clearing up TC

Is anybody following this thread? I think I've got it figured out now.

There are two versions...

Strong: All instances of a pitch in a scale's "Cartesian" cross-set occupy
the same scale-degree position in the row or column in which they fall.

Weak: All pitches appearing in both the original scale and any one of the
transposed scales in the "Cartesian" cross-set occupy the same scale
position in their respective scales.

The question...

If [S -TC-> S*f] and [S -TC-> S*g], must [S -TC-> S*(f-g)]?

Example, S = 1/1 5/4 3/2
If [S -TC-> S*5/4] and [S -TC-> S*3/2], must [S -TC-> S*6/5]?

...asks if these two versions are actually independent. Here's the
"Cartesian cross-set" of the C major scale in 12tET...

B C# D# E F# G# A#
A B C# D E F# G#
G A B C D E F#
F G A Bb C D E
E F# G# A B C# D#
D E F# G A B C#
C D E F G A B

...the "original" scale of C major is TC with all the transposed scales in
the set. However, the F and B transposed versions are not TC with
eachother. So the answer to the question above is "NO!", and the Strong
and Weak versions above really are independent.

What does this mean? It means there are scales that are Weak TC in some
modes but not in others. Why? Because B does get compared to the original
scale when the cross set of the Lydian mode is taken. Is anybody following
this?

Strong TC is by definition a property of all modes. It is also equivalent
to CS. I've said that before...

>>It seems to me, if I've understood correctly how you've defined it, that
>>in most cases it would be most meaninful to consider TC shorthand for
>>"TC by all intervals within the scale itself".
>
>You mean only those intervals measured up from the root, or measured from
>anywhere? If you mean the latter, then your suggested definition is
>equivalent to CS (constant structureness).

Here's an outline of my reasoning...

1. CS means all instances of a pitch in a scale's "diamondic" cross-set
occupy the same scale position.

2. Strong TC means all instances of a pitch in a scale's "Cartesian"
cross-set occupy the same scale position.

3. Multiplying a scale by n/d changes its accidentals in the same way (but
in opposite directions) as multiplying the scale by d/n.

4. So making a "Cartesian" cross set makes a "virtual" diamond, which is
equivalent as far as common tones go.

5. I realized that all strictly proper scales are CS. Paul Erlich realized
that all strictly proper scales are TC. That seems to add weight to the
argument.

Conclusions? I like Weak TC, because I think it better reflects how this
property works in music. Although CS now takes on a new level of interest
for me.

The future? I'd like to learn which of these properties, if any, all MOS's
have. Strong TC / CS is out, since the diatonic scale in 12tET is MOS.
That leaves Weak TC. After that, I'll want to return to lattice TC, and
search for periodicity blocks that have it.

-Carl

>It's sounds like you're retreading -- albeit in a generalized way -- much of
>good old fashioned twelve-tone theory. You might find the use of "row box"
>diagonals in late Stravinsky to be of interest.

Is there a reference?

>I've posted this observation several times before, but does anyone else find
>the shared structure of Partch's diamond and the Schoenbergian 12-tone matrix
>to be more than a music-historical coincidence?

I must have missed these posts. I'm afraid I don't know what a
"Schoenbergian 12-tone matrix" is...

-Carl

Carl wrote,

>Is anybody following
>this?

Carl, I'm really trying, but your language is very confusing. Can you please
demonstrate this:

>It means there are scales that are Weak TC in some
>modes but not in others. Why? Because B does get compared to the original
>scale when the cross set of the Lydian mode is taken.

>Carl, I'm really trying, but your language is very confusing. Can you please
>demonstrate this:
>
>>It means there are scales that are Weak TC in some modes but not in others.
>>Why? Because B does get compared to the original scale when the cross set
>>of the Lydian mode is taken.

Here's the cross-set of the Cmaj scale in 12tET...

