2. 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

3.  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.

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.

5.  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”.

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.

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:

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.

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

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

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.

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.

13.  We will begin our analysis with a passage and its diatonic fragment that corresponds to measures 41 and 45.

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

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 

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.

18. The signature transformation is: Of course, we can look at it as: as well.

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

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.

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

22.  Measures 251, 252, 254, and 255 are in A Aeolian, which has no accidentals in the key signature.

23.  Measures 257, 258, 260, and 261 are in G  Aeolian, which has five accidentals in the key signature.

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.

25.  The transformation is from G  Aeolian to F  Aeolian in measures 263,264 and 266 and 267.

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.

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

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 

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.

31.  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.

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.

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 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.

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  } 

35.  Audétat and Junod arrange the 66 proper spelled heptachords in their diatonic bell which they arrange in order of “closeness” to the diatonic heptachord. The 462 modes of the diatonic bell The 462 modes of the diatonic bell. http://www.cloche-diatonique.ch/atlas/preview.php?lang=en&c=12&d=7

36. 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 }  3 ({C #,D #,E, F , G #, A,B # }) ={A #,B # C #,D ,E, F , G  } (131131: Class GYP in Hook’s  -heptachord classes)

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 Hook’s line of fifths and heptachord classes.  Hungarian-Gypsy scale: 2131131 (step- wlse) wlse)  131131: Class GYP in Hook’s line of fifths and heptachord classes line 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.  Mela Dhenuka scale (mode 4, lidian): 2131122 (step-wise) 2131122 (step-wise)  112114: Class NMIN in Hook’s line of of fifths and line of of fifths and heptachord classes heptachord classes

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#

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#

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

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)

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.  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).

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  ?

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).