A Periodicity Block view of MODMOS's - coming up with a finite list to autogenerate for temperaments

There have been, so far, three fundamental characteristics that any
MODMOS will have. To recap, they are:

1) The interval that you have modded the MOS by - the chroma c=L-s,
the hyperchroma h=|s-c|, etc
2) How many alterations you have made to the MOS to arrive at the
current scale (or the fewest number of alterations possible that will
bring you here)
3) The number of levels you have to go down the scale tree to find an
MOS that will support the resultant scale

By specifying parameters for 1, 2, and 3, a finite list of MODMOS's is
generated. One can then prune the list afterward to only incorporate
proper MODMOS's as a further refinement of the procedure.

So how do we make sense of all of this? We do so by noting that an MOS
is an unmapped rank-2 periodicity block. It consists of a generator
and a period, and since it is an MOS, due to Gene's proving of Paul's
famous Hypothesis, it is a rank-2 periodicity block. Let's look at the
diatonic scale to see what this might look like (view fixed width).

Sharp first note in generator chain, width 7
C D E F# G A B C - another mode of the diatonic scale

0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 ...
.__.----------------------.______________________.-----
|F |C G D A E B F# | C# G# D# A# E# B# Fx | ...
'¯¯'----------------------'¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯'-----

Sharp second note in generator chain, width 9
C# D E F G A B C# - melodic minor scale

1 0 1 1 1 1 1 2 1 2 2 2 2 2 ...
.--.__.----------------.__.---.______________.-----
| F| C| G D A E B |F#| C#|G# D# A# E# B#| ...
'--'¯¯'----------------'¯¯'---'¯¯¯¯¯¯¯¯¯¯¯¯¯¯'-----

Sharp third note in generator chain, width 10
C D E F G# A B C - harmonic minor scale

1 1 0 1 1 1 1 2 2 1 2 2 2 2 ...
.------._.-------------._____.--.____________.-----
| F C |G| D A E B |F# C#|G#|D# A# E# B# | ...
'------'¯'-------------'¯¯¯¯¯'--'¯¯¯¯¯¯¯¯¯¯¯¯'-----

Sharp fourth note in generator chain, width 11
C D# E F G A B C - Ionian #2

1 1 1 0 1 1 1 2 2 2 1 2 2 2 ...
.---------._.----------.________.--._________.-----
| F C G |D| A E B |F# C# G#|D#|A# E# B# | ...
'---------'¯'----------'¯¯¯¯¯¯¯¯'--'¯¯¯¯¯¯¯¯¯'-----

Sharp fifth note in generator chain, width 12!!
C D E F G A# B C - Ionian #6

1 1 1 1 0 1 1 2 2 2 2 1 2 2 ...
.------------._.------.____________.--.______.-----
| F C G D |A| E B | F# C# G# D#|A#|E# B# | ...
'------------'¯'------'¯¯¯¯¯¯¯¯¯¯¯¯'--'¯¯¯¯¯¯'-----

NOTE - The last one had width 12. This means that the above one is the
largest mode that will actually fit in the next-highest MOS up, which
is 12 notes. If we keep going, we end up with this:

Sharp sixth note in generator chain, width 13
C D E# F G A B C - Ionian #3

1 1 1 1 1 0 1 2 2 2 2 2 1 2 ...
.---------------._.---._______________.--.___.-----
| F C G D A |E| B | F# C# G# D#|A#|E#|B# | ...
'---------------'¯'---'¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯'--'¯¯¯'-----

Sharp seventh note in generator chain, width 14
C D E F G A B# C - Ionian #7

1 1 1 1 1 0 1 2 2 2 2 2 1 2 ...
.------------------.__.__________________.--..-----
| F C G D A E |B | F# C# G# D#|A#|E#|B#|| ...
'------------------'¯¯'¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯'--''-----