B C# D# E F# G# A#
A B C# D E F# G#
G A B C D E F#
F G A Bb C D E
E F# G# A B C# D#
D E F# G A B C#
C D E F G A B

...the question is, if the column on the far left is TC with all other
columns, are all columns necessarily TC with all other columns? The answer
is no, since the far left column _is_ TC with all the others, and the columns
starting on F and B are not TC with eachother. So we have two seperate
measures, Weak and Strong. This is trivial, and I gather you understood it.

It is also trivial that the Weak version varies across the modes of a scale,
since the column starting on F would be in the far left if the cross set of
the lydian mode had been taken instead of the ionion. The Scala user can
check this by doing the following...

1. equal 12 ; creates 12tET
2. mos -> 7 -> 4 ; removes "black notes"
3. key 4 ; you're in the ionian mode
4. show transpose ; observe that the two leftmost columns are the same
5. undo ; now back to the lydian mode
6. show transpose ; observe that the two leftmost columns differ

So we have...

1. CS= Are there no ambiguous intervals (in the diamond)?
2. Strong TC= No ambiguous intervals in the cross set?
3. Weak TC= No ambiguous intervals between a mode and its transpositions?

Was there an error in my reasoning that 1 & 2 are the same?

I asked if all MOS's have at least one mode that is Weak TC. That's the
same as asking if no MOS has at least one ambiguous interval appearance in
every mode. Last year on November 19th I had asked if any proper scale
could have this property. I answered the question affirmatively on
December 29th, and the example I gave had Myhill's property...

C Eb Gb
2nds 3 3 *6
3rds *6 *6 9
4ths 12 12 12

...so it seems not all MOS's have at least one Weak TC mode. However, this
is a fairly degenerate case -- the generator has not yet spanned the IE. I
haven't been able to find a "serious" MOS whose modes weren't mostly TC.

Now, could you (Paul Erlich) explain this...

>>When you take a cross set of a scale and itself, you only multiply by the
>>intervals measured up from the root. So each mode of a scale has a
>>different cross set, right?
>
>Different only up to a transposition, which shouldn't matter.

-Carl

Carl wrote,

>>>It means there are scales that are Weak TC in some modes but not in
others.
>>>Why? Because B does get compared to the original scale when the cross
set
>>>of the Lydian mode is taken.

I wrote,

>>Carl, I'm really trying, but your language is very confusing. Can you
please
>>demonstrate this:

Carl wrote,

>Here's the cross-set of the Cmaj scale in 12tET...

> B C# D# E F# G# A#
> A B C# D E F# G#
> G A B C D E F#
> F G A Bb C D E
> E F# G# A B C# D#
> D E F# G A B C#
> C D E F G A B

>...the question is, if the column on the far left is TC with all other
>columns, are all columns necessarily TC with all other columns? The answer
>is no, since the far left column _is_ TC with all the others, and the
columns
>starting on F and B are not TC with eachother. So we have two seperate
>measures, Weak and Strong. This is trivial, and I gather you understood
it.

That's very strange, but basically you've changed the definition of TC to a
_relation_ between two absolute pitch sets, and you're saying that F major
and B major are not TC with one another because the interval between E and
A#/Bb is a fourth in one and a fifth in the other. Am I following you so
far? What would be different if you took the cross set of the lydian mode?

>It is also trivial that the Weak version varies across the modes of a
scale,
>since the column starting on F would be in the far left if the cross set of
>the lydian mode had been taken instead of the ionion.

But that is not all that would change:

>Now, could you (Paul Erlich) explain this...

>>>When you take a cross set of a scale and itself, you only multiply by the
>>>intervals measured up from the root. So each mode of a scale has a
>>>different cross set, right?
>
>>Different only up to a transposition, which shouldn't matter.

I already explained that. If you followed that explanation, you'd know that
the Cartesian square of the C major scale, which you showed above, is
identical to the Cartesian square of the Bb lydian mode, provided you rotate
the numbering of the rows and columns:

5 6 7 1 2 3 4

5 B C# D# E F# G# A#
6 A B C# D E F# G#
7 G A B C D E F#
1 F G A Bb C D E
2 E F# G# A B C# D#
3 D E F# G A B C#
4 C D E F G A B

So what you said about the column on F seems incorrect.

