Well, after all the interesting comments about diatonics, I owe it a small

explanation of how I work.

The rules I put out in my article were 'rules of thumb', that, given the

other ET scales I looked at, produced the diatonics that we know from C19

and C31, and C26. They also gave rise to the 11 tone C20 diatonic, and all

of the others that I have found.

***This does not mean that other scales which can be built from grids or

lattices are not valid as creative source material.***

As for coherency and incoherency, this is my attempt at a rule of thumb to

allow me to detect easily those scales where the number of 'semitone' steps

inside a given scale step is consistent over the scale. To take Balzano's

example of the left-handed scale, p72: "The size 7 scale given previously

[C12: PCs: 0,1,2,3,4,5,6,7], for example, contains distances of one and two

scalesteps that are smaller in semitones, than distances of three, four and

five scalesteps." And for sake of argument, I choose understand the term

semitone to mean a difference of 1 in PC terms, and so 'generalize' it to

include the same PC difference in all scales of other Cn. Hence my 'rule'

about groups of adjacent PCs.

As I stated in my introduction:

"This paper approaches these and other 'diatonic' scales from the viewpoint

of a composer seeking new materials for creative work, rather than trying

for rigorous mathematical proof."

I do have an awareness of the mathematical basis for tuning theory, but the

scope of my studies was to derive scales that have _musical_ properties that

bore resemblance to the one diatonic scale that (in whatever form) does

exist. Shape and structure guided what I looked for, and in doing so I may

have 'broken' some 'rules'. As for my own 'rules': I am happy break those

too.

Mark

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> As I stated in my introduction:

>

> "This paper approaches these and other 'diatonic' scales from the

viewpoint

> of a composer seeking new materials for creative work, rather than

trying

> for rigorous mathematical proof."

>

> I do have an awareness of the mathematical basis for tuning theory,

but the

> scope of my studies was to derive scales that have _musical_

properties that

> bore resemblance to the one diatonic scale that (in whatever form)

does

> exist. Shape and structure guided what I looked for, and in doing

so I may

> have 'broken' some 'rules'. As for my own 'rules': I am happy break

those

> too.

>

> Mark

make no mistake: i feel exactly the same way you do about this. i

will take a look at your ideas again when i have an opportunity --

meanwhile, i hope you will take a look at my 'gentle introdution to

fokker periodicity blocks' -- talk about 'shape' and 'structure'!