Atomic projection

We can define what I've called a "notation"--a pair of unimodular
matries inverse to each other, one of which is a row matrix of monzos
and another a column matrix of vals--for atomic in various ways. The
simplest is

[|-67 35 5>, |-15 8 1>, |161 -84 -12>]
[<12 19 28|, <0 -1 7|, <5 8 11|]

While the most common way to project intervals and vals is to toss
octaves, we can do an atomic projection for ets by tossing result of
mapping the atom with an et, and an atomic projection for intervals by
ignoring the results of the <5 8 11| val. Below I give 5-limit ets and
intervals in the atomic coordinates; leaving off the last coordinate
does the projection. The interval |-67 35 5> is of course
approximately 100 cents; more precisely, 99.9936 cents.

Just as with projecting away the octave, we can plot the results of
projecting away the atom on a plane diagram; anyone who wants to
ponder the results of that is welcome to do so. Of course we get lines
on both kinds of diagrams, with either ets or intervals as points; for
example 55-43-31-19 is [0 -1 11]-[-1 -1 11]-[-2 -1 11]-[-3 -1 11].

5-limit ets in atomic

1 <13 3 -31|
2 <-4 -1 10|
3 <9 2 -21|
4 <-13 -3 32|
5 <5 1 -11|
6 <18 4 -42|
7 <-4 -1 11|
8 <14 3 -32|
9 <-8 -2 21|
10 <5 1 -10|
12 <1 0 0|
15 <10 2 -21|
19 <-3 -1 11|
22 <6 1 -10|
31 <-2 -1 11|
34 <7 1 -10|
46 <8 1 -10|
53 <4 0 1|
65 <5 0 1|
84 <2 -1 12|
87 <11 1 -9|
99 <12 1 -9|
118 <9 0 2|
152 <16 1 -8|
171 <13 0 3|
205 <20 1 -7|
236 <18 0 4|
289 <22 0 5|
323 <29 1 -5|
441 <38 1 -3|
559 <47 1 -1|
612 <51 1 0|
730 <60 1 2|

5-limit commas in atomic

|161 -84 -12> [0 0 1]
|71 -99 37> [7 -358 -30]
|-90 -15 49> [7 -358 -31]
|-17 62 -35> [-6 307 26]
|144 -22 -47> [-6 307 27]
|-107 47 14> [1 -51 -5]
|54 -37 2> [1 -51 -4]
|-36 -52 51> [8 -409 -35]
|37 25 -33> [-5 256 22]
|-53 10 16> [2 -102 -9]
|91 -12 -31> [-4 205 18]
|1 -27 18> [3 -153 -13]
|-16 35 -17> [-3 154 13]
|38 -2 -15> [-2 103 9]
|-52 -17 34> [5 -255 -22]
32805/32768 [0 1 0]
19073486328125/19042491875328 [3 -152 -13]
6115295232/6103515625 [-2 104 9]
1224440064/1220703125 [-2 105 9]
1600000/1594323 [1 -48 -4]
15625/15552 [1 -47 -4]
2109375/2097152 [1 -46 -4]
393216/390625 [-1 57 5]
78732/78125 [-1 58 5]
2048/2025 [0 10 1]
81/80 [0 11 1]
3125/3072 [1 -36 -3]
128/125 [0 21 2]
250/243 [1 -26 -2]
648/625 [0 32 3]
25/24 [1 -15 -1]
135/128 [1 -4 0]
16/15 [1 6 1]
27/25 [1 17 2]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> We can define what I've called a "notation"--a pair of unimodular
> matries inverse to each other, one of which is a row matrix of monzos
> and another a column matrix of vals--for atomic in various ways. The
> simplest is
>
> [|-67 35 5>, |-15 8 1>, |161 -84 -12>]
> [<12 19 28|, <0 -1 7|, <5 8 11|]

It doesn't give the atomic mapping, but if we only want the projection
a much simpler choice would be 135/128 in place of |-67 35 5>. This
gives us

[|-7 3 1>, |-15 8 1>, |161 -84 -12>]
[<12 19 28|, <48 77 105|, <5 8 11|]

It is interesting that 135/128 and the schisma allow us to represent
the 5-limit up to atomic equivalence. We can also relate this to an
Agmon/meantone system, since <48 77 105| = 11<5 8 11|-<7 11 16|. Since
meantone is a*h12 + b*h7, it is a*[1,0,0]+b*[0,-1,11] =
[a,-b,11b] in these coordinates.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> We can define what I've called a "notation"--a pair of unimodular
> matries inverse to each other, one of which is a row matrix of
monzos
> and another a column matrix of vals--for atomic in various ways. The
> simplest is
>
> [|-67 35 5>, |-15 8 1>, |161 -84 -12>]
> [<12 19 28|, <0 -1 7|, <5 8 11|]

* * * * *
"5-limit commas in atomic" checks out beautifully, but "5-limit ets
in atomic" only works for the initial value (number of steps for 2,
or the octave). Is this intentional?

