Manuel M. Ponce’s piano Sonata No. 2 (1916): An Analysis Using Signature Transformations and...
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Manuel M. Ponce’s piano Sonata No. 2 (1916): An Analysis Using Signature Transformations and Spelled Heptachords International Congress on Music and Mathematics Puerto Vallarta, México November 26-29, 2014 Mariana Montiel Department of Mathematics and Statistics Georgia State Universiy
Manuel M. Ponce’s piano Sonata No. 2 (1916): An Analysis Using Signature Transformations and Spelled Heptachords International Congress on Music and Mathematics
Manuel M. Ponces piano Sonata No. 2 (1916): An Analysis Using
Signature Transformations and Spelled Heptachords International
Congress on Music and Mathematics Puerto Vallarta, Mxico November
26-29, 2014 Mariana Montiel Department of Mathematics and
Statistics Georgia State Universiy
Slide 3
Manuel M. Ponces sonata no. 2 has a nationalist character. The
two themes of the first movement come from two folksongs: El
sombrero ancho and Las maanitas The date of this composition, 1916,
falls in what is still considered Ponces romantic period as opposed
to his modern style of later years However, the first movement of
the sonata is full of non-traditional chord progressions, of
dissonance, and the influence of the impressionism of his admired
Debussy.
Slide 4
Within the neo-Riemannian focus there have arisen several forms
of carrying out theoretical analysis of a score by means of
mathematical transformation groups. There is an undeniable
coincidence among these forms. However, each one offers unique
aspects that privilege the specificities of the piece itself and
the needs of the analyst. In this work we began to make use of
signature transformations, fruit of the theoretical development of
Julian Hook, a tool we thought could serve to analyze
transformations that Ponce carries out during the development of
the sonata.
Slide 5
Hooks signature transformations, that capture tonality in the
seven diatonic modes, offers the possibility of tracing diatonic
organization Interesting to experiment with Ponces piano sonata no.
2, which shows the characteristics of twentieth century musical
modernity, although classified as belonging to his romantic period.
Doctoral thesis in musicology that classified the work as
modal.
Slide 6
Signature transformations act on the set of fixed diatonic
forms. Fixed diatonic forms are equivalence classes of fragments of
diatonic music, with a key signature and a clef. These fragments
are in the same equivalence class if their pitch-class content is
the same (modulo 12), and if their key signatures are equivalent up
to enharmonic equivalence. For example, C and D major. For example,
C and D major.
Slide 7
We will use the notation S n, for the number n of sharps that
are added (or flats that are subtracted) and the number n of sharps
that are subtracted (or flats that are added), with n N The
operation of adding sharps (or subtracting flats) is positive The
operation of subtracting sharps (or adding flats) is negative. S -6
reduces the key signature by 4 sharps, and then we continue to
count negatively by adding flats:
Slide 8
The signature transformations form a cyclic group of 84
elements, generated by S 1 They pass through the twelve pitches of
the chromatic scale and the seven diatonic modes (although it is
not expected that 84 sharps should be added to the key signature!)
Sn and S-n can be reached through compositions with the chromatic
and diatonic transposition operators Tn and tn.
Slide 9
Adding seven sharps to a key signature will transpose the
diatonic collection a semitone (for example, from C major to C
major). Therefore, S 7 acts as T 1 Analogously, S -7 acts as T 11
Hence the validity of compositions such as and the perspective of
composition with Schritts
Slide 10
while the chromatic transposition operator implicitly changes
the key signature as well as the actual notes, the diatonic
transposition operator does not change the key signature. while the
chromatic transposition operator implicitly changes the key
signature as well as the actual notes, the diatonic transposition
operator does not change the key signature. that is, the diatonic
transposition operator transposes within its diatonic scale (but
can change the mode). If t 1 is applied to a diatonic fragment or
diatonic form-, without changing the key signature, we have the
same pattern in the pitches but transposed up a scale step.
However, if we apply S 12 we also transpose a scale step
Slide 11
Every transposition operator, whether chromatic or diatonic,
can be written as an S n for some n. Any S n can be written as a
composition of some T n and t n as the generator S 1 can be
obtained by: T n and t n as the generator S 1 can be obtained by:
Signature transformations can explain transformational aspects of
music that translates (transposes) its content between different
diatonic forms. This means that the transformations always occur
within a diatonic context that must be identified, something that
is not a requisite for other neo-Riemannian type transformations,
such as P,L, and R.
