# © 1999 by Paul Erlich & Joe Monzo

```
Onelist Tuning Digest 397
From:	Paul H. Erlich
Sent:	Friday, September 24, 1999 6:22 PM
To:	'tuning@onelist.com'

Joe Monzo wrote,

> 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 Hindu system of 22 srutis, in its common JI form is essentially a
Fokker periodicity block, but not one of the good ones I'm looking for.
Since this is a 5-limit system, we need two unison vectors. The first one
is the diaschisma, ratio 2048/2025 or 3^-4 * 5^-2, as we've discussed
before, there's evidence that the same pitch (sruti #2) functioned as
135/128 and as 16/15 (or in ma-grama, as 45/32 and as 64/45). That's about
19.6 cents. The other unison vector is large: the difference between sruti
#1 in the two gramas, 3^-9 * 5 or 68.7 cents. So the Fokker matrix is

-4    -2
-9     1

The inverse of this matrix is

-1/22        -1/11
-9/22         2/11

If we transform the set of lattice points with this matrix, we can define
a periodicity block within any 1 X 1 square of the transformed lattice.
The usual approach is to use the "unit square" from the origin to (1,1),
but of course we are free to translate this square wherever we want since
the result will still tile the plane with the unison vectors as
generators. So let us choose the square where the position along each
dimension is greater than -1/2 and less than or equal to 1/2, so the
origin will serve as a central note rather than a corner. Taking the
resulting 22 points

-5/11         1/11
-9/22         2/11
-4/11         3/11
-7/22         4/11
-3/11         5/11
-5/22        -5/11
-2/11        -4/11
-3/22        -3/11
-1/11        -2/11
-1/22        -1/11
0            0
1/22         1/11
1/11         2/11
3/22         3/11
2/11         4/11
5/22         5/11
3/11        -5/11
7/22        -4/11
4/11        -3/11
9/22        -2/11
5/11        -1/11
1/2           0

and transforming them back to the lattice (using the original Fokker
matrix, as explained here), we get
[~cents]

-1          -1         112
0          -1         814
1          -1         316
2          -1        1018
3          -1         520
-5           0          90
-4           0         792
-3           0         294
-2           0         996
-1           0         498
0           0           0
1           0         702
2           0         204
3           0         906
4           0         408
5           0        1110
-3           1         680
-2           1         182
-1           1         884
0           1         386
1           1        1088
2           1         590

in lattice form:

(All lattices here follow the "triangular" convention, and
ratios are indicated on all of them by their cents-values.)

680---182---884---386---1088--590
/ \   / \   / \   / \   / \   / \
/   \ /   \ /   \ /   \ /   \ /   \
90---792---294---996---498----0----702---204---906---408---1110
\   / \   / \   / \   / \   /
\ /   \ /   \ /   \ /   \ /
112---814---316---1018--520

Now of course all the properties of the block are preserved if we
transpose one (or more) note(s) by one of the unison vectors used to
create the block. This corresponds to distorting the edges of the block in
parallel as I've discussed before. Let's take (-3,1) (680.5 cents) and
lower it by the unison vector (-9,1) (68.7 cents) to obtain the note (6,0)
(612 cents):

182---884---386---1088--590
/ \   / \   / \   / \   / \
/   \ /   \ /   \ /   \ /   \
90---792---294---996---498----0----702---204---906---408---1110--612
\   / \   / \   / \   / \   /
\ /   \ /   \ /   \ /   \ /
112---814---316---1018--520

This is the scale of srutis according to most reputable sources, such as
S. Ramanathan, etc. etc.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Paul responded to me [private communication]:

Joe Monzo [in 5-Limit Implications of Ancient Indian Tuning]:

> Donald Lentz, in Tones and Intervals of Hindu Classical
> Music, gives a description of ancient Indian tuning as a
> series of "perfect 4ths" and "perfect 5ths", which would give
> a "Pythagorean" or 3-Limit system of 22 notes ("srutis")
> to the "octave". He then gives a tabulation of the ratios,
> but, interestingly, it includes 5-Limit ratios as replacements
> for some which should be 3-Limit.

