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Messiaen's Transposition invariance inclusion lattice

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Messiaen's Transposition invariance inclusion lattice

Postby organsRgreat » Fri Mar 24, 2017 7:20 am

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.pdf

Can anyone help? Thanks :-)
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Re: Messiaen's Transposition invariance inclusion lattice

Postby telemanr » Fri Mar 24, 2017 7:52 am

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Re: Messiaen's Transposition invariance inclusion lattice

Postby organsRgreat » Fri Mar 24, 2017 8:19 am

Thanks Rob – I'd already seen this, but the writer says “. . . These scales are known as the Modes of Limited Transposition, of which there are seven. We will cover all seven before venturing into their harmonic usages and the Transposition Invariance Inclusion Lattice”. So far as I can see, he doesn't get as far as explaining the lattice; he says “This concludes the introduction to Messiaen’s Modes of Limited Transposition. Eventually, I will create a Part 2 and beyond to fully cover the subject” - so presumably the information I'm looking for will be in that part 2.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby telemanr » Fri Mar 24, 2017 8:48 am

Are you interested in composing using Messiaen's methods or are you just intellectually interested. Because unless I were analyzing his pieces as a theory exercise I can't see it adding much when just listening to the music. But perhaps it does for you?
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Re: Messiaen's Transposition invariance inclusion lattice

Postby mstng67 » Fri Mar 24, 2017 11:01 am

I'm all for cool graphs and such, but I don't recall ever studying a "lattice" in grad school when we covered Messiaen and his compositional techniques in depth.

My wife, who teaches theory, points out that it looks like more recent theoretical research.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Fri Mar 24, 2017 4:20 pm

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.pdf

Can 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.
Last edited by OrganoPleno on Fri Mar 24, 2017 5:26 pm, edited 1 time in total.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Fri Mar 24, 2017 4:40 pm

OrganoPleno wrote: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.}


For the two mysterious vertical lines connecting above and below item 6-30, we can follow the derivations this way:

Item 4-25 contains the notes C,D,F#,G#, and then C again.
The intervals (measured in chromatic semitones) are 2,4,2,and 4.

If we split each step of 4 semitones into a step of 3 semitones plus 1 semitone,
the sequence then becomes 2,3,1,2,3,1.

This corresponds to the intervals in item 6-30 which contains the notes C#,D#,F#,G,A,C, and C# again.
Notice that we had to shift the starting point one step away from C to make the matched pattern visible.

If we now split each step of 3 semitones into a step of 2 semitones plus 1 semitone,
the sequence then becomes 2,2,1,1,2,2,1,1.

This corresponds to the intervals in item VI which contains the notes D,E,F#,G,G#,A#,C,C# and D again.
Again notice that we had to shift the starting point now two steps away from C to make the matched pattern visible.

So this gives some justification for those two vertical lines in the Lattice.

Studying and splitting the Sequences this way may account for all of the other connecting lines in the Lattice as well, but I have not yet had the chance to look into this.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Fri Mar 24, 2017 4:56 pm

OrganoPleno wrote:Studying and splitting the Sequences this way may account for all of the other connecting lines in the Lattice as well, but I have not yet had the chance to look into this.


Yes, this works too.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Fri Mar 24, 2017 5:13 pm

OrganoPleno wrote:.Based strictly on looking at the patterns inherent in the stated Lattice...
The first number tells how many different notes are in the Scale. The second number I do not yet understand.


Well, no wonder. Because it is NOT inherent in the stated Lattice. Rather, it refers to Allen Forte's collection of atonal sets.

https://en.wikipedia.org/wiki/Forte_number

"The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches."

That's fine too, so long as we remember that this whole business is rather arbitrary.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby organsRgreat » Fri Mar 24, 2017 6:08 pm

Rob: partly intellectual curiosity - I love Messiaen's harmonies, and would like to understand better the way he achieves his effects. For instance, knowing that Debussy made extensive use of whole tone scales helps me understand the static quality of some of his music – there are only two possible versions of a whole-tone scale; further starting notes only produce transposed versions of the first two. Something else I need to look into is the way Messiaen uses his system to create music which also makes sense in terms of “old-fashioned” diatonic harmony.

I entirely agree that knowing the theory behind a piece of music doesn't necessarily add to enjoyment – or even lead to a better performance. After playing the great Schumann Fantasy (Opus 17) for forty years I suddenly realised that the opening figuration, in the left hand, is based on the same notes as the theme with which the right hand subsequently enters – A G F E D (this video shows the score along with the music - https://www.youtube.com/watch?v=l5cmBah0F20 ; in the LH Schumann omits the E, presumably because it would be too dissonant in a passage that needs the sustaining pedal).

Knowing the structure hasn't affected the way I play that section. The knowledge that helps there is of Schumann's deep love for his piano teacher's daughter, whom he subsequently married.

mstng67: Thank you for your contribution, and your wife's. I am reassured that my failure to understand the diagram will not prevent me understanding Messiaen's composing technique.

organopleno: Thanks for taking so much trouble to explain all this! I know that a post of this length takes a while to write, quite apart from the thought behind it; and you've since added two further points. I can now see why the diagram didn't make sense to me – when I was at school maths teaching didn't include lattices! Fortunately I know a retired maths teacher, so I'll be able to get some one-to-one tuition. I expect that to someone who understands lattices this chart could be helpful, but I just found it confusing.

So – the quest continues, but I already understand a lot more than I did at the start of the day. Thanks :-)
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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Fri Mar 24, 2017 6:40 pm

organsRgreat wrote:I've been reading about Messiaen's modes of limited transposition, and find that concept easy enough to understand.


