Re: [tuning-math] Digest Number 124

I've had to skip quite a bit on these lists recently. Is there anything
already written that someone could point me to, that would help me start
to understand the couple of things I mention below?

Graham wrote, responding to Gene:

> > 5-limit
> >
> > 16/15, 25/24, 81/80
>
> And these are the usual intervals for defining a 5-limit diatonic
> scale!

I know where these intervals crop up in comparing combinations of
intervals in the usual J.I. scale to one another, but how, precisely, do
the three intervals "define" the scale? Are they like a "basis" for
5-limit diatonic space, and I could just as well take three other linearly
independent intervals like 9:8 10:9 4:3 to define the scale, in a similar
way to however Gene's three do?

> > 11-limit
> >
> > 100/99, 121/120, 441/440, 3025/3024, 9801/9800
>
> These give
>
> [ 46 26 8 -14 15]
> [ 73 41 13 -22 24]
> [107 60 19 -32 35]
> [129 73 23 -39 42]
> [159 90 28 -48 52]
>
> So all bar the largest define 46-equal.

How do you obtain the matrix from those intervals? What, then, can you
read off of it (i.e. what does it tell you)? What other operations can you
do with the matrix?

Thanks for any help, however quick and concise --Jon Wild

--- In tuning-math@y..., jon wild <
wild@f...> wrote:
>
> I've had to skip quite a bit on these lists recently. Is there anything
> already written that someone could point me to, that would help me start
> to understand the couple of things I mention below?
>
> Graham wrote, responding to Gene:
>
> > > 5-limit
> > >
> > > 16/15, 25/24, 81/80
> >
> > And these are the usual intervals for defining a 5-limit diatonic
> > scale!
>
> I know where these intervals crop up in comparing combinations of
> intervals in the usual J.I. scale to one another, but how, precisely, do
> the three intervals "define" the scale? Are they like a "basis" for
> 5-limit diatonic space, and I could just as well take three other linearly
> independent intervals like 9:8 10:9 4:3 to define the scale, in a similar
> way to however Gene's three do?

The two intervals 25:24 and 81:80
"define" the diatonic scale in a
very precise sense. Have you
studied the "Gentle Introduction
to Periodicity Blocks"? If that
doesn't help, perhaps we should
meet for coffee and I'll give you a
copy of my paper, _The Forms of
Tonality_.

--- In tuning-math@y..., jon wild <wild@f...> wrote:

> > > 16/15, 25/24, 81/80

> > And these are the usual intervals for defining a 5-limit diatonic
> > scale!

> I know where these intervals crop up in comparing combinations of
> intervals in the usual J.I. scale to one another, but how,
precisely, do
> the three intervals "define" the scale?

These three intervals and 9/8, 10/9, 16/15 are very closely related.
We have 10/9 = 25/24 16/15 and 9/8 = 81/80 10/9 = 81/80 25/24 16/15.
Looked at in matrix form, we have an upper diagonal transformation
matrix:

[1 1 1]
[0 1 1]
[0 0 1]

Transforming the basis (81/80)^a (25/24)^b (16/15)^c, which
represents everything in the 5-limit, to another basis
(9/8)^a' (10/9)^b' (16/15)^c' which does the same (as we can see by
the fact that the transformation is unimodular, with determinant 1.)
The first system has jargon wherein 81/80 and 25/24 are "commatic
unison vectors" and 16/15 is a "chromatic unison vector" in a
situation where we are seeking a 7-note "periodiity block" scale; the
7 can be deduced from

[-4 4 -1]
det [-3 -1 2] = 7a + 11b + 16c,
[a b c]

where we have replaced the largest of the three "unison" intervals
with indeterminates a, b, c. The result represents the 7-equal
division of the octave, or "7-tet = 7-et"; it has 7 steps in an
octave, takes 11 steps to approximate 3 and 16 steps to approximate
5. The scales we get are approximations to this 7-et, and hence are
reasonable as scales. We can always pass in this way from a step
system of representation like 9/8, 10/9, 16/15, to a comma system
like 16/15, 25/24, 81/80, and back; the comma system is more useful
in a number of ways and tends to be taken as fundamental here, though
most people focus on the steps instead.

Are they like a "basis" for
> 5-limit diatonic space, and I could just as well take three other
linearly
> independent intervals like 9:8 10:9 4:3 to define the scale, in a
similar
> way to however Gene's three do?