>
> While the most common way to project intervals and vals is to toss
> octaves, we can do an atomic projection for ets by tossing result of
> mapping the atom with an et, and an atomic projection for intervals
by
> ignoring the results of the <5 8 11| val.

* * * * *
Would it be possible to demonstrate these two cases? (tossing the
result of mapping the atom with an et, and 2) ignoring the results of
the <5 8 11| value?

Below I give 5-limit ets and
> intervals in the atomic coordinates; leaving off the last coordinate
> does the projection. The interval |-67 35 5> is of course
> approximately 100 cents; more precisely, 99.9936 cents.

Okay, I guess.
>
> Just as with projecting away the octave, we can plot the results of
> projecting away the atom on a plane diagram; anyone who wants to
> ponder the results of that is welcome to do so. Of course we get

* * * * *
I'd like to ponder the results, if I could only understand them. What
exactly is projecting away the atom on a plane diagram?

lines
> on both kinds of diagrams, with either ets or intervals as points;
for
> example 55-43-31-19 is [0 -1 11]-[-1 -1 11]-[-2 -1 11]-[-3 -1 11].
>
>
> 5-limit ets in atomic
>
> 1 <13 3 -31|
> 2 <-4 -1 10|
> 3 <9 2 -21|
> 4 <-13 -3 32|
> 5 <5 1 -11|
> 6 <18 4 -42|
> 7 <-4 -1 11|
> 8 <14 3 -32|
> 9 <-8 -2 21|
> 10 <5 1 -10|
> 12 <1 0 0|
> 15 <10 2 -21|
> 19 <-3 -1 11|
> 22 <6 1 -10|
> 31 <-2 -1 11|
> 34 <7 1 -10|
> 46 <8 1 -10|
> 53 <4 0 1|
> 65 <5 0 1|
> 84 <2 -1 12|
> 87 <11 1 -9|
> 99 <12 1 -9|
> 118 <9 0 2|
> 152 <16 1 -8|
> 171 <13 0 3|
> 205 <20 1 -7|
> 236 <18 0 4|
> 289 <22 0 5|
> 323 <29 1 -5|
> 441 <38 1 -3|
> 559 <47 1 -1|
> 612 <51 1 0|
> 730 <60 1 2|
>
>
> 5-limit commas in atomic
>
> |161 -84 -12> [0 0 1]
> |71 -99 37> [7 -358 -30]
> |-90 -15 49> [7 -358 -31]
> |-17 62 -35> [-6 307 26]
> |144 -22 -47> [-6 307 27]
> |-107 47 14> [1 -51 -5]
> |54 -37 2> [1 -51 -4]
> |-36 -52 51> [8 -409 -35]
> |37 25 -33> [-5 256 22]
> |-53 10 16> [2 -102 -9]
> |91 -12 -31> [-4 205 18]
> |1 -27 18> [3 -153 -13]
> |-16 35 -17> [-3 154 13]
> |38 -2 -15> [-2 103 9]
> |-52 -17 34> [5 -255 -22]
> 32805/32768 [0 1 0]
> 19073486328125/19042491875328 [3 -152 -13]
> 6115295232/6103515625 [-2 104 9]
> 1224440064/1220703125 [-2 105 9]
> 1600000/1594323 [1 -48 -4]
> 15625/15552 [1 -47 -4]
> 2109375/2097152 [1 -46 -4]
> 393216/390625 [-1 57 5]
> 78732/78125 [-1 58 5]
> 2048/2025 [0 10 1]
> 81/80 [0 11 1]
> 3125/3072 [1 -36 -3]
> 128/125 [0 21 2]
> 250/243 [1 -26 -2]
> 648/625 [0 32 3]
> 25/24 [1 -15 -1]
> 135/128 [1 -4 0]
> 16/15 [1 6 1]
> 27/25 [1 17 2]

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > We can define what I've called a "notation"--a pair of unimodular
> > matries inverse to each other, one of which is a row matrix of
> monzos
> > and another a column matrix of vals--for atomic in various ways. The
> > simplest is
> >
> > [|-67 35 5>, |-15 8 1>, |161 -84 -12>]
> > [<12 19 28|, <0 -1 7|, <5 8 11|]
>
> * * * * *
> "5-limit commas in atomic" checks out beautifully, but "5-limit ets
> in atomic" only works for the initial value (number of steps for 2,
> or the octave). Is this intentional?

I said "vals", not "ets"; all of them are vals.

>
> >
> > While the most common way to project intervals and vals is to toss
> > octaves, we can do an atomic projection for ets by tossing result of
> > mapping the atom with an et, and an atomic projection for intervals
> by
> > ignoring the results of the <5 8 11| val.
>
> * * * * *
> Would it be possible to demonstrate these two cases? (tossing the
> result of mapping the atom with an et, and 2) ignoring the results of
> the <5 8 11| value?