Slide 12
There are 477 measures in the first movement of the sonata,
without counting repetitions. The first 399 measures have a key
signature with four sharps, corresponding to C minor, or to C
Aeolian. In measure 400 the key signature acquires three more
sharps, for a total of seven, corresponding to C major, or C
Ionian. In the coda, that begins at measure 453, there is a return
to C minor until the end at measure 477.
Slide 13
We will begin our analysis with a passage and its diatonic
fragment that corresponds to measures 41 and 45.
Slide 14
To travel from G Locrian to D Locrian we can, of course, use
However, to illustrate the signature transformation perspective, we
can first go from G Locrian to G Phrygian by means of S 1, given
that G Locrian has 3 sharps and G Phrygian has the four sharps of D
locrian Then the Diatonic translation t 4 is applied to move from G
to D Hence the signature transformation that carries out the group
action is
Slide 15
G Locrian: G A B C D E F G D Locrian: D E F G A B C D G
Phrygian: G A B C D E F
Slide 16
Slide 17
Measures 227 and 228 are in E Aeolian, which only has one
sharp. Measures 231 and 232 are in G Aeolian which has five sharps.
To travel from E Aeolian to G Aeolian four sharps must be added, as
G aeolian has five sharps. This is done by S 4 which goes to E
Lydian, and then t 2 which transposes diatoncally by two
tones.
Slide 18
The signature transformation is: Of course, we can look at it
as: as well.
Slide 19
Measures 241 and 242 are in B Aeolian, which has two sharps.
Thus the signature transformation from G Aeolian with five sharps
to B Lydian is S -3 t 2 given that the diatonic transposition is a
generic third and the number of sharps is reduced by three
Slide 20
Hence the signature transformation is S -3 First t 2 is
applied, leading to B Lydian. Then three sharps are reduced to
arrive a B Aeolian.
Slide 21
If we were to commute, the operation could not be carried out
because no diatonic mode with tonic G can only have two sharps.
This just tells us that the mathematical possibility is not
relevant in this particular application. Once again, we could
arrive of course by
Slide 22
Measures 251, 252, 254, and 255 are in A Aeolian, which has no
accidentals in the key signature.
Slide 23
Measures 257, 258, 260, and 261 are in G Aeolian, which has
five accidentals in the key signature.
Slide 24
We arrive to G Aeolian, once again in order stipulated. If we
were to commute, the operation would not work. None of the seven
modes that have A as a tonic can have an A in their signature.
work. None of the seven modes that have A as a tonic can have an A
in their signature.
Slide 25
The transformation is from G Aeolian to F Aeolian in measures
263,264 and 266 and 267.
Slide 26
In the spirit of the signature transformation perspective, we
get the following composition: There are six diatonic steps from G
to F and the number of sharps in the key signature is reduced by
two The transformation passes through G Locrian which has three
sharps in the key signature and forecasts the D naturals in
measures 263, 264 (and 266, 267) of the original score. in measures
263, 264 (and 266, 267) of the original score.
Slide 27
Slide 28
Measures 89-91 are in C Mixolydian Measures 354-356 seem like
an ideal candidate for a signature transformation that would change
the mode while leaving the tonic fixed
Slide 29
The signature transformation should be S -1 given one sharp is
eliminated However, the eliminated sharp is A ; it is not possible
to go from 6 to 5 sharps removing A diatonically, it should be
E
Slide 30
In measures 354-356 with C as tonic we do not have any of the 7
diatonic modes We have the Hindu scale, or the Dorian mode of the
acoustic scale, or the fifth mode of the melodic minor scale, whose
pattern is 2212122 While not within the diatonic scheme, there is
definitely a voice leading phenomenon from 2122212 to 2212122.
Slide 31
Hooks work on spelled heptachords addresses non-diatonic
collections and actually classifies a rotation of the pattern of
the scale identified in measures 354, 355 y 356 under the name of
MMIN (for melodic minor). In this generalization of the signature
transformations, until now, there is no overarching mathematical
function that represents the change from a diatonic context to a
non-diatonic one (we always have a set bijection). However it can
be categorized within the theory developed in this article on
spelled hexachords.
Slide 32
' Spelled heptachords Spelled heptachords are sets of seven
pitch classes in which each letter name only appears one time. Any
diatonic scale is a spelled heptachord. Many almost diatonic scales
are spelled heptachords which are proper: free of enharmonic
doublings or voice crossings. The melodic minor (acoustic, Hindu),
harmonic minor, gypsy, mela dhenuka scales, among others, are
spelled heptachords.