No; to say the scale is "defined" by the fact that 16/15 is
a "chromatic unison vector" (something of a misnomer, I fear) and
81/80 and 25/24 are "commatic unison vectors" tells us how to
construct "periodicity block" scales in a way even more tightly
controlled than "defining" a system by requiring it to be constructed
from steps of size 9/8, 10/9 and 16/15; the two ideas are closely
related but are not the same and do not always lead to the same
results.

>
> > > 11-limit
> > >
> > > 100/99, 121/120, 441/440, 3025/3024, 9801/9800
> >
> > These give
> >
> > [ 46 26 8 -14 15]
> > [ 73 41 13 -22 24]
> > [107 60 19 -32 35]
> > [129 73 23 -39 42]
> > [159 90 28 -48 52]
> >
> > So all bar the largest define 46-equal.

> How do you obtain the matrix from those intervals?

He factored the intervals I gave into primes, wrote them out as row
vectors (so that 100/99 = 2^2 3^(-2) 5^2 7^0 11^(-1) becomes
[2 -2 2 0 -1]) and then took the matrix inverse of the resulting
square matrix. The columns of this matrix can be thought of as ets,
applied to the rows they are the ets that represent one of the
intervals by a step (downward in one case, 14) and the rest by a
unison.

What, then, can you
> read off of it (i.e. what does it tell you)?

For one thing, we can look at the linear span of subsets of the
columns, which gives us ets with certain properties, having some
elements as commas in common--for instance 46+26=72, and adding the
first two columns together gives us the 72-et; like 46 and 26 it will
have 441/440, 3025/3024 and 9801/9800 in its kernel (or nullspace.)
This tells us we can "temper out" these intervals, leading for
instance to 46 and 26 tone subsets of the 72 et, and tells us how to
construct such temperings. We can also immediately see how to
construct a 46 tone periodicity block, other blocks and tempered
scales, and so forth.

What other operations can you
> do with the matrix?

Stick around and I'm sure you'll see more! Maybe I'll do that 72-et
calculation for Joseph Pehrson's benefit.

jon wild wrote:

> I know where these intervals crop up in comparing combinations of
> intervals in the usual J.I. scale to one another, but how, precisely,
> do
> the three intervals "define" the scale? Are they like a "basis" for
> 5-limit diatonic space, and I could just as well take three other
> linearly
> independent intervals like 9:8 10:9 4:3 to define the scale, in a
> similar
> way to however Gene's three do?

You could use 9:8, 10:9 and 16:15. I'd prefer not to include 4:3 because
it means you can't get the diatonic scale by only adding intervals.
(That's *only* adding, not subtracting or inverting and then adding.)
(Also means "adding intervals" is multiplying ratios.) The thing about
16:15, 25:24 and 81:80 is that you have a "diatonic" step, a "chromatic"
step and a comma.

> > > 11-limit
> > >
> > > 100/99, 121/120, 441/440, 3025/3024, 9801/9800
> >
> > These give
> >
> > [ 46 26 8 -14 15]
> > [ 73 41 13 -22 24]
> > [107 60 19 -32 35]
> > [129 73 23 -39 42]
> > [159 90 28 -48 52]
> >
> > So all bar the largest define 46-equal.
>
> How do you obtain the matrix from those intervals? What, then, can you
> read off of it (i.e. what does it tell you)? What other operations can
> you
> do with the matrix?

First step is to convert the intervals to matrix form. 100/99 is [2 -2 2
0 -1], 121:120 is (-3 -1 -1 0 2] and so on. The full matrix is

[ 2 -2 2 0 -1]
[-3 -1 -1 0 2]
[-3 2 -1 2 -1]
[-4 -3 2 -1 2]
[-3 4 -2 -2 2]

Then you take the matrix inverse. The first column of the result defines
46-equal. So the top entry tells you there are 46 steps to the octave.
The next one that there are 73 steps to the approximation of 3:1. Then
that there are 107 steps to the approximation of 5:1, and so on. This is
the temperament you get by tempering out all the unison vectors except the
first one, [2 -2 2 0 -1] or 100/99.

The second column defines 26-equal. You get that by tempering out
everything except 121:120. Temper out everything except 121:120 and
100:99 and you have a linear temperament covering 26- and 46-equal, and
therefore 72-equal (because 26+46=72). A while back on the big list, I
probably suggested a notation for this temperament, but I can't remember
myself.

