Yahoo Tuning Groups Ultimate Backup crazy_music exponential remainders

Friends,

Let me bounce some ideas off you all here.

I am trying to come up with good english terms and
symbols for a useful math operator I want to introduce
to the tuning language. It's something many of us have
worked with but I am not aware of a good general term
for it.

IDEA #1: REMAINDERING FOR NON-INTEGERS

As you know, there is regular real number division:

7 / 2 = 3.5

and there is integer division:

7 / 2 = 3 remainder 1

which can also be expressed using the div and mod
'operators':

7 div 2 = 3
7 mod 2 = 1

(div is the integer division operation and mod is
integer remainder operation).

Another way to show how remainder works is as
follows:

7 mod 2 = 7 - ((7 div 2) * 2) = 7 - (3 * 2) = 7-6 = 1

But why should there not also be remainders from
fractional and real numbers using the same mechanism:

4.5 div 3 = 1
4.5 mod 3 = 1.5

...since this idea of having something 'leftover'
or remaindered is generally useful.

IDEA #2: TUNING INTERVALS AND POWER OPERATIONS

OK, next - lets look at the frequently discussed tuning
operation of deriving the syntonic comma - the place
where the just octave 2/1 and the just 3/2 fifth don't
line up -- the difference between 12 fifths and 7
octaves is a small interval - the syntonic comma.

The ratio of 12 fifths is (3/2) ^ 12, where the caret
is the symbol or operator we use to indicate 'raised to
the power of'. Some people also use the symbol '**',
kind of showing that exponentiation (raising to a
power) is a kind of hyper-multiplication: (3/2) ** 12

Likewise, the ratio of 7 octaves is 2/1 multiplied by
itself 7 times, or (2/1)^7, or (2/1) ** 7.

To get the difference between two ratios, you divide
them, so the difference here is:

(3/2) ^ 12
---------- = syntonic comma
(2/1) ^ 7

...which it turns out is about 23.46 cents.

Now the thing is that the operation on the fractions
is rather unwieldy and inconvenient, and yet it is
something we do often, especially if we are working
with meantone or MOS scales.

It would be GREAT to have an operator so we could write
this stuff a little more concisely. Sometimes I call
this idea of letting one interval chain up and then be
cut by another 'folding'. But technically, it isn't
really folding as much as it is shuffling, do you know
what I mean? Anyway, I would like to be able to write:

(fifth ^ 12) fold_into octave

and have it come out with 'syntonic comma'.
(What is 'it'? 'It' is my interval calculator
application.)

Now already, I have 'fifth' predefined as 3/2 and
octave defined as 2/1, so what I wrote above is really:

((3/2) ^ 12) fold_into (2/1) => syntonic comma

And we are left with wondering how this 'fold_into'
operator works. Well, it works like remainder actually.

Consider it in cents, where it really IS remainder:

ratio_to_cents(3/2) = 701.955 cents
ratio_to_cents(2/1) = 1200.000 cents

ratio_to_cents((3/2)^12) = 12 * 701.955 cents =
8423.460 cents.

8423.460 cents mod 1200.000 cents = 23.46 cents

Now you see why I started in about remainders of
nonintegers -- it's a useful thing to be able to talk
about in the tuning world.

The syntonic comma is the REMAINDER of DIVIDING an
OCTAVE into twelve fifths. Obviously an octave 'divides
into' twelve fifths seven times, but not evenly, there
is a remainder left over -- and that remainder is the
syntonic comma.

In other words, comma is wery much like a remainder,
and when working in cents (the log of frequency
ratios), it is exactly like remainder.

But what do we call it when working in 'ratio space'?
What is this operator called? Well, it is basically a
'exponential remainder' (frequency ratio is the
exponent of cents just as cents is the logarithm of
frequency ratio).

So we have the 'linear remainder' - mod, and the
'exponential remainder'.

Here is my question -- what terms do we use for talking
about and what symbols do we like for writing this
operation when working in the frequency ratio domain?

Here are some ideas:

syntonic comma = (fifth ^ 12) fold into octave
syntonic comma = (fifth ^ 12) fold octave
syntonic comma = (fifth ^ 12) shuffle into octave
syntonic comma = (fifth ^ 12) shuffle octave
syntonic comma = (fifth ^ 12) comma octave
syntonic comma = (fifth ^ 12) , octave

syntonic comma = (fifth ^ 12) emod octave
syntonic comma = (fifth ^ 12) emainder octave
syntonic comma = (fifth ^ 12) %% octave

I kind of like the %% since in C, the linear mod
operator is written '%' -- and exponential mod is to
linear mod as power (**) is to multiplication (*).

Of course, there should also be an exponential div
operator: I favor the following:

(fifth ^ 12) // octave = 7
(fifth ^ 12) ediv octave = 7

Here of course, the '//' is a sort of hyperdiv
operator.

Of course, that equation can be rearranged usefully to
say things like: "7 is the floor of the logarithm base
octave of fifth". But how do we read it as above? I'd
like to say "Twelve fifths exponentially divided by an
octave gives 7 with a remainder of a syntonic comma".
Isn't that better? I really think this is a better way
of saying it, more precise and clear! The only issue is
what to call the operation. What does everyone think?
Any suggestions? Does this seem a good thing?

Other language issues:

What if we call these 'arithmetic remainder' and
'geometrical remainder' -- would that make sense? Or is
'linear remainder' and 'exponential remainder' better?

Feedback please!!

These are math operations we learn about in junior high
here (logarithms & exponents) and elementary school
(remainder and dividend), but I harldy ever talk about
it or write to people about it and on the tuning lists
and books we have to resort to english language
gymnastics 'the syntonic comma is the difference
between seven octaves and 12 fifths' - which is
miseading since 'difference' and 'seven' and 'fifth' do
not mean the same thing here they would to most people
in the world.

- Jeff

> Other than that, I've noticed at times you seem to be on the
> brink of imagining a *complete* new math. I haven't quite
> figured out where you're going though so. Not sure how to
> advise here.

OH, it's not that big a deal. Here's the summary of my
various issues:

1) Base-12 ? YES I LOVE BASE 12. Way better than silly
decimal. I have my own system of counting in base-12
and have memorized much of the base-12 multiplication
tables. I LOVE BASE 12.

OK, I admit it, I am a freak here, THOUGH I AM RIGHT
THAT BASE-12 IS WAY BETTER THAN BASE-10 and those who
think otherwise have been BRAINWASHED.

OK, fine, yes, I am typing on a Dvorak keyboard right
now which is BETTER than QWERTYUIOP - why do you ask?

2) Prime-factorization notation. I think it's very
silly to thing that using a base made up out of prime
numbers is useful for doing what amounts to arithmetic
(tuning stuff). But I do agree with Nash that base
prime is cool for mystical and ritualistic stuff.

3) Exponential remainders: I use these all the time and
its time to come up with a word for it -- I guess I
have -- but also a symbol for it because I've decided
to add an operator for it to my tuning software and
want to know what people think makes sense.

I really like %% and //, and also ediv, emod, and
emainder -- maybe I'll just add all of them?

Any comments on the operators?
On the names?

I will have to alter the current lexing of '//' -- it
is currently accepted as a line comment, like in C and
C++. (Yes C too, it was formally added to the official
C standard three years ago.)

- Jeff

exponential remainders - math help requested

🔗X. J. Scott <xjscott@...>

🔗Orphon Soul, Inc. <tuning@...>

🔗X. J. Scott <xjscott@...>

🔗Orphon Soul, Inc. <tuning@...>