note the two identical halves, 602 cents apart.

Now we're talkin'. That's exactly why I like the
lattice diagrams: you can see interesting properties
like this at a glance! This is much easier for me
to digest than a table of numbers (altho it's important
to provide the actual datatoo). It also makes what one
hears in music that uses a particular (JI) tuning that
much easier to grasp.

-monz