can anyone please help?

hi.

i am a 14 year old musician, and as it were, i am researching a project on just
intonation and tuning systems.

can anyone please help me explain the math behind all of these? most of what i
do not understand is on the vector and matrix things. so can someone please
explain these to me?

-

hi piano_maestro3000,

--- In tuning-math@yahoogroups.com, "» «" <piano_maestro3000@m...>
wrote:

> hi.
>
> i am a 14 year old musician, and as it were, i am
> researching a project on just intonation and tuning systems.
>
> can anyone please help me explain the math behind all of these?
> most of what i do not understand is on the vector and
> matrix things. so can someone please explain these to me?

the vectors (which Gene Ward Smith and i call "monzos")
are lists of the exponents of the prime-factors of the
just-intonation ratios. some prefer to simply call them
"vectors", but a "monzo" is defined as a vector with
this specific meaning.

they are notated with a square bracket on the left and
a right angle bracket (i.e., "greater-than" sign) on
the right. commas are used after the exponent of 3,
and after every third exponent after that.

the prime series goes: 2, 3, 5, 7, 11, 13 ... etc.

so, for example, the octave, with ratio 2/1, has a monzo
which is written: [1 0, 0 0 0, ...etc. >

the pythagorean perfect-5th, with ratio 3/2, is: [-1 1, 0 0 0, ... >

the just major-3rd, with ratio 5/4, is: [-2 0, 1 0 0, ... >

for a more complicated example, 63/64 is: [-6 2, 0 1 0, ... >

the pythagorean-comma, 531441/524288, is: [-19 12, 0 0 0, ... >

it is only necessary to carry the monzo out to the last
non-zero exponent ... i've included the zeros above just
to make it easier for you to see how they work.

so the pythagorean-comma only needs to be: [-19 12 >
the 63/64 only needs to be: [-6 2, 0 1 > , etc.

there are other types of vectors which we use, such as
"vals" and "wedgies", but those are connected with
temperaments. for just-intonation, monzos are all you need.

the great value in using this notation is that it allows
the reader to immediately pinpoint where that ratio fits
in a prime-space lattice-diagram.

a matrix is simply a way of arranging several vectors
so that complicated mathematical procedures may be
performed more easily, using standard matrix algebra.

if you need more info on this stuff, you should consult
my Encyclopedia:

http://tonalsoft.com/enc

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> for a more complicated example, 63/64 is: [-6 2, 0 1 0, ... >

oops ... what i wrote there is correct, but i meant to
use the inverse of that ratio as my illustration. so ...

the monzo (or vector) of 64/63 is: [6 -2, 0 -1 >

-monz