about the terminology used here:

– for ease of reading, the modes are referred to according to their ‘transposition’: where MESSIAEN calls M3 3 the third transposition of mode 3 and M6 5 the fifth transposition of mode 6, we use DM3 (=F#M3=BbM3) and EM6 (=BbM6) / D and E are not ‘tonics’, but serve as a basis from which to construct the modes (even if for some modes this reference note can indicate a feeling of gravity) / for the sake of convenience, MESSIAEN starts from C to classify the modes, ‘but other notes can be used. MOLT have no initial tone’ ( TRCO, page 107 ) / the term ‘initial’ is used here to designate the reference note

– about the notion of transposition: let’s take the example of mode 4, which has 6 transpositions: according to the classification used by MESSIAEN, the initial disposition of CM4 is also its first transposition: CC#DFF#GG#B ( no transposition 0 ) / for the sake of sanity, we systematically choose the ‘transposition’ of the mode closest to C: FM7 is always preferred to BM7, DbM6 to GM6, EM4 to BbM4, DM2 to BM2, CM1 to AbM1, EbM3 to GM3, etc… F#M4 will always be replaced by CM4, AM6 will always be replaced by EbM6, etc… / for MOLT with added notes, the MOLTexplorer program uses circular reading / example: the set of notes C C# D E F F# G# A# B will be named CM6 add C#, but transposed by a tritone the set C D E F# G G# A# B will be ( wrongly ) named F#M6 addG ( see chapter on modes with added notes )


-intervallic terminology :

m2 = minor second

M2 = major second

m3 = minor third

M3 = major third

P4 = perfect fourth

#4/#11 = tritone, diminished fifth

P5 = perfect fifth

m6 = minor sixth, augmented fifth

M6 = major sixth

m7 = minor seventh

M7 = major seventh


– within a mode, notes are called ‘degrees’: E is the 3rd degree of CM6, G is the 6th degree of CM3, F is the 4th degree of CM4, etc… ‘by degree’ will be used to signify the movement of a note towards the note that precedes or succeeds it within a given mode/ e.g. for CM4, the ascending ‘by degree’ movement will be C Db D F F# G Ab B C, and the descending ‘by degree’ movement will be C B Ab G F# F D Db C.

– use of enharmony: the ‘tonal lisibility’ of intervals and chords is favoured / so, more often than not, B-A# will be preferred to B-Bb, B-D# will be preferred to B-Eb, etc. / for pragmatic and subjective reasons, Cb, Fb, E# and B# disappear from the language: D-B will therefore be preferred to D-Cb ( even if the reference is Eb min harmonic ), B-F is preferred to B-E# even if the reference is B maj #11 ), etc…. most often (but not always) F#-F is preferred to Gb-F (even if the reference chord is Gb maj) / other subjective choices are also made to make certain hybrid chords or very dense passages more lisible / attention is also paid to the left/right hand dissociation for the piano, by making certain triads/tetrads or groupings with a referenced tonal character recognisable… but it’s impossible to be systematic, given the abundance of language and the many possible interpretations, so subjectivity and interpretation will always favour common sense

–  use of ‘italics’ always indicate quotes from MESSIAEN (also from his writings or technical terms from his musical vocabulary), they are usually integrated into the narrative, and are immediately referenced to make reading smoother / as the web presentation format scrolls the text uninterruptedly, longer references generally appear at the end of the paragraphs or passages concerned / when a reference is missing, it’s because it’s been quoted from memory, or has been taken on the fly without noting the source / all references can be found on the ‘sources/inspirations’ page.

– some terms are ‘borrowed’ from set-theory and transformational theory (*1):

PC ( Pitch Class ) (in French : CH = classes de hauteurs)  : PC represents each of the 12 notes, regardless of enharmony or register. 12 chromatic notes = 12 PC: C=0, C#=1,…, B=11.

PC set (Pitch Class Set) (in French : ECH = ensemble de classes de hauteurs ): PC set represents the list of PCs contained in a set of notes, regardless of order or frequency of appearance, but ordered as close as possible to C. PC set of C Maj = [ 0, 4, 7 ], PC set of CM4 = [ 0, 1, 2, 5, 6, 7, 8, 11 ].

IC (Interval Class) ( in French : CI = classes d’intervalles ) : IC is the numerical representation of the number of semitones contained in an interval. M2=2, m2=1, #11=6, etc…

SI ( intervallic structure = structure intervallique in French ) :  within a given group of notes, SI lists the intervals from one note to the next in the order in which they appear, including the interval separating the last note from the first (*2) / for example, SI in CEG = ( 4, 3, 5 ).

