Scala command script example

In this post I shall give a short lesson of how to create a Scala command
file that calculates a scale as a function of some parameters.

Bela Bartok has devised an axis-system of notes. It consists of steps of
subdominant-tonica-dominant triplets.
There is a main step (Dutch: hoofdtrap):

f c g
S T D

and secondary steps (Dutch: neventrappen) where the mediant of the previous
step becomes the dominant:

d a e b f# c# ab eb bb
S T D S T D S T D

Arranged in axis crosses (assenkruis, Achsenkreuz):

S T D

f c g

d ab a eb e bb

b f# c#


The middle cross is used for instance for the global tonality structure of
Music for Strings, Percussion and Celesta.
Reference:
E. Lendvai: Einfu"hrung in die Formen- und Harmoniewelt Barto'ks, 1957.

We can (ab)use this system to create scales with, by taking the size of
dominant and mediant as parameters. The subdominant will be the octave
inversion of the dominant.
So if we for instance start with the major triad 4:5:6 we get the scale
25/24 10/9 125/108 5/4 4/3 25/18 3/2 125/81 5/3 125/72 50/27 2/1
which ratios have lattice structure (to the right: 3, upwards: 5,
the zero is C):

* * *
* * *
* * *
* 0 *

This is sort of a slanting version of Euler's genus bichromaticum [33555].
Here comes a solution for the command file:

clear 0
echo enter the mediant
append ?
echo enter the dominant
append ?

First we have to enter the values for mediant and dominant. A question mark
is substituted with what the user enters. The working scale is cleared and
the two notes entered into it.

calculate/noout 2-%2
append $0

The value of the subdominant is calculated by subtracting the second note
(referenced with a % sign) from the octave. That value, which is stored in
pitch memory number 0 (referenced with a $ sign), is added to the scale.

calculate/noout 2+%1-%2
delete 1
copy 0 1

Now we calculate what the shift must be to get the next step. It is O+M-D,
or S+M (from C to A). This value is now in pitch memory 0.
The mediant is deleted so the working scale contains now T-D-S. This scale
is saved in scale memory number 1.

equaltemp 1 $0 3
product 1

A cycle of three steps (so four notes) of the shift value is created.
The 1 indicates it is undivided (division of 1).
Now comes the trick: the Carthesian product of this is taken with the
saved T-D-S triplet, giving the 3 x 4 = 12 notes at once.

clear 1
reduce 2/1
sort
append 2/1

Scale memory 1 is no longer needed and cleared. All the notes are octave
reduced and sorted in order and the octave is added to the resulting scale.
So instead of having to write expressions for each separate note, we make
use of the structure of the scale and use the proper Scala command that can
make it.

Taking a quarter-comma fifth for the dominant (Scala expression: 3/2-$k\4 )
and 5/4 for mediant gives us a regular mean-tone scale in the key of B-flat.

All the command files supplied with Scala may serve as other examples.
In particular the ones for creating tritriadic, tetratriadic, tritetradic,
etc. scales, for creating Fokker's single- and double-tied circular
mirrorings of triads and tetrads, for creating duodenes, straight and
reversed triadic diamonds, hexany diamonds and mean-tone scales.

Manuel Op de Coul coul@ezh.nl

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