This is a list of all of the modes that we can get by sharpening one
and only one interval in a diatonic scale. However, to get the FULL
list of alterations, we also have to look at the mirror images of
these scales - AKA look at the periodicity blocks reflected around the
x-axis. You can generate this by taking every instance of a sharpened
scale and come up with a symmetrical "flattened" scale; e.g. the
symmetrical counterpoint of Ionian with the third note in the
generator chain sharpened (which is G, which works out to C D E F G# A
B C a mode of harmonic minor) is Ionian with the third last note in
the generator chain flattened (which is A, which works out to C D E F
G Ab B C which is harmonic major). This applies to all of the modes
except for the diatonic and the #2nd note in chain mode, which are
symmetric. I can draw some ASCII diagrams of the flattened mirror
image modes if you'd all like, but perhaps it'll be clear enough from
what I do have.

Anyway, once one has a list, one can continue this process for
additional alterations by simply recursively applying this algorithm
to the scales it finds. Run the algorithm N times for N max
alterations.

So here's the algorithm:
1) Decide how many alterations you want to calculate.
2) Decide what the "chromatic" MOS for this scale is. Is it just the
next MOS down the scale tree? Or is it two MOS's down, as in the case
of porcupine[7] or mavila[7]?
3) Find all of the MODMOS's that have that many alterations and that
fit in the scale by the algorithm above.
4) You now have a finite list of MODMOS's for every temperament. Yay!
5) Now prune this finite list for only those that are proper,
optionally, and you have the Golden Scale list for this temperament in
a nutshell.

Phew! Hope that's clear enough. Those diagrams were a pain in the ass to do.