If we set semitone = 2^(1/12) and schisma = (648/625)^(1/32), then the
atomic projection for intervals leads to a tempering, which is quite
different than octave equivalence classes. Nevertheless, formally they
look the same--I take the results of applying <12 19 28| and <0 1 -7|
and ignore <5 8 11|, which corresponds to ignoring <1 0 0| when we do
octave classes. To get the tempering, take semitone^a schisma^b, where
a is the result of applying <12 19 28| to the interval, and b is the
result of applying <0 1 -7|.

The projection for vals works so long as the val sends the atom to
zero--as, for instance, 12-et, 612-et, 4296-et. Otherwise the best you
can do is an approximation, which gives floating point results which
round off, so long as we don't go too far afield, to the correct et
value. You can also toss the result of applying the val to an atom,
but the results won't make a whole lot of sense. So, for instance, 612
gives the semitone a value of 51 and the schisma a value of 1, so the
projected et is <51 1|. By comparison <22/12, 1/20| works, more or
less, like 22, but this isn't algebra, it's an approximation which is
only valid after rounding and in a restricted sense.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

By comparison <22/12, 1/20| works, more or
> less, like 22, but this isn't algebra, it's an approximation which is
> only valid after rounding and in a restricted sense.

<22/12 1/16| might be better:

(22/12)<12 19 28| + (1/16)<0 1 -7| = <22 34.896 50.896|

If you round off, it has a limited ability to act like 22; what good
this is, if any, is another question.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> <22/12 1/16| might be better:
>
> (22/12)<12 19 28| + (1/16)<0 1 -7| = <22 34.896 50.896|
>
> If you round off, it has a limited ability to act like 22; what good
> this is, if any, is another question.

There's another approach, which would work in any practical case,
which is to lift the atomic projection back to the 5-limit first. This
becomes easier to do the more accurate the temperament, since the
lifting involves finding the preimage value with the least height. In
the case of atomic, for any reasonable situation there will be only
one lifted value which makes sense, because the numerator and
denominator of the atom is so large.

For instance, 27/25 is |1 17> in terms of the atomic projection,
meaning one semitone of st = |-65 35 5> and 17 schismas (32805/32768.)
If we look at st * schisma^17 * atom^i for various i, only i=2 leads
to a value which does not have huge numerators and denominators, so we
lift |1 17> easily to 27/25. We can likewise lift anything with
numerators and denominators less than 48 digits by this means. Once
lifted, of course, we can do anything we like with it, such as find
out its value in 22-equal.

For less accurate temperaments, or higher limit temperaments, this
will of course not work in quite this strong a way. For instance
ennealimmal, with commas of 2401/2400 and 4375/4374, can lift products
of (27/25)^a (21/20)^b to the 7-limit so long as the numerator and
denominator are three digits or less.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > We can define what I've called a "notation"--a pair of
unimodular
> > > matries inverse to each other, one of which is a row matrix of
> > monzos
> > > and another a column matrix of vals--for atomic in various
ways. The
> > > simplest is
> > >
> > > [|-67 35 5>, |-15 8 1>, |161 -84 -12>]
> > > [<12 19 28|, <0 -1 7|, <5 8 11|]
> >
> > * * * * *
> > "5-limit commas in atomic" checks out beautifully, but "5-limit
ets
> > in atomic" only works for the initial value (number of steps for
2,
> > or the octave). Is this intentional?
>
> I said "vals", not "ets"; all of them are vals.

Your heading is "5-limit ets in atomic". For example, 31-et has
vector (-2 -1 11) which generates a val of (31 51 58). Only the first
value is correct (was expecting (31 49 72)).
>
> >
> > >
> > > While the most common way to project intervals and vals is to
toss
> > > octaves, we can do an atomic projection for ets by tossing
result of
> > > mapping the atom with an et, and an atomic projection for
intervals
> > by
> > > ignoring the results of the <5 8 11| val.
> >
> > * * * * *
> > Would it be possible to demonstrate these two cases? (tossing the
> > result of mapping the atom with an et, and 2) ignoring the
results of
> > the <5 8 11| value?
>
> If we set semitone = 2^(1/12) and schisma = (648/625)^(1/32), then
the
> atomic projection for intervals leads to a tempering, which is quite
> different than octave equivalence classes. Nevertheless, formally
they
> look the same--I take the results of applying <12 19 28| and <0 1 -
7|
> and ignore <5 8 11|, which corresponds to ignoring <1 0 0| when we
do
> octave classes. To get the tempering, take semitone^a schisma^b,
where
> a is the result of applying <12 19 28| to the interval, and b is the
> result of applying <0 1 -7|.
>
> The projection for vals works so long as the val sends the atom to
> zero--as, for instance, 12-et, 612-et, 4296-et. Otherwise the best
you
> can do is an approximation, which gives floating point results which
> round off, so long as we don't go too far afield, to the correct et
> value. You can also toss the result of applying the val to an atom,
> but the results won't make a whole lot of sense. So, for instance,
612
> gives the semitone a value of 51 and the schisma a value of 1, so
the
> projected et is <51 1|. By comparison <22/12, 1/20| works, more or
> less, like 22, but this isn't algebra, it's an approximation which
is
> only valid after rounding and in a restricted sense.

Thanks!