Here's a discussion of Messaien's "Modes of Limited Transposition", based on what I've learned here today.

We start with a series of intervals, measured in chromatic semitones. The simplest would be 1,1,1,1,1,1,1,1,1,1,1,1 which would represent the Chromatic Scale. Messaien does not use this as such. We could say it is because twelve notes is too many for a scale according to his preference (representing as it were the full universe of possibilities rather than any selected subset).

Another possibility would be 1,5,1,5. This still totals 12 semitones, but Messaien does not use this. We could say it is because four notes is not enough for a scale according to his preference (being rather more like a chord than an actual scale).

So we cannot make a proper scale using a "5" or any larger number.

Using a "4", we could get 2,4,2,4 (not enough notes), or 1,1,4,1,1,4 (finally we've found a good one).

Using a "3", we might try 1,2,3,1,2,3 but Messaien rejects this. We could say it has too many kinds of steps, since none of Messaien's Modes use more that two sizes of steps.

We could consider a scale such as 1112111211 or 111211122, but Messaien introduces a further requirement for Symmetry. Neither of these scales is built up of any repeating unit, therefore they do not qualify.

So here are the Scales which Messaien selects:

1,1,4,1,1,4 This has a repeated unit of 1,1,4 (containing three steps representing three modes) which is repeated twice (representing two axes of symmetry). Called Mode V.

1,1,1,3,1,1,1,3 This has a repeated unit of 1,1,1,3 (containing four steps representing four modes) which is repeated twice (representing two axes of symmetry). Called Mode IV.

1,1,1,1,2,1,1,1,1,2 This has a repeated unit of 1,1,1,1,2 (containing five steps representing five modes) which is repeated twice (representing two axes of symmetry). Called Mode VII.

2,2,2,2,2,2 This has a repeated unit of 2 (containing one step representing one mode) which is repeated six times (representing six axes of symmetry). Called Mode I.

1,2,1,2,1,2,1,2 This has a repeated unit of 1,2 (containing two steps representing two modes) which is repeated four times (representing four axes of symmetry). Called Mode II.

1,1,2,1,1,2,1,1,2 This has a repeated unit of 1,1,2 (containing three steps representing three modes) which is repeated three times (representing three axes of symmetry). Called Mode III.

1,2,2,1,1,2,2,1 This has a repeated unit of 1,2,2,1 (containing four steps representing four modes) which is repeated twice (representing two axes of symmetry). Called Mode VI.

These seven Modes appear to exhaust all of the possibilities within the given constraints.

As to terminology, what the PDF link refers to as "Modes" are actually the number of unique TRANSPOSITIONS of the scale, NOT the number of Modes (which are the different starting points possible within a given sequence). The ARTICLE linked above seems to be more accurate in its terminology.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby mstng67 » Fri Mar 24, 2017 7:01 pm

Of all the modes, the most frequently used with Messiaen and his contemporaries is number 2 (octatonic). I recall writing an short organ piece using it many years ago as an assignment for a doctoral theory class on Messiaen and his Contemporaries that I titled "À minuit avec ma petite fille." (I was often awake at midnight with my baby daughter - great time to compose.) Writing it helped me understand the Octatonic scale - and Messiaen - much better.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Fri Mar 24, 2017 7:17 pm

mstng67 wrote:Of all the modes, the most frequently used with Messiaen and his contemporaries is number 2 (octatonic).


And of course the Simple Symmetry of the Sequence (1,2,1,2,1,2,1,2) is very pleasing as well.
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Re: Messiaen's Transposition invariance inclusion lattice

Postby mdyde » Sat Mar 25, 2017 5:40 am

[Topic moved here.]
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Martin.

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Re: Messiaen's Transposition invariance inclusion lattice

Postby OrganoPleno » Sun Mar 26, 2017 12:06 am

OrganoPleno wrote:For the two mysterious vertical lines connecting above and below item 6-30, we can follow the derivations this way... So this gives some justification for those two vertical lines in the Lattice.


Here's a better discussion of these two remaining connections in the Lattice. They work in the same way as the other connections shown, in that the lower item is a subset of the upper item for each connection, however for these two cases the relationship only becomes clear after a Transposition.

For the lowest item here in question, called "4-25", its members are C,D,F#, and G#.
Let us transpose up a semitone and obtain now Db,Eb,G,and A. We see that this now matches the members given for item "6-30" except for missing the C and the F# contained in the latter item. As previously, a Diminished Fifth has here been omitted.

Now take the members of item "6-30", which are C,Db,Eb,F#,G,and A.
Let us transpose up a semitone and obtain now Db, D,E,G,Ab,and Bb. We see that this now matches the members given for item "8-25 VI" except again for missing the C and the F# contained in the latter item. As in the previous cases, a Diminished Fifth (or Augmented Fourth, if you prefer) has been omitted.

So, allowing for a semitone transposition in these two cases, we can understand the entire Lattice as displaying the relationships between items, such that for every connection shown, the lower item is a proper subset of the upper item, being reduced from the upper item by either two notes (which are always a Diminished Fifth apart) or by three notes (which always form an Augmented Triad). Noting also that the Diminished Fifth divides the Octave into two equal parts, and the Augmented Triad divides the Octave into three equal parts... these being the bases of the Symmetry expressed throughout this entire Lattice, and therefore reflected in the Seven Modes used by Messaien (as displayed in the Lattice).

Thank you to all for this stimulating discussion, which provided the opportunity to dig into these Musical Relationships a little more deeply.
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