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 sonata no. 2 has a nationalist
character.
The two themes of the first movement come from
two folksongs: El sombrero ancho and Las mañanitas
The date of this composition, 1916, falls in what is
still considered Ponce’s “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.
 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.
 Hook’s signature transformations, that
capture tonality in the seven diatonic modes,
offers the possibility of tracing diatonic
organization
 Interesting to experiment with Ponce’s 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”.
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.
 We will use the notation Sn, 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:
The signature transformations form a cyclic
group of 84 elements, generated by S1
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 .
Adding seven sharps to a key signature will
transpose the diatonic collection a semitone (for
example, from C major to C major).
Therefore, S7 acts as T1
Analogously, S-7 acts as T11
Hence the validity of compositions such as
and the perspective of composition with Schritts
 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 t1 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 S12 we also transpose a scale step
Every transposition operator, whether chromatic or
diatonic, can be written as an Sn for some n.
Any Sn can be written as a composition of some
Tn and tn as the generator S1 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.
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.
We will begin our analysis with a passage and its
diatonic fragment that corresponds to measures 41
and 45.
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
S1, given that G Locrian has 3 sharps and G Phrygian has the
four sharps of D locrian
Then the Diatonic translation t4 is applied to move from G to
D
Hence the signature transformation that carries out the group
action is
G Locrian: G A B C D E F G
G Phrygian: G A B C D E F
D Locrian: D E F G A B C D
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 S4 which goes to E Lydian, and then t2
which transposes diatoncally by two tones.
The signature transformation is:
Of course, we can look at it as:
as well.
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 t2
given that the diatonic transposition is a generic
third and the number of sharps is reduced by
three
 Hence the signature transformation is S-3
 First
t2
is applied, leading to B Lydian.
 Then three sharps are reduced to arrive a B Aeolian.
 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
Measures 251, 252, 254, and 255 are in A
Aeolian, which has no accidentals in the key
signature.
Measures 257, 258, 260, and 261 are in G
Aeolian, which has five accidentals in the key
signature.
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.
The transformation is from G Aeolian to F
Aeolian in measures 263,264 and 266 and 267.
 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.
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
The signature transformation should be S-1
one sharp is eliminated
given
However, the eliminated sharp is A ; it is not possible to
go from 6 to 5 sharps removing A diatonically, it should
be E
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 .
 Hook’s 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.
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.
 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 Hook’s “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.
Let k represent la transposition by fifths according to the
table.
Then
G }
3 ({C#,D#,E,
F , G#, A ,B#}) ={A # ,B# C#,D ,E, F ,
 Audétat and Junod arrange the 66 proper spelled heptachords in
their diatonic bell which they arrange in order of “closeness” to
the diatonic heptachord.
http://www.cloche-diatonique.ch/atlas/preview.php?lang=en&c=12&d=7
The 462 modes of the diatonic bell.
The Hungarian Gypsy scale 2131131 (step pattern), modulating from C to A
3 ({C#,D#,E, F , G#, A ,B#}) ={A # ,B# C#,D ,E, F , G }
(131131: Class GYP in Hook’s -heptachord classes)
 Indeed, if we look at the following passages from the sonata, we
find constant transformations between heptachords, both intraand 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 Hook’s line of
fifths and heptachord classes.
 Hungarian-Gypsy scale: 2131131 (stepwlse)
 131131: Class GYP in Hook’s
line of fifths and heptachord classes
 Mela Dhenuka scale (mode 4, lidian):
2131122 (step-wise)
 112114: Class NMIN in Hook’s
line of of fifths and
heptachord classes
 Mixolidian mode diatonic scale: 2212221
(step-wise)
 111111: Class DIA in Hook’s generalized
circle of fifths and heptachord classes.
Dorian mode (mode 2) of the ascending Melodic Minor scale- Lydian mode of the Acoustic scale.
G#
Hungarian-Gypsy scale: 2131131
Mela Dhenuka scale (mode 4, Lydian): 2211312
C#
F#
Dorian mode (mode 2) of the Melodic Minor scale- Lydian mode of the Acoustic scale.
E#
Hungarian-Gypsy scale: 2131131
Mixolidian mode diatonic scale
A#
D#
 Clearly, as above:
({G#, A , B,C, D, E,F#}) ={E #, F#, G#, A, B,C#,D #} is an intrascale 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 #.
3
 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 intrascale changes (Acustic, Hungarian-Gypsy), the transposition
is 3
 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)
 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 .
 Audétat and Junod show (visually and auditively)
the 462 modes of the diatonic bell, that is, 66
representatives of Hook’s -classes and their 7
rotations (modes).
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 ?
 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).