-MIke

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> There have been, so far, three fundamental characteristics that any
> MODMOS will have. To recap, they are:
>
> 1) The interval that you have modded the MOS by - the chroma c=L-s,
> the hyperchroma h=|s-c|, etc
> 2) How many alterations you have made to the MOS to arrive at the
> current scale (or the fewest number of alterations possible that will
> bring you here)
> 3) The number of levels you have to go down the scale tree to find an
> MOS that will support the resultant scale
>
> By specifying parameters for 1, 2, and 3, a finite list of MODMOS's is
> generated. One can then prune the list afterward to only incorporate
> proper MODMOS's as a further refinement of the procedure.
>
> So how do we make sense of all of this? We do so by noting that an MOS
> is an unmapped rank-2 periodicity block. It consists of a generator
> and a period, and since it is an MOS, due to Gene's proving of Paul's
> famous Hypothesis, it is a rank-2 periodicity block. Let's look at the
> diatonic scale to see what this might look like (view fixed width).
>
> Sharp first note in generator chain, width 7
> C D E F# G A B C - another mode of the diatonic scale
>
> 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 ...
> .__.----------------------.______________________.-----
> |F |C G D A E B F# | C# G# D# A# E# B# Fx | ...
> '¯¯'----------------------'¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯'-----
>
>
>
> Sharp second note in generator chain, width 9
> C# D E F G A B C# - melodic minor scale
>
> 1 0 1 1 1 1 1 2 1 2 2 2 2 2 ...
> .--.__.----------------.__.---.______________.-----
> | F| C| G D A E B |F#| C#|G# D# A# E# B#| ...
> '--'¯¯'----------------'¯¯'---'¯¯¯¯¯¯¯¯¯¯¯¯¯¯'-----
>
>
>
> Sharp third note in generator chain, width 10
> C D E F G# A B C - harmonic minor scale
>
> 1 1 0 1 1 1 1 2 2 1 2 2 2 2 ...
> .------._.-------------._____.--.____________.-----
> | F C |G| D A E B |F# C#|G#|D# A# E# B# | ...
> '------'¯'-------------'¯¯¯¯¯'--'¯¯¯¯¯¯¯¯¯¯¯¯'-----
>
>
>
> Sharp fourth note in generator chain, width 11
> C D# E F G A B C - Ionian #2
>
> 1 1 1 0 1 1 1 2 2 2 1 2 2 2 ...
> .---------._.----------.________.--._________.-----
> | F C G |D| A E B |F# C# G#|D#|A# E# B# | ...
> '---------'¯'----------'¯¯¯¯¯¯¯¯'--'¯¯¯¯¯¯¯¯¯'-----
>
>
>
> Sharp fifth note in generator chain, width 12!!
> C D E F G A# B C - Ionian #6
>
> 1 1 1 1 0 1 1 2 2 2 2 1 2 2 ...
> .------------._.------.____________.--.______.-----
> | F C G D |A| E B | F# C# G# D#|A#|E# B# | ...
> '------------'¯'------'¯¯¯¯¯¯¯¯¯¯¯¯'--'¯¯¯¯¯¯'-----
>
>
> NOTE - The last one had width 12. This means that the above one is the
> largest mode that will actually fit in the next-highest MOS up, which
> is 12 notes. If we keep going, we end up with this:
>
>
>
> Sharp sixth note in generator chain, width 13
> C D E# F G A B C - Ionian #3
>
> 1 1 1 1 1 0 1 2 2 2 2 2 1 2 ...
> .---------------._.---._______________.--.___.-----
> | F C G D A |E| B | F# C# G# D#|A#|E#|B# | ...
> '---------------'¯'---'¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯'--'¯¯¯'-----
>
>
>
> Sharp seventh note in generator chain, width 14
> C D E F G A B# C - Ionian #7
>
> 1 1 1 1 1 0 1 2 2 2 2 2 1 2 ...
> .------------------.__.__________________.--..-----
> | F C G D A E |B | F# C# G# D#|A#|E#|B#|| ...
> '------------------'¯¯'¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯'--''-----
>
>
> This is a list of all of the modes that we can get by sharpening one
> and only one interval in a diatonic scale. However, to get the FULL
> list of alterations, we also have to look at the mirror images of
> these scales - AKA look at the periodicity blocks reflected around the
> x-axis. You can generate this by taking every instance of a sharpened
> scale and come up with a symmetrical "flattened" scale; e.g. the
> symmetrical counterpoint of Ionian with the third note in the
> generator chain sharpened (which is G, which works out to C D E F G# A
> B C a mode of harmonic minor) is Ionian with the third last note in
> the generator chain flattened (which is A, which works out to C D E F
> G Ab B C which is harmonic major). This applies to all of the modes
> except for the diatonic and the #2nd note in chain mode, which are
> symmetric. I can draw some ASCII diagrams of the flattened mirror
> image modes if you'd all like, but perhaps it'll be clear enough from
> what I do have.
>
> Anyway, once one has a list, one can continue this process for
> additional alterations by simply recursively applying this algorithm
> to the scales it finds. Run the algorithm N times for N max
> alterations.
>
> So here's the algorithm:
> 1) Decide how many alterations you want to calculate.
> 2) Decide what the "chromatic" MOS for this scale is. Is it just the
> next MOS down the scale tree? Or is it two MOS's down, as in the case
> of porcupine[7] or mavila[7]?
> 3) Find all of the MODMOS's that have that many alterations and that
> fit in the scale by the algorithm above.
> 4) You now have a finite list of MODMOS's for every temperament. Yay!
> 5) Now prune this finite list for only those that are proper,
> optionally, and you have the Golden Scale list for this temperament in
> a nutshell.
>
> Phew! Hope that's clear enough. Those diagrams were a pain in the ass to do.
>
> -MIke
>
Thanks Mike, this is beautifully done. From my end, I have found that 57 of the 66 septachord types get covered as MODMOS (or just plain MOS). That is to say, all
septachord types are covered except these nine, which are essential chromatic type
collections:

0123456
0123457
0123458
0123459
012345.10
0123467
012349.10
0123567
012389.10

So most of the septachord types are findable with ordinary accidentals. Now if you just
apply the M7 relation to these you get scales which are covered (Trick for M7 relation:
keep evens fixed and move odds by a tritone (+6). This will change the basis from
semitones to perfect fifths)

PGH