Slide 33
In the previous example the mod-7 musical material does not
change at all; only the heptachord H changes, from DIA(+6) to
MMIN(+5). This field change (from DIA to MMIN) is similar to Hooks
field transposition (which changes the mode but maintains the
tonic), but it cannot literally be this type of transposition,
since it's not the same type of field (heptachord). Indeed, there
are 66 -classes (fields) of proper spelled heptachords.
Slide 34
Let k represent la transposition by fifths according to the
table. Then 3 ({C #,D #,E, F, G #, A,B # }) ={A #,B # C #,D,E, F, G
} Then 3 ({C #,D #,E, F , G #, A,B # }) ={A #,B # C #,D ,E, F , G
}
Slide 37
Indeed, if we look at the following passages from the sonata,
we find constant transformations between heptachords, both intra-
and inter-classes. Dorian mode (mode 2) of the ascending Melodic
Minor scale- Lydian mode of the Acoustic scale: 1212222 (step-wise)
211112: Class MMIN in Hooks line of fifths and heptachord classes.
Hungarian-Gypsy scale: 2131131 (step- wlse) wlse) 131131: Class GYP
in Hooks line of fifths and heptachord classes line of fifths and
heptachord classes Mixolidian mode diatonic scale: 2212221
(step-wise) 111111: Class DIA in Hooks generalized circle of fifths
and heptachord classes. Mela Dhenuka scale (mode 4, lidian):
2131122 (step-wise) 2131122 (step-wise) 112114: Class NMIN in Hooks
line of of fifths and line of of fifths and heptachord classes
heptachord classes
Slide 38
Dorian mode (mode 2) of the ascending Melodic Minor scale-
Lydian mode of the Acoustic scale. Hungarian-Gypsy scale: 2131131
Mela Dhenuka scale (mode 4, Lydian): 2211312 G#G#G#G# C#C#C#C#
F#F#F#F#
Slide 39
Hungarian-Gypsy scale: 2131131 Dorian mode (mode 2) of the
Melodic Minor scale- Lydian mode of the Acoustic scale. Mixolidian
mode diatonic scale E#E#E#E# D#D#D#D# A#A#A#A#
Slide 40
Clearly, as above: 3 ({G #, A, B,C, D, E,F # }) ={E #, F #, G
#, A, B,C #,D # } is an intra- scale transformation that goes from
the G # Dorian mode (mode 2) of the ascending Melodic Minor scale-
Lydian mode of the Acoustic scale, to its transposition to E #. The
same occurs with 3 applied intra the Hungarian Gypsy scale in the
passage we just heard. It is interesting to note that in all these
non- diatonic intra- scale changes (Acustic, Hungarian-Gypsy), the
transposition is 3
Slide 41
As we saw, there we do not have right now a mathematical
transformation to represent change between fields ( -classes of
spelled heptachords) ; However, the proper spelled heptachords have
similar symmetric properties to the diatonic collection (this is
not true for non-proper heptachords or subsets of spelled pitch
class space of other sizes)
Slide 42
The 66 -classes of proper spelled heptachords, plus the 462
spelled pitch class structures that are generated by complete
diatonic structures, provides a formal, mathematical and, above
all, detailed, way to analyze music that has often been labeled as
chromatic, without any further classification. Audtat and Junod
show (visually and auditively) the 462 modes of the diatonic bell,
that is, 66 representatives of Hooks -classes and their 7 rotations
(modes).
Slide 43
Questions However, is it possible to find algebraic
mathematical functions that represents the changes between
different classes of spelled heptachords (scales) ? In other words,
can we formalize at another level the work done till now? The
signature transformations are restricted to DIA; The -classes are
classified in terms of the fifth transpositions; However, every
proper spelled heptachord has seven modes. In DIA we can use
Sturmian morphisms to generate the modes. Can we generalize to the
spelled heptachords? If this was the case, how would these
morphisms relate to ?
Slide 44
The change can also be represented in terms of the two or three
dimensional lattices developed by Tymoczko (2011). In his figure,
page 111, an adjacency between F # diatonic and B acoustic is
generated. In our case, the change can also be conceived between F
# diatonic (given that C # mixolidian is a rotation of F # ionian)
and B acoustic (given that the Hindu scale that begins on C # is a
rotation of the acoustic scale that begins on B). These are the two
heptachords that Hook would label DIA(+6) and MMIN(+5).