IV ( Interval vector ) ( in French : VI = vecteur intervallique ) :  within a given group of notes, IV lists the ascending order of intervals existing between all the notes in the group / considering the equivalence between an interval and its complement ( m2=M7, M2=m7, m3=M6, M3=+5, P4=P5, #11=#11 ), CEG’s IV ( intervals CE, CG, EG, EC, GC, GE ) = < 0 0 1 1 1 0 > ( 0 m2, 0 M2, 1 m3/M6 -EG/GE-, 1 M3/+5 -CE/EC, 1 P4/P5 -GC/CG- )

IFUNC (Interval Function = fonction intervallique in French): within a given set of notes, IFUNC identifies the existing intervals between all the notes in the group (from C to E, from C to G, from E to G, from E to C, from G to C, from G to E), but clearly distinguishes between an interval and its complement: between m2 and M7, between M3 and +5, etc / the IFUNC of CEG = [ 3 0 0 1 1 1 0 1 1 1 0 0 ] ( 3 P8 -CC/EE/GG-, 0 m2, 0 M2, 1 m3 -EG-, 1 M3 -CE-, 1 P4 -GC-, 0 #11, 1 P5 -CG-, 1 +5 -EC-, 1 M6 -GE-, 0 m7, 0 M7 ) (*3)

PC set, PF, SI, IV and IFUNC are different analytical readings of the same group of notes (*4)

for the CEG set: PCs = [ 0, 4, 7 ], SI = ( 4, 3, 5 ), IV = < 0 0 1 1 1 0 >, IFUNC = [ 3 0 0 1 1 1 0 1 1 1 0 0 ] (*5)

(*1) : set theory introduces the notion of Prime Form ( PF, basic structure ), which can be defined as the reduction of a set of notes to its most compact form, to its essence / this theory was developed mainly by Allen Forte ( ‘the structure of atonal music’ ) and is essentially an excellent analytical tool / suggested reading for a precise explanation of the concept of prime form:  the structure of atonal music'( ALLEN FORTE ), ‘analytical approaches to 20th century music’ ( JOEL LESTER ),…/ for a clear and compact introduction to ‘set-theory’ and ‘transformational theory’ (in French): read MORENO ANDREATTA, ‘une introduction à la set-theory’ / the few definitions, explanations, references and classifications used here are inspired by them and serve mainly as a basis for analysis

(*2): there is a difference here with the system used by JOHN O’GALLAGHER who, in his great book ‘twelve tone improvisation, a method for using tone rows in jazz’, recommends, for practical reasons, reducing the SI of a triad to two intervals, ignoring the interval between the third and the first note / so CEG = 4 + 3 ( 4 3 )

(*3): the first number corresponds to the octave interval, so here it represents the cardinality of the set, i.e. the number of notes present

(*4): IV and IFUNC define the interval content of a set of notes. IV has the advantage of analytical conciseness, but in practice IFUNC distinguishes between an interval and its complement (between m2 and M7, between P4 and P5, etc.), so it is somewhat closer to musical ‘reality’

 (*5): the way in which sets and structures are presented often differ from one author, article or method to another / we chose for:

PC set : between brackets [ square braquets ] , with commas

SI : between brackets ( regular braquets ), with commas

IFUNC : between brackets [ square braquets ] , without commas

IV : between brackets < angle braquets >, without commas

PF: between brackets (regular braquets), with commas

nb: MOLTexplorer uses T (ten) for 10 and E (eleven) for 11


– acronyms/abbreviations :

MOLT = mode(s) of limited transposition


TLM = traité de mon langage musical

TRCO = traité de rythme, de couleur et d’ornithologie

DC = dodecaphonic complement

CM = complementary modes

CCR = chords of contracted resonance ( accords à résonance contractée )

CTI = chords of transposed inversion on the same bass note ( accords à renversements transposés sur une même note de basse )

RC = revolving chords ( accords tournants )

TS = tonal signatures

MS = modal signatures

DTS = disjoint tonal subsets

TC = talea color

t : transposition ( t+1 : a 1/2 tone higher, t-2 : a tone lower, etc…)

P: prime form (*)

R: retrograde movement

I: inversion

RI: retrograde movement of inversion

(*): the reference series in a dodecaphonic matrix ( P, R, i and Ri )  are numbered ( 0 to 11 ) according to their transposition /  the initial series is P0, unlike MESSIAEN which lists the initial configuration of a MOLT as its first ‘transposition’ ( there is no transposition 0 in MESSIAEN’s terminology )