Re: Paul's Message #17

Paul,

In tuning-math message #17, in describing your analysis of Monz's "24-tone periodicity block you" (Monz) "came up with to derive the 22-shruti system of Indian music" you stated:

< 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.>

I have been calculating the tonal generators and unison vectors for the 12-tone scale made up of the interval ratios 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, and 15/8 (which is a subset of Monz's periodicity block related to the Indian shruti system (1996), and which consists of the 12 interval ratios which are centered (around the mid-point between the 1/1 and 3/2 interval ratios) in a 3x4 rectangle of intervals within Monz's matrix of interval ratios found on page 131 of his "JustMusic: A New Harmony", 1996. The commatic unison vectors I get are syntonic (81/80), diaschismic (2048/2025), and a second syntonic (81/80), representing an (apparently) "ill-conditioned" situation for linear algebraic analysis (as a 5-limit system containing 3, as opposed to 2, commatic unison vectors, thus leaving an opening for a "chromatic" unison vector equal to the minimal step size in the primal basis of 2, 3, and 5.

I have read (and partially absorbed) your "The Indian Sruti System as a Periodicity Block" paper of 1999. My questions are:

(1) Could you explain further the meaning and implications of your above statement, "Now, if _two_ of anything is a unison vector, then the thing itself must be either a unison or a half-octave."; and

(2) What can be done in tempering such a scale in order to reduce the number of commatic unison vectors to 2, instead of 3 consisting of a "pair" of syntonic commas? I recognize that there may be no one simple answer to this question, but would appreciate your thoughts in general regarding what you might endeavor to choose to do in such a case as this.

Sincerely, J Gill

--- In tuning-math@y..., J Gill <JGill99@i...> wrote:
> Paul,
>
> In tuning-math message #17, in describing your analysis of Monz's
"24-tone
> periodicity block you" (Monz) "came up with to derive the 22-shruti
system
> of Indian music" you stated:
>
> < 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.>
>
> I have been calculating the tonal generators and unison vectors for
the
> 12-tone scale made up of the interval ratios 1/1, 16/15, 9/8, 6/5,
5/4,
> 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, and 15/8 (which is a subset of
Monz's
> periodicity block related to the Indian shruti system (1996), and
which
> consists of the 12 interval ratios which are centered (around the
mid-point
> between the 1/1 and 3/2 interval ratios) in a 3x4 rectangle of
intervals
> within Monz's matrix of interval ratios found on page 131 of his
> "JustMusic: A New Harmony", 1996. The commatic unison vectors I get
are
> syntonic (81/80), diaschismic (2048/2025), and a second syntonic
(81/80),
> representing an (apparently) "ill-conditioned" situation for linear
> algebraic analysis (as a 5-limit system containing 3, as opposed to
2,
> commatic unison vectors,

Hmm . . . have you read the _Gentle Introduction to Periodicity
Blocks_, including the "excursion" (the third webpage in the series)?
It seems to me you may be confused about something.

> (1) Could you explain further the meaning and implications of your
above
> statement, "Now, if _two_ of anything is a unison vector, then the
thing
> itself must be either a unison or a half-octave.";

Well, if two of something is a unison vector, then two of something is
close to 0 cents, or close to 1200 cents, or close to 2400 cents, etc.
(assuming the usual situation where the octave is the interval of
equivalence). Then the something itself must be close to 0 cents, or
close to 600 cents, or close to 1200 cents (which is equivalent to 0
cents), etc. I.e. it must be either a unison of a half-octave.

and
>
> (2) What can be done in tempering such a scale in order to reduce
the
> number of commatic unison vectors to 2, instead of 3 consisting of a
"pair"
> of syntonic commas?

Tempering does not decrease the number of unison vectors, though I
still think you're confused about something that's leading you to
count three unison vectors. 5-limit space with octave-equivalence is
two-dimensional, so there can only be two different independent unison
vectors for any periodicity blocks within that space (though
additional, non-independent unison vectors, formed by "adding" or
"subtracting" the two you start with, can be found, and in fact are
relevant in many situations -- both the "excursion" and _The Forms Of
Tonality_ give clear (I hope) examples of this.

> I recognize that there may be no one simple
answer to
> this question, but would appreciate your thoughts in general
regarding what
> you might endeavor to choose to do in such a case as this.
>
Perhaps you could supply some crude diagrams to help clarify what
you're seeing? It would help me a lot in trying to answer your
questions.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Hmm . . . have you read the _Gentle Introduction to Periodicity
> Blocks_, including the "excursion" (the third webpage in the
series)?