>That's very strange, but basically you've changed the definition of TC to
>a _relation_ between two absolute pitch sets, and you're saying that F major
>and B major are not TC with one another because the interval between E and
>A#/Bb is a fourth in one and a fifth in the other. Am I following you so
>far?

Yes, except that TC has always been a relation between two absolute pitch
sets. First there was lattice TC, where a set of connected lattice points
could be transposed by any one of the intervals connecting it without any of
the (absolute pitch) common tones between the original and tranposed scales
changing scale positions. Last I remember, we had decided that if you add
connectedness to your search criteria, then lattice TC was basically a weaker
version of your search -- which hardly seemed desirable.

So I proposed a version where the scale -- connected or not -- could be
rooted on any one of its members without having common tones jump scale
positions. I called this self TC. There are two versions: weak and strong.
Weak means you can take a scale and root it on any of its members without
getting ambigous intervals between the two scales. Say you're playing along
in Cmaj -- you can switch to Fmaj without a hangup. You can't switch to F#
without a hangup, but you'd never think of switching to F# because there
isn't one in the current scale. You can switch to D first -- then you'll
have an F# -- but each of the switches will be cool.

Strong TC says, "I'm going to be perverse, and demand that TC hold over more
than single depth of switch. It has to hold from C thru D to F#. Because a
listener could still have C in his mind when F# comes around."

>What would be different if you took the cross set of the lydian mode?

A single transposition could turn up an ambiguity.

>>It is also trivial that the Weak version varies across the modes of a
>>scale, since the column starting on F would be in the far left if the
>>cross set of the lydian mode had been taken instead of the ionion.
>
>But that is not all that would change:

Yikes!

>>Different only up to a transposition, which shouldn't matter.
>
>I already explained that. If you followed that explanation, you'd know that
>the Cartesian square of the C major scale, which you showed above, is
>identical to the Cartesian square of the Bb lydian mode, provided you rotate
>the numbering of the rows and columns:

Obviously the relations in the cross set stay the same, which is why Strong
TC is defined for scales, not modes. So Bb Lydian isn't Strong TC either --
no mode of the major scale is. Nor is Bb Lydian Weak TC, as Bb Ionion and C
Ionion are.

Hopefully all this is clear by now. I feel like a bit of an ass -- I've
managed to confuse everyone here, starting with myself -- over something
rather unremarkable.

Clearly the missing ingredient in all these TC's is attention to the
proportion of common tones under transposition. So basically, what it boils
down to is: "Paul, you could narrow your search by insisting that the
periodicity blocks be connected on the triangular lattice, and that
transposition by at least one of the connecting intervals gives a scale with
many common tones."

I'm fairly certain that such a search would find scales with high Rothenberg
efficiency.

-Carl

Carl, I'm happy to drop the subject if you are, but there are still some
things I don't understand. Like why you're using the Cartesian square in the
first place. Many of my favorite tunes modulate from C major to Eb major or
Ab major -- at least as convincing as a modulation to E or A.

>Carl, I'm happy to drop the subject if you are, but there are still some
>things I don't understand. Like why you're using the Cartesian square in the
>first place.

Well, don't let that stop you. It was an arbitrary choice.

>Many of my favorite tunes modulate from C major to Eb major or Ab major -- at
>least as convincing as a modulation to E or A.

These all involve either 3 or 4 accidentals. I'm suggesting they are less
of a jump (on average) than modulations with 5 or 6 accidentals. The idea
behind TC was that one should also ask if the chromatic shifts are confined
to scale degrees. I think it's interesting that any scale that has 1 TC
accidental when modulated by x is an MOS of x. I'm convinced that harmonic
scales with high efficiency are connected, oblong structures on the lattice.

-Carl