Make a new meantone scale , The calculation

Make a new mean tone scale

How to use: Just enter a new value for the comma or the position of the fifth in the cycle, and the scale will get generated at once. You can also enter values for any of the intervals or wolf intervals. Values are read as cents unless they are ratios.

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

The number for the position of the wolf fifth refers to the cycle of fifths starting from C as 0.

 Gb Db Ab Eb Bb F  C  G  D  A  E  B  F# C# G# D# A#
-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9  10

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.

Note to programmers: You are welcome to modify this code and copy it - it is free source. View Source, and cut and paste.

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The calculation

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 in the tuning for the open strings of cello + violin if one uses pure fifths.

Each pure fifth multiplies by 3/2, so with C' on the 'cello as 1/1, we get the open e string on the violin as (3/2) ⁴ = 81/16.

Now let's try a major third followed by octaves.

This time we have C' C' c e. We want a pure major third, so the e = 5/4 above c, and so with C' as the 1/1 we get e = 5/1 (or 80/16).

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

Since we want the interval c to e to be a pure 5/4, 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. In fact, each one needs to be flattened by a quarter of a syntonic comma - which of course is why these scales are called quarter comma.

The reason they are called meantone scales is 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.

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 of fifths that goes like this:

Ab Eb 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 at the two ends of the cycle of fifths one has to have one wolf fifth that is far sharper than normal, rather than flat as it is in pythagorean tunings.

Now, 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#. Then, 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.

It's not that hard to do with a calculator but involves many steps - so the applet saves a lot of work in this.

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