Talk:Modes of limited transposition

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Part of me really wants to add a section about the relationship of these scales to the linear temperaments that have a 12-note MOS, (namely "augmented", "diminished", "diaschismatic" or "srutal", and an unnamed one with a sixth-octave period) because it's so beautiful the way it works out, but part of me says no, because I don't have any published, verifiable sources. —Keenan Pepper 00:35, 6 May 2006 (UTC)

Write an article about it, post it online (Google Sites is good), and link it right here. I'd love to read your thoughts on it. (talk) 22:15, 15 June 2009 (UTC)
Mode 2 has 3 tranpositions, not 2.

Slurs in first example[edit]

The description of the whole-tone scale talks about six groups of two, which I suppose refers to the six sequential major seconds. But the slurring doesn't correspond to the text. Can someone go back to the original and put in the six slurs? Alternatively, the text should explain. David Brooks 00:11, 4 September 2007 (UTC)

Fixed. DerHindemith (talk) 12:19, 29 January 2008 (UTC)

Is the fourth mode wrongly illustrated?[edit]

It seems to me that the intervals for the fourth mode ought to be (in terms of semitones):

1 1 3 1, 1 1 3 1.

But on the picture, it reads:

1 2 2 1, 1 1 3 1.

Not symmetrical as it's claimed to be.

I don't see a problem. Compare the mode examples with, for instance, this. They seem to agree, and the online journal predates this example so can't be derived from it. The fourth mode intervals are 1 1 3 1, 1 1 3 (1). Transpose to F and you get the same notes. David Brooks 22:39, 7 September 2007 (UTC)
David, you are right.
My misreading.Xplorr 03:42, 11 September 2007 (UTC)

more than messiaen[edit]

1) tritones

T T - C Gb C
x - - - - - x - - - - - x
1 mode
6 keys

2) major thirds

M3 M3 M3 - C E Ab C
x - - - x - - - x - - - x
1 mode
4 keys

3) minor thirds

m3 m3 m3 m3 - C Eb Gb A C
x - - x - - x - - x - - x
1 mode
3 keys

4) Messiaen's 1st mode (Wholetone scale/Major seconds)

1 1 1 1 1 1 - C D E Gb Ab Bb C
x - x - x - x - x - x - x
1 modes
2 keys

5) Messiaen's 2nd mode (Diminished Scale/Semitone-tone scale/Tone-semitone scale)

1 ½ 1 ½ 1 ½ 1 ½ - C D Eb F Gb Ab A B C
x - x x - x x - x x - x x
2 modes
3 keys

6) Messiaen's 3rd mode

½ ½ 1 ½ ½ 1 ½ ½ 1 - C Db D E F Gb Ab A Bb C
x x x - x x x - x x x - x
3 modes
4 keys

7) Messiaen's 4th mode

½ ½ ½ m3 ½ ½ ½ m3 - C Db D Eb Gb G Ab A C
x x x x - - x x x x - - x
4 modes
6 keys

8) Messiaen's 5th mode

½ M3 ½ ½ M3 ½ - C Db F Gb G B C
x - - - x x x - - - x x x
3 modes
6 keys

9) Messiaen's 6th mode

½ ½ 1 1 ½ ½ 1 1 - C Db D E Gb G Ab Bb C
x x x - x - x x x - x - x
4 modes
6 keys

10) Messiaen's 7th mode

½ ½ ½ ½ 1 ½ ½ ½ ½ 1 - C Db D Eb E Gb G Ab A Bb C
x x x x x - x x x x x - x
5 modes
6 keys

11) ½ 1 m3 ½ 1 m3 - C Db Eb Gb G A C
x - x - - x x - x - - x x
3 modes
6 keys

12) M3 1 M3 1 - C E Gb Bb C
x - - - x - x - - - x - x
2 modes
6 keys

13) P4 ½ P4 ½ - C F Gb B C
x - - - - x x - - - - x x
2 modes
6 keys

14) m3 ½ m3 ½ m3 ½ - C Eb E G Ab B C
x - - x x - - x x - - x x
2 modes
4 keys

15) m3 1 ½ m3 1 ½ - C Eb F Gb A B C
x - - x - x x - - x - x x
3 modes
6 keys

Maybe some of the above needs a mention??? Are any of them legal, i'm not sure, i've been scouring his book, but i don't quite know what he means by truncated.Dominant7flat9 (talk) 01:22, 4 May 2009 (UTC)Dominant7flat9 (talk) 20:26, 3 May 2009 (UTC)