> I would argue, however, that the theoretical implications
> of the ancient Indian tuning would allow even more 5-limit
> ratios than Lentz's description.

There may be a subtle reason why Lentz included particular 5-limit ratios
but not others that are suggested by schismatic equivalencies. I've
discussed these issues a bit in my paper and on the list, but it may be time
to revisit them with an eye toward possibly linking my periodicity-block
analysis of the srutis with your page here.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I'm glad you included my statement [the last paragraph above]
but it was really meant as an off-page question to you:
What ratios did Lentz himself actually include?

Here are the actual 5-limit ratios Lentz substituted for
some of the ones given in his tuning paradigm of 3-1...11,
given in the same lattice format Paul used above:

182---884---386---1088--590
/ \   / \   / \   / \   / \
/   \ /   \ /   \ /   \ /   \
90---792---294---996---498----0----702---204---906---408---1110
\   / \   / \   / \   / \   / \   /
\ /   \ /   \ /   \ /   \ /   \ /
610---112---814---316---1018--520

There is only one note different from either of Paul's lattices:
64/45 (= ~610 cents) is included here, instead of the 40/27
(= ~680 cents) in Paul's first lattice, and instead of the
729/512 (= 36 = ~612 cents) in his second one.

[Lentz 1961, p 5]
Three different small intervals called sruti are the basis for
the Indian tonal system.  The names used in this book for these
small intervals are those of the southern area of india, although
the names used in other areas could be substituted.  The size
of the interval is the important factor.  The intervals, listed
according to ratio, are:

1. 81/80. Pramana sruti (or comma diesus): equal to
22 cents or 5.4 savarts.  This is the difference between a
major and minor tone.  (9/8 ÷ 10/9 = 81/80.)

2. 25/24. Nyuna sruti: equal to 70 cents or 17.73 savarts.
This is the difference between a minor tone and a semitone.
(10/9 ÷ 16/15 = 25/24.)

3. 256/243. Purana sruti (or limma): equal to 90 cents
or 22.65 savarts.  This is the difference between a perfect fourth
and two major tones.  Other rare sruti, such as 27 cents,
have been considered but do not seem to be commonly used in India.

[Note that Lentz's use of "savarts" is the older meaning,
equivalent to "heptamerides".  More precise values are
~5.395031887, ~17.72876696, and ~22.63369171 heptamerides,
so Lentz's figure for the purana sruti is slightly incorrect.]

Lentz then describes in detail the tuning procedure of a series of
11 ascending '5ths' which are then 'octave'-reduced, then 11
ascending '4ths' which are then 'octave'-reduced [Lentz 1961, p 6].
Then he concludes this with:

[Lentz 1961, p 7]
When all the tones have been reduced to within the ambit of an
octave it will be seen that each tone derived from the cycle of
fifths is an 81/80 (Pramana sruti) higher than the contiguous
tone in the cycle of fourths.

Of course we tuning theorists know that things here are not
exactly as Lentz describes: the difference is not the syntonic comma
as Lentz says it is, but rather the Pythagorean comma.  But he
tabulates the scale of 22 srutis as given in the lattice above
and here makes no comment whatever about the schismatic differences
between them and the entirely Pythagorean system derived from the
tuning procedure he describes.

On page 13, Lentz also discusses the 68-cent sruti Paul used as
a unison vector, but it arises only in connection with notes outside
of his lattice that Lentz says are 'used only rarely'.

Then, he says:

[Lentz 1961, p 13]
Large ratios have been reduced [in Lentz's 'comparative chart],
such as 1024/729 to 45/32, or 729/512 to 64/45. ... The discrepancies
are so slight as to be negligible and do not affect the understanding
of the system.