Yes, but my "absorption factor" remains "less than unity".
I will be re-reading it, with particular emphasis on your
periodicity block which includes those very interval ratios
(in the 12-tone JI system of which you ask is Ramos, and
Monz points out he believes it was from De Caus, with Manuel
Op de Coul pointing out it also Ellis' "Harmonic Duodene",
and your "excursion" in Part 3, as you suggested.

> It seems to me you may be confused about something.

> > Tempering does not decrease the number of unison vectors,

one *might* add, "but does alter the individual values of those
unison vectors, whose number is determined by the prime limit of the
system"???

> though I
> still think you're confused about something that's leading you to
> count three unison vectors. 5-limit space with octave-equivalence
is
> two-dimensional, so there can only be two different independent
unison
> vectors for any periodicity blocks within that space (though
> additional, non-independent unison vectors, formed by "adding" or
> "subtracting" the two you start with, can be found, and in fact are
> relevant in many situations -- both the "excursion" and _The Forms
Of
> Tonality_ give clear (I hope) examples of this.

I get the feeling that I may be simply be missing a fundamental rule
in the process of deriving unison vectors. I will run you through
what I have done, for your inspection and comment, below:

"octave-reduced" interval ratios ordered in ascending numerical value
are: 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8.

"tonal generators" (with redundancies omitted, and in ascending
value) are: 25/24, 135/128, 16/15, 27/25.

"unison vectors" (as I imagine them...with reduncies RETAINED) are:
81/80, 2048/2025, and 81/80.

Could it be that I am simply unaware that redundant results
for "unison vectors" (as in the case of the "tonal generators") are
to be OMITTED? Thus, I would report 2, rather than 3, resultant
vectors emerging from the above calculation...???

Sincerely, J Gill

--- In tuning-math@y..., "J Gill" <JGill99@i...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Hmm . . . have you read the _Gentle Introduction to Periodicity
> > Blocks_, including the "excursion" (the third webpage in the
> series)?
>
> Yes, but my "absorption factor" remains "less than unity".
> I will be re-reading it, with particular emphasis on your
> periodicity block which includes those very interval ratios
> (in the 12-tone JI system of which you ask is Ramos, and
> Monz points out he believes it was from De Caus, with Manuel
> Op de Coul pointing out it also Ellis' "Harmonic Duodene",
> and your "excursion" in Part 3, as you suggested.
>
> > It seems to me you may be confused about something.
>
> > > Tempering does not decrease the number of unison vectors,
>
>
> one *might* add, "but does alter the individual values of those
> unison vectors, whose number is determined by the prime limit of the
> system"???

Exactly.

>
> I get the feeling that I may be simply be missing a fundamental rule
> in the process of deriving unison vectors. I will run you through
> what I have done, for your inspection and comment, below:
>
> "octave-reduced" interval ratios ordered in ascending numerical value
> are: 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8.
>
> "tonal generators" (with redundancies omitted, and in ascending
> value) are: 25/24, 135/128, 16/15, 27/25.

I'm not familiar with this concept of "tonal generators". Perhaps a term that would make more
sense to me would be "step sizes"?
>
> "unison vectors" (as I imagine them...with reduncies RETAINED) are:
> 81/80, 2048/2025, and 81/80.

How are you obtaining that?
>
> Could it be that I am simply unaware that redundant results
> for "unison vectors" (as in the case of the "tonal generators") are
> to be OMITTED?

I'm pretty sure, if I'm understanding you correctly, that the answer is "yes". However, for the
12-tone scale you mention, I'm surprised you didn't find 128/125 as one of the unison vectors.

> Thus, I would report 2, rather than 3, resultant
> vectors emerging from the above calculation...???

There should be 2 . . . although 81/80, 125/128, and 2048/2025 are three unison vectors for
this scale, each of these is just a linear combination of the other two, as can be seen very easily
from the vector notation of these three intervals:

81/80 = (4 -1)
128/125 = (0 -3)
2048/2025 = (-4 -2)

So clearly,

81/80 = 128/125 "-" 2048/2025,
128/125 = 81/80 "+" 2048/2025,
2048/2025 = 128/125 "-" 81/80,

where "+" is vector addition but multiplication of the ratios,
and "-" is vector subtraction but division of the ratios.