Re : Basic tuning math

Thanks, John, for your method. It's very precise. Her's one for dummies, in
case you don't want to buy a logaritm calculator if this is all you use it
for and you just want to find the whole number of cents from a ratio:
use 1.0594631 as a multiplication parameter on a normal calculator. Count
the number of times you have to press the = button (for every 100 cents)
before you pass the
ratio.
Then use 1.0005778 , do the same (press once for every cent) and get as
close as you can. For exemple: a 3/2 is 7 times, using 1.0594631 and 2 times
using 1.0005778. The result: 1.5000388

Did you find information about the guzheng? I know about a Yueh chin, which
has an octagonal body and is tuned f-f and c'-c' (two double strings). Since
chinese ideograms are pronounceable in many different ways, especially
"chen" and "chang", it might be the same instrument.

----------
>De�: "John A. deLaubenfels" <jadl@idcomm.com>
>� : tuning@onelist.com
>Objet�: [tuning] Basic tuning math
>Date�: Ven 13 ao� 1999 16:42
>

> From: "John A. deLaubenfels" <jadl@idcomm.com>
>
>
> There has been some interest expressed in having tutorial(s) on the
> math involved in tuning. So, here we go...
>
> Imagine this scene: you're at a party. A lovely young woman arrives,
> and somehow you find yourself in conversation with her, and somehow it
> comes around to tuning, and she asks, "But isn't equal temperament the
> perfect way to tune? Why would you want to change it?". Are you ready?
> Could grab a calculator and dazzle her with math?
>
> Or are you left stammering? Dont'cha HATE it when that happens? Well,
> now you can be math literate, because Uncle John has prepared...
>
>
> BASIC TUNING MATH: converting frequency ratios to cents and back
>
>
> A conceptual introduction to logarithms
>
> The piano provides the perfect model, as well as an illustration of why
> we have to do these transformations. On the one hand there is the row
> of strings: with each octave higher, the frequency doubles, from A440 to
> 880 to 1760 hertz (cycles/second) as we go up to higher A's; 220, 110,
> and 55 hertz as we go down to lower A's. On the other hand, the keys
> move in linear progression, twelve semitones per octave.
>
> Logs are a mathematical way of relating doubling to linear motion.
> Exponents (powers) go the other way, relating linear motion to doubling.
>
> Whenever the whole process gets confusing, think of the piano. But,
> even if you're lost or indifferent so far, you can still do
>
>
> The math
>
> First you need a calculator with logs and exponents. A key might be
> labeled "log" or "ln" (for log natural). The other key we want is
> something like "X^Y", where Y is probably a superscript to X, indicating
> that the key will raise X to the Y power.
>
> Don't have a calculator with those keys? Get down to the store and buy
> one! They're cheap. Yes, you did vow most solemnly you'd never own
> anything of the sort, but holding out this long is enough to have made
> your point. Now buy (or borrow) one.
>
> Let's walk through some basic calculations. You gotta get this to work
> before we can go on. Confess your lack of knowledge to someone who
> might know, if you're stuck.
>
> Raise 2.0 to the 3.0 power; get 8.0. I wish I could help you with more
> specific instructions, but calculators are all different.
>
> Next exercise: divide the log of 1.5 by the log of 2. It'll go one of
> two ways:
>
> If your calculator uses natural logs:
>
> log(1.5) = 0.4055
> log(2.0) = 0.6931
> log(1.5) / log(2.0) = 0.5850
>
> If your calculator uses base 10 logs:
>
> log(1.5) = 0.1761
> log(2.0) = 0.3010
> log(1.5) / log(2.0) = 0.5850
>
> Note that the final answer comes out the same, so it doesn't matter;
> we're actually calculating "log to base 2" by either path. By the way,
> 0.5850 happens to be the fraction of an octave representing a perfect
> fifth (frequency ratio 1.5).
>
> Cents review: a cent is a measure defined to be 1/100 of a 12 tone
> equal temperament semitone. So, a cent is 1/1200 of an octave.
>
> Now you're ready!
>
> To convert frequency ratio to cents interval:
>
> Delta_cents = 1200 x log(freq_ratio) / log(2)
>
> OK, walk thru with an example: a major third, a 5/4 ratio, divides out
> to 1.25 frequency ratio; take the log of that; take the log of 2 and
> divide; you should see 0.3219 (if your calculator displays intermediate
> results); multiply by 1200 and get 386.31 cents (compare with 400 cents
> for a major third in 12-tET).
>
>
> To convert cents interval to frequency ratio:
>
> Freq_ratio = 2 ^ (delta_cents / 1200)
>
> That's "two to the power". Again, do an example: a 12-tET minor third,
> 3 semitones or 300 cents, divided by 1200 gives 0.25 (octaves). Two
> (2.0) raised to that power gives 1.1892; compare to 1.2000 for a perfect
> 6/5 minor third ratio.
>
> You are SO cool. Put a mortar board on your head!
>
> Note that everything we've talked about relates TWO notes. A single
> note in space, such as A440, may be a starting point for forming
> intervals, but there's little we can do with it by itself.
>
> JdL
>
>
>
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