For comparison, here's the pitch-set I derived in my own analysis of
Indian tuning, drawn in the same format as the others illustrated here:

182---884---386---1088--590--- 92---794
/ \   / \   / \   / \   / \   / \   /
/   \ /   \ /   \ /   \ /   \ /   \ /
294---996---498----0----702---204---906---408
/ \   / \   / \   / \   / \   / \   /
/   \ /   \ /   \ /   \ /   \ /   \ /
1108---610---112---814---316---1018--520

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I had originally called the above diagram a 'periodicity block',
but Paul Erlich corrected me and said that it was not.

I explained to Paul that I had determined the finity of this
pitch-set by the unison vectors which form the intervals of a
skhisma [= 3^8 * 5^1 = ~2.0 cents] and a diesis [= 5^-3 = ~41.1 cents].

Then I found that the determinant of this matrix is

8    1
0   -3

= 24 notes.  So I asked Paul to find the periodicity block
defined by this matrix, and here's his answer:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Paul Erlich, private communication:

By the way, using the exact same formalism on _your_ matrix

8    1
0   -3

notes -- one in the upper-left corner (680¢), and one in the
lower-right corner (22¢), making it a perfect parallelogram.
Without those two extra notes, it doesn't have any of the
properties of a periodicity block.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The periodicity block I derived is therefore (in cents):

680---182---884---386---1088--590--- 92---794
/ \   / \   / \   / \   / \   / \   / \   /
/   \ /   \ /   \ /   \ /   \ /   \ /   \ /
294---996---498----0----702---204---906---408
/ \   / \   / \   / \   / \   / \   / \   /
/   \ /   \ /   \ /   \ /   \ /   \ /   \ /
1108---610---112---814---316---1018--520----22

Here is a table showing it as a scale, with a pitch-height
graph of the notes:

Here is a graph of the step-sizes:

So now I'm wondering if perhaps those two 'rare' intervals mentioned
by Lentz can be fitted into my interpretation, so that it can be
demonstrated that Indian musicians have all along conceived of
the finity of their system as a 22-tone subset of my 24-tone
periodicity block.

In fact, when I first added this analysis to my book back around
1996, I intuitively felt that the 'block' of pitches should be
a perfect parallelogram of 24 notes (it was a rectangle in my old
diagram) as it is here.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Paul's response to this echoed precisely the observation I noticed
while making my original lattice in 1996:

If you wish, you can "explain" the reduction from 24 to 22 by
saying  that the 1/1 and 3/2 were the most important notes in
or something), and that the tones a comma away from those two
were discarded, because, for instance, if 1/1 and 3/2 formed a
drone, either of these 'rare' tones would sound very
out-of-tune against the drone.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Here is a response to the above paragraph which I received
in a private email:

[There are] some misinformed remarks concerning Indian
music.

Of course the exact pitch of a paricular tone in a particular raga has
always been a point of debate even among musicians. Also, much depends
on the tone-quality, the harmonic content. I study a very old style of
vocal  music through lessons with a great singer from a family that has
been keeping this tradition for 20 generations, and they are noted for
extreme meticulousness in intonation and timbre-control. But as their
manners of verbal expression, understanding, experiencing akouphenomena
differ so much from our own, it is not possible to share the
information. Westerners such as myself are in a different postion as
they can combine both approaches.

correct. It means 1st and 2nd most important tones in a certian raga,
which can be Sa and Pa (1st and 5th) but this depends on the character
of the raga. In raga Shri it is Pa and very low Re, in Multani the
raised 4th and high 7 and so fourth.

-Martin Spaink (Amsterdam)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

After Gene Edward Smith made an analysis of the (0 -3),(8 1)
Yahoo Tuning-Math List, message 2170 (Wed Dec 26, 2001 8:56am,
Subject: Microtemperament and scale structure).  Some
selected quotes:

>> ... the unison-vectors which are not tempered out are not
>> quarter-tones, but rather the syntonic comma 81:80 = (-4 4 -1)
>> = ~21.5 cents, and the diaschisma = (11 -4 -2) = ~19.55 cents.
>>
>> ...
>>
>> So the four step-sizes between degrees of the scale,
>> in ~cents, are:
>>
>>    90.225 _purana_ sruti
>>    70.672 _nyuna_ sruti
>>    21.506 } both of these are conflated
>>    19.553 } into the _pramana_ sruti
>>
>> and with an anomalous interval of ~92.179 cents between the
>> highest degree and the "8ve".
