organsRgreat wrote:I've been reading about Messiaen's modes of limited transposition, and find that concept easy enough to understand. However I can't see what the “Transposition invariance inclusion lattice” is trying to tell me, and I haven't been able to find an explanation online. The chart is at on the SECOND PAGE of:

http://lulu.esm.rochester.edu/rdm/pdfli ... and.Tn.pdfCan anyone help? Thanks

Here's a go at deciphering the "Transposition Invariance Inclusion Lattice" which is related to Messaien's "Modes of Limited Transposition"... based strictly on looking at the patterns inherent in the stated Lattice.

The top item, "U" means Union as in "All-Inclusive". Let this represent a 12-note Chromatic Scale in standard Western usage.

The bottom item, "{}" represents the "Empty Set", that is a Scale so far reduced that there is nothing left. This Scale contains no notes at all.

Each of the fourteen other items on the Lattice corresponds to one of the fourteen Scales illustrated below the Lattice. The first number tells how many different notes are in the Scale. The second number I do not yet understand. The Roman Numerals (I through VII) indicate which of the Scales correspond to the ones used by Messaien in his System.

The Lattice is called an "Inclusion" lattice because each item "Includes" all the members of the items connected below it. This works if we consider that two of the vertical lines (connecting 6-30 up to VI, and connecting 6-30 down to 4-25) are spurious. {If anybody can justify the existence of those two vertical lines, I'd be happy to hear about it.}

From the bottom up, it is called an "Inclusion Lattice", but it works just as well from the top down as an "Exclusion Lattice". Let us consider each item in this fashion.

From the top "U" (12-note chromatic scale) down to the item III, there are three notes missing which are D#, G, and B which are an Augmented Triad (an Equal Division of the Octave).

From item III to the item 6-20, there are three notes missing which are D, F#, and A#, again an Augmented Triad.

From item 6-20 down to item 3-12, there are three notes missing which are Db, F, and A, again an Augmented Triad.

To get from item 3-12 down to the Empty Set "{}" we just omit the remaining three notes C, E, and G# , again an Augmented Triad.

From the top "U" down to item VII, two notes are missing which are F and B which are a Diminished Fifth (again an Equal Division of the Octave).

From item VII down to item II, two notes are missing which are D and G#, again a Diminished Fifth.

From item VII down to item VI, two notes are missing which are D# and A, again a Diminished Fifth.

From item VII down to item IV, two notes are missing which are E and A#, again a Diminished Fifth.

From item VI down to item I, two notes are missing which are Db and G, again a Diminished Fifth.

From item VI down to item V, two notes are missing which are E and A#, again a Diminished Fifth.

From item 6-30 down to item 4-28, two notes are missing which are Db and G, again a Diminished Fifth.

From item 6-30 down to item 4-9, two notes are missing which are D# and A, again a Diminished Fifth.

From item 4-28 down to item 2-6, two notes are missing which are D# and A, again a Diminished Fifth.

From item 4-25 down to item 2-6, two notes are missing which are D and G#, again a Diminished Fifth.

From item 4-9 down to item 2-6, two notes are missing which are Db and G, again a Diminished Fifth.

From item 2-6 down to the Empty Set "{}", we just omit the remaining two notes C and F#, again a Diminished Fifth.

Other connections shown:

From item II down to 6-30 we omit two notes E and A#, again a Diminished Fifth.

From item IV down to 6-30 we omit two notes D and G#, again a Diminished Fifth.

From item I down to 4-25 we omit two notes E and A#, again a Diminished Fifth.

From item V down to 4-25 we omit two notes Db and G, again a Diminished Fifth.

From item IV down to item V we omit two notes D# and A, again a Diminished Fifth.

From item V down to item 4-9 we omit two notes D and G#, again a Diminished Fifth.

So we have two Families of relationships in the Lattice. Along the left, reductions by an Augmented Triad, and for the rest of the Lattice, reductions by a Diminished Fifth (or Augmented Fourth, which is equivalent when using an equal-tempered system of twelve notes to the Octave).

Three lines are shown making a connection between the Families:

Item III includes all the notes in item 2-6, being C and F# (a Diminished Fifth).

Likewise Item VII includes all the notes in item 3-12 (being C, E, and G#, an Augmented Triad),

and Item I also includes all the notes in item 3-12 (again being C, E, and G#... an Augmented Triad).

This fully accounts for the entries and connections shown in the Lattice, so far as I understand it right now. Any further discussion or illumination is most welcome.