If I understood correctly, truncation is just removing notes. Except that the result has to also be symmetrical. So C Eb Gb A is a truncation of the C diminished scale, for example. But some of your examples seem to be valid... I can't find the augmented scale, for instance, as a truncation of any of the ones he mentions. (It's number 14 on your list). (talk) 16:29, 18 June 2010 (UTC)

do these two modes fulfill messiaen's criteria??[edit]


½ 1 m3 ½ 1 m3 - C D♭ E♭ G♭ G A C 3 modes 6 Transpositions


m3 1 ½ m3 1 ½ - C E♭ F G♭ A B C 3 modes 6 Transpositions

I'm really not sure, i've taken them out of the page because the groups aren't symmetrical? Although you could argue that neither is messiaen's 4th mode. They are 'modes of limited transposition' though! Can anyone qualify this. Dominant7flat9 (talk) 01:04, 4 May 2009 (UTC)

They do fulfill the criteria. If you examine the scales closely, you see that they have only 3 modes each, and transpose 5 times, making 6 different versions in all - even though they are not included in Messiaen's list. Contrary to a comment in the article, Messiaen's list is not complete. Another scale that meets the requirements is the so-called augmented scale: example C D# E G Ab B C. It has three modes, and four possible transpositions (including the original version).
Messiaen's 4th mode is as symmetrical as any of the others are, although I can see two quite distinct meanings that "symmetrical" can have when applied to scales, so I don't think it's a good term to use, although it's probably so entrenched now that it may never be possible to change it.
In my own analysis in the past, I have used the terms "periodic" and "reflective" for the two types. The Messiaen type of scale is periodic, because it can be divided into two or more periods which duplicate each other exactly in intervallic make-up. A reflective scale (which is symmetrical in a sense) is one which when mirror-reversed ends up in the same scale. A Dorian scale beginning on D, for example, is a reflective scale, but certainly not periodic in the sense that I described.
If the terms "periodic" and "reflective" could become standard terminology for the two types of scale, the ambiguous term "symmetrical" could be avoided altogether, since every scale that can be described as symmetrical would fall into either or both of the categories "periodic" and "reflective". M.J.E. (talk) 15:58, 12 August 2009 (UTC)


Ok, i've researched this and will write a section on truncation which incorporates the other modes i posted above. Dominant7flat9 (talk) 12:46, 4 May 2009 (UTC) Right, i've put all the truncation stuff in, and as far as i'm concerned this page is DONE - that is until all the complaints start flooding in....... Dominant7flat9 (talk) 18:04, 4 May 2009 (UTC)