>>
>> If you didn't before, I think now you can see why Paul calls
>> this "not a well-behaved periodicity-block"... at any rate,
>> now *I* finally understand!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

From Paul Erlich, 2001.12.26:

I ended up convincing Gene that this does not in fact result in
quartertones. He then realized that his error was because of
something called "torsion". If you temper out the unison vectors,
you actually get 12-tone equal temperament, not 24-tone equal
temperament.

Another reason that this is not a well-behaved periodicity block
is that the 3:2, for example, is not subtended by the same number
of PB steps everywhere it occurs -- even if you only look at 3:2s
within the Indian Diatonic scale. Thus, I feel that your PB
interpretation of the Indian sruti system is quite poor, because the sruti
system prides itself on being able to represent all the 3:2s (at least
the commonly used ones) by the same number of srutis.

In fact, you might want to reconsider using the term "24-tone
periodicity block" on your webpage. As the block includes more than
one member of each equivalence class, or else must be seen using the
unmusical notion of torsion, it shouldn't be called a periodicity
block at all. A torsional block, perhaps.

[I have since taken up Paul's suggestion.  -monz]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Below is a post by Paul Erlich, referring to my
my periodicity-block analysis of the Indian 22-sruti tuning.

> Message #17:
>
> --- In tuning-math@y..., "monz"  wrote:
> >
> > --- In tuning-math@y..., "Paul Erlich"  wrote:
> >
> > http://groups.yahoo.com/group/tuning-math/message/14
> >
> > > --- In tuning-math@y..., graham@m... wrote:
> > >
> > > > Temper out the schisma from the periodicity block above.
> > > > You end up with a 24-note schismic scale. No way can that
> > > > have two step sizes!
> > > >
> > > > That looks like a refutation with the definitions I have.
> > >
> > > I think the problem is that, as you said before, the scale
> > > really has 12 pitch classes, not 24, due to the syntonic comma
> > > squared vanishing.
> > >
> > >
> >
> > Can you guys please illustrate all this with lattices and
>  > other tables and diagrams?
>
> Hi Monz.
>
> What we're discussing here is the 24-tone periodicity block
> you came up with to derive the 22-shruti system of Indian music.
>
> The unison vectors of that periodicity block are the schisma
> and the diesis.
>
> As you can see, half the notes in that periodicity block
> differ from the other half by a syntonic comma. You can
> see that either in the lattice diagram or in the list of ratios.
>
> But here's the rub. If the schisma is a unison vector, and
> the diesis is a unison vector, then the schisma+diesis
> (multiply the ratios) is a unison vector. But you can verify
> that the ratio for the schisma times the ratio for the diesis
> is the square of the ratio of the syntonic comma. In other
> words, it represents _two_ syntonic commas.
>
> Now, if _two_ of anything is a unison vector, then the
> thing itself must be either a unison or a half-octave.
> But in your scale, the syntonic comma separates pairs
> of adjacent pitches, so it's clearly not acting as a
> half-octave. So it must be a unison. In a sense, it's
> logically contradictory to say that the schisma and diesis
> are both unison vectors while maintaining syntonic comma
> differences in the scale. The scale is "degenerate", or
> perhaps more accurately, it's a "double exposure" -- it
> seems to have twice as many pitch classes than it really has.

Here is a diagram illustrating what Paul is saying.
The diesis and skhisma are the two unison-vectors which
define "my" 24-tone periodicity-block.  If they are
added together (i.e., multiply their ratios), they produce
a third unison-vector which is the sum of two syntonic commas.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

REFERENCE

Lentz, Donald.  1961.
Tones and Intervals of Hindu Classical Music.

```

Here's my paper on Indian tuning.

also see Paul's A Gentle Introduction to Fokker Periodicity Blocks

Updated: 2002.2.5, 2001.12.26, 2000.10.20, 2000.1.24, 2000.1.20
By Joe Monzo

 For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.