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How to calculate the cents values for a mean tone scale from the size 
of the comma

Let's start with the quarter comma meantone scale.

Our objective is to temper the fifths, in order to get as many pure 
major thirds as possible.

Start with a cycle of four fifths, such as

C' G' D A e

(e.g. in tuning for cello + violin using pure fifths)

Each pure fifth multiplies by 3/2, so with C' as 1/1, we get e as 
(3/2)^4 = 81/16.

Now try a major third followed by octaves.

This time we have

C' E' E e

We want E' = 5/4 above C', and so e = 5/1 = 80/16.

The ratio between (3/2)^4 and 5/1 is the syntonic comma of 81/80.

Now, all the fifths in C' G' D A e need to be flattened slightly in 
order to achieve 5/1 for the interval from C' to e.

So each one needs to be flattened by a quarter of a syntonic comma. 
Which of course is why this scale is called quarter comma meantone.

These are called meantone scales because all whole tones are the same 
size, and in particular, the major third is divided into two equal 
sized whole tones, rather than the unequal sized 9/8 and 10/9 of just 
intonation.

A pure fifth is 701.955 cents, and the syntonic comma is 21.5063 
cents.

So each fifth uas to be flattened by about five cents, and more 
exactly, one wants to flatten it by 21.5063/4 = 5.376575 cents to give 
701.955- 5.376575 = 696.578425 cents.

Of couse, the quarter comma meantone isn't the only meantone scale of 
interest, and one will want to be able to find the cents values for any 
 meantone scale.

So more generally, the size of the fifth is 701.955-21.5063*x cents, 
where x is the amount of the comma.

To find the size of the tempered fifth, enter the amount of the comma 
as a fraction:

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Equal temperament can be thought of as a special case of a meantone 
scale, and that may be a helpful way of thinking of it here.

For equal temperament, objective is to have Ab and G# the same in the 
sequence

Ab Db Bb F C G D A E B F# C# G#

So we compare twelve fifths (3/2)^12 = 531441/4096 with seven octaves 
2^7 = 128 and the ratio of these two gives the Pythagorean comma. Each 
fifth needs to be flattened by a twelth of the Pythagorean comma 
531441/524288 = 23.46001 cents

So in equal temperament, all the fifths are flattened by 23.46001/12 = 
 1.95500083 cents, or roughly one eleventh of a syntonic comma.

We see that quarter comma meantone scale has fifths that are flattened 
 more than they are for equal temperament. In fact, they are flattened 
by so much that it has to have one wolf fifth that is far sharper than 
normal, rather than flat.

To construct the scale, we have to decide where to place the wolf 
fifth.

A common choice is to put it between G sharp and D flat.

So lets do that.

Now all we need to do is to start from C as 1/1, and keep adding our 
tempered fifth of 696.578425 cents until we reach G#.

Also working backwards, start from C and keep subtracting 696.578425 
cents. until we reach D flat.

Then transpose all the notes into the same octave by adding or 
subtracting multiples of 1200 cents, and we are done.

As usual, the easy way is to make a little program to do this.

Here we are.

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Enter the amount of the comma as a ratio, and you can find the cents 
values for all the notes.

For the position of the wolf fifth, I use the position of its first 
note in the cycle of fifths starting from C as 0.

So G# is the 8th note in the cycle, counting upwards from C. If 
instead you want the wolf fifth between C# and Ab, position the wolf 
fifth at the 7th note in the cycle.

Results show the size of the wolf fifth, and also the wolf major and 
minor thirds. Since the intervals of a tritone have to add up to the 
octave, you end up with four wolf major thirds, one in each tritone. 
You also have three wolf minor thirds - one in each diminished seventh 
chord.

Try 1/11 for the amount of the comma to check that the result is close 
 to equal temperament.

Code is in javascript, so embedded in the html - programmers who want 
to use it can use View Source, and cut and paste.