Relating to all the detailed discussion of truncated modes: while that is so as far as it goes, I think the importance of truncation is overestimated, and I think the article makes it seem more significant in a fundamental sense than it really is.
In reality, I believe it can be shown quite objectively that the distinction between truncated and non-truncated modes is almost entirely arbitrary. Let's consider the chromatic scale: it is certainly a mode of limited transposition, even though the article doesn't mention it at all, because it meets the stated criteria: it has only one mode, and it cannot be transposed without giving the exact same set of notes. If you consider that scale, *every* other scale or mode is a truncated form of that - and it seems to me to render the distinction quite arbitrary and rather irrelevant.
Years ago I did a detailed analysis of what I call periodic scales and their properties and relationships, which are quite intricate. What I call periodic scales are exactly the same as modes of limited transposition; but, for my own use, I avoid using Messiaen's name for them, because there are many more such scales than the seven Messiaen listed, and I feel that Messiaen's name can be applied properly only to the seven he catalogued and named that way. They are sometimes called "symmetrical scales", and I used to call them that once; but I stopped when at some time I realized that there are two quite different ways in which a scale can be considered symmetrical, and so that made the name ambiguous. "Periodic scale", as I use it, signifies that type of symmetry in which a scale can be divided into two or more equal portions, all of which are identical in intervallic make-up. (The other sense is that a scale can be inverted, mirror-fashion, to yield the same scale, as is the case with the Dorian mode, for example - which, as a 7-note scale, is certainly not a periodic scale or mode of limited transposition, thus showing that we are talking about two quite different forms of symmetry here.)
It is possible to draw a complicated flowchart showing what periodic scales are subsets of another (that is, truncated versions of them), and indeed I did that once. Doing this shows that every single periodic scale other than the chromatic scale is a truncated version of at least one other periodic scale, usually of several different ones. This chart is available on-line, actually - I posted it, intending to accompany it with a detailed article containing the findings of my analysis of all periodic scales, but one thing and another happened, and I never did manage to do that - but the image is still there, completely orphaned - here, if anyone's interested:
Study of this flowchart definitely shows which scales are truncated ones (all but one are), and what other scales they are truncated from. But, rather than classifying scales, in black-and-white fashion, as truncated or not, it instead (and far more usefully, I believe) shows the exact relationships involved, and what other scales a particular one is descended from (truncated from).
There are 16 periodic scales in all, containing from 2 to 12 notes - or 17, if you also include a hypothetical or theoretical empty scale or zero-note scale - which is meaningless in a practical musical sense, but does make a certain theoretical sense, and gives an extra degree of symmetry to the entire structure of periodic scales. These are shown in my flowchart as circles containing numbers, with the numbers representing their intervallic make-up, counted in semitones, plus an abbreviated name which is intended as a mnemonic for the basic nature of the scale. I arranged the numbers in circular form to avoid making any one of them appear as an arbitrary starting point, thus implying a particular tonic note or starting note for the scale. (Messiaen's 7 scales are marked in the diagram as "M1", "M2", etc.) Straight lines join some circles to each other, and not others. If you look at one circle (scale), it is a truncated version of any other scale joined to it with a line that lies further down the diagram. Scales higher up the diagram which are joined to it are truncated versions of it. (In constructing the diagram, I thought of scales as being formed by adding notes to simpler scales rather than by subtracting notes from more complex scales - so that's why you have to go upwards rather than downwards to find a truncated version of a given scale.)
Obviously, without sources other than myself, none of this stuff can be put into the article - while I did do this research on my own, without reference to any other sources, I don't for one moment think it is likely I am the first to have done this, so it may not be original research in a strict sense - but I am sure it *is* original research in the meaning of that term in the Wikipedia context. But I mention it here, behind the scenes, to explain why I believe the article makes too much significance of truncation as a property of a scale which is either present or not, in a black-and-white fashion - I would believe the article should be modified to reduce this inappropriate slant. I'm not sure I would take it on myself to do that, though - maybe there's someone who knows more about the subject especially in the Messiaen context who could do this without making it original research. M.J.E. (talk) 10:17, 9 December 2013 (UTC)

What is a mode?[edit]

The article uses the word "mode" to mean two different things:

  1. A particular cyclic sequence of musical intervals, of which Messiaen defined seven. The white notes of a standard piano, extended indefinitely, are one transposition of a single mode by this definition.
  2. A particular phase of such a cyclic sequence. The Gregorian modes, major mode and minor mode are (more or less) modes by this definition.

It may be the case that Messian used the term in a non-standard way, but I'm not sure that the article adequately addresses the use of the term to mean two different things. As a result, the statement "Therefore mode 2 has two modes" sounds somewhat puzzling.

What's the best way to address the distinction between the two meanings? Should we be using different terms for the two things, or would it suffice to write a sentence or two to note the ambiguity of the word? — Smjg (talk) 23:12, 21 February 2016 (UTC)

Mode 5 as truncation of 6[edit]

It's more obviously a truncation of 4; I haven't played around with 6 enough to confirm one of its modes isn't equivalent to 5 after truncation - but it lacks the G natural as written. 5 is very obviously a truncation of 4, though.