II DIESIS

 DIESIS

Now we will become acquainted with a third comma, diesis.

The first number (Pythagorean comma)was 12 (ratio 12 fifths / octave)

next (syntonic comma) 4 (ratio 4 fifths / third).

The last number is 3, the ratio of 3 thirds / octave

If we add the three pure major thirds "on top of" each other

Ab- C

C-E

E-G#

we will again get an unwanted additional tone.

 G# is almost one quarter tone below compared with the enharmonic Ab and this is the largest comma of the three.

We must «lift»(enlarge) some of the thirds in order to maintain a pure prim / octave.

The video that follows in the link first display three pure thirds horizontally (only temporary ) and we will see how the latter end up far below the starting tone. (An diesis below)

So the G# will rise in two rounds. Last time G# becomes in tune with Ab, and the comma which must be distributed we see is as much as 1260 TU.

After that I am again using an example from Vallotti.

From the main diagram we saw that

Ab-C = 540 TU,

C-E = 180 TU and

E-G # = also 540 TU

Since I want the thirds vertically as in the «main diagram», I lift them until they are in a vertically line.

Then we can listen to how these three thirds sounds.

Anywhere in this diagram where you add three thirds vertically you will always get 1260 TU, the comma called diesis.

Video: Let us hear and see the Diesis

The top and bottom fifth-line is therefore entirely parallel, they are enharmonic ie same tone on the keyboard but with different names. (eg B# and C)

All fifth-lines have the same shape, but the fifth-lines are moved four fifths to the left relative to the nearest below.

And constantly it is 660 TU between a tone and the same tone on the line below.

The fifth-lines falls with 720 TU, from left to right.

And with diesis we now have three commas more or less visible in a diagram.

Picture: Pythagorean, Syntonic comma and Diesis in one diagram

What is new about this is that we get an overview of the distribution of diesis for all the major thirds.

We have looked at the thirds Ab-C-E-G #

Have a look and check the numbers for the third chain Db-F-A-C#

Db-F = 660 TU (Pythagorean third)

F A = 180 TU

A-C # = 420 TU (same as an equal-tempered third)

660 + 180 + 420 = 1260

This diagram shows the same intervals several places, just like the long diagram(2 octaves) does.

We find the third C-E three places.

We have two Db-F, and one C# -E#.

One advantage of this is that you can choose for instance to focus on F# instead of Gb etc.

And in the section «for the advanced» where I shows the analysis of chord progressions, one will easily find a good place to do this.

Now I hope you can also find something in the Diesis diagram.

.................................

Not important at all:

The foregoing comma is really called "the lesser diesis".

One could also make a chart that showed 4 minor thirds on top of another (eg C# -E-G-Bb-Db.)

Here would usually vectors point downwards because the minor third is almost always less than pure.

The distance between the C#(-E-G-Bb-)Db as a whole minor third chain would be -1920 TU (-480 x 4), called the greater diesis.

Such a diagram does not exist here, but one can get something approximating by following the dotted lines in the example below.

From top to bottom we then get

3 of the 4 minor thirds in the greater diesis. Same color belong to the same dim-chord (4 minor thirds)

The vertical distance (not the length of the dotted) corresponds to the deviation from pure minor third.

Picture: Minor third (greater diesis)