Oh, and it's second on my list of 11-limit temperaments

18/59

basis:
(0.5, 0.152673565758)

mapping:
([2, 0], ([5, 8, 5, 6], [-6, -11, 2, 3]))

primeApprox:
([72, 46], [(114, 73), (167, 107), (202, 129), (249, 159)])

highest interval width: 15
notes required: 31
highest error: 0.002008 (2.409 cents)

Graham

--- In tuning-math@y..., genewardsmith@j... wrote:

> The first system has jargon wherein 81/80 and 25/24 are "commatic
> unison vectors" and 16/15 is a "chromatic unison vector" in a
> situation where we are seeking a 7-note "periodiity block" scale;

Something's wrong here . . . in a 7-tone PB, specificially the
diatonic scale, 81:80 is the commatic unison vector, 25:24 is a
chromatic unison vector, and 16:15 is not a unison vector at all, but
a "step vector".

> No; to say the scale is "defined" by the fact that 16/15 is
> a "chromatic unison vector" (something of a misnomer, I fear) and
> 81/80 and 25/24 are "commatic unison vectors"

Misnomer because it's incorrect!

Gene -- do you have any response here? Have you been misunderstanding
something all along?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > The first system has jargon wherein 81/80 and 25/24 are "commatic
> > unison vectors" and 16/15 is a "chromatic unison vector" in a
> > situation where we are seeking a 7-note "periodiity block" scale;
>
> Something's wrong here . . . in a 7-tone PB, specificially the
> diatonic scale, 81:80 is the commatic unison vector, 25:24 is a
> chromatic unison vector, and 16:15 is not a unison vector at all,
but
> a "step vector".
>
> > No; to say the scale is "defined" by the fact that 16/15 is
> > a "chromatic unison vector" (something of a misnomer, I fear) and
> > 81/80 and 25/24 are "commatic unison vectors"
>
> Misnomer because it's incorrect!

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

> Gene -- do you have any response here? Have you been
misunderstanding
> something all along?

It's the same thing which made it so hard for me to get the
terminology in the first place--in the tempered situation, we have
*two* scale-step vals. For instance, we could have h12 and h7, and
then we would have h12(16/15) = h12(25/24) = 1, but h7(16/15) = 1 and
h7(25/24) = 0. So 25/24 is a "unison" according to h7 but it is also
a "step" on the piano keyboard--according to h12.

We have (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. For the purpose of
constructing a JI scale approximating to h7, 16/15 is a step and
25/24 is a comma. If we temper out 81/80, we can pick an et of the
form n h7 + m h5, and then in that et 16/15 will be n intervals and
25/24 will be m intervals, but in the scale 16/15 is still one step
and 25/24 not an allowed step; it is still a unison so far as h7 is
concerned.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Gene -- do you have any response here? Have you been
> misunderstanding
> > something all along?
>
> It's the same thing which made it so hard for me to get the
> terminology in the first place--in the tempered situation, we have
> *two* scale-step vals. For instance, we could have h12 and h7, and
> then we would have h12(16/15) = h12(25/24) = 1, but h7(16/15) = 1
and
> h7(25/24) = 0. So 25/24 is a "unison" according to h7 but it is
also
> a "step" on the piano keyboard--according to h12.

What if we substituted the word "second" instead of "step"?

Better yet, can we avoid bringing 12 into this at all? I see no
reason it should be brought in.

> We have (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. For the purpose
of
> constructing a JI scale approximating to h7, 16/15 is a step and
> 25/24 is a comma. If we temper out 81/80, we can pick an et of the
> form n h7 + m h5, and then in that et 16/15 will be n intervals and
> 25/24 will be m intervals, but in the scale 16/15 is still one step
> and 25/24 not an allowed step; it is still a unison so far as h7 is
> concerned.

Right . . .

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

> Better yet, can we avoid bringing 12 into this at all? I see no
> reason it should be brought in.

Since we aren't tempering, leaving it out is the thing to do.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Better yet, can we avoid bringing 12 into this at all? I see no
> > reason it should be brought in.
>
> Since we aren't tempering, leaving it out is the thing to do.

Well . . . 81:80 being a commatic unison vector sort of implies
tempering . . . but in no way selects 12 over other meantones like
19, 31, 43, 55, or 74.

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

> Well . . . 81:80 being a commatic unison vector sort of implies
> tempering . . . but in no way selects 12 over other meantones like
> 19, 31, 43, 55, or 74.

Those are all of the form n h7 + m h5, which is what I was talking
about.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Well . . . 81:80 being a commatic unison vector sort of implies
> > tempering . . . but in no way selects 12 over other meantones
like
> > 19, 31, 43, 55, or 74.
>
> Those are all of the form n h7 + m h5, which is what I was talking
> about.

Gene, I really liked your n h7 + m h5 discussion!