Rhythmic design in Mozart's piano sonata VII K.309 and Beethoven's Op.110

In Charles Rosen’s “The Classical Style” the author comments on “two principal sources of musical energy… dissonance and sequence – the first because it demands resolution, the second because it implies continuation”.

Naturally the main discussion of sonata form is concerned with pitch and key, and rhythm often takes a second or third place. This blog puts a filter on pitch and key and looks primarily at the rhythmic play in Mozart’s Piano Sonata No.VII.  There are 14 rhythmic cells in the exposition, only four of these are used in the development and the design of the recap is nearly identical to the exposition. In terms of bar length the exposition is 58, the recapitulation 58 + 4 for the coda; the development can be taken as 28 bars (just short of half the outer sections) with 8 bars of play on figures a and b as a transition to the recap. The opening bars are subdivided into two figures (a and b) as the second is used to generate momentum, particularly in the development. Some of the rhythms are varied, always with values being shortened to propel the music forward; there is one example of a phrase in diminution (bar 45 plays on the second theme – bar 35) which links to LH /RH semiquavers leading us to expect the cadence which is humorously delayed, which is just as well as Mozart gives us a delightful syncopated rhythm in the codetta. One may also note that the transition has the most pedestrian of the rhythms which only highlights the character of the principal phrases. There are a few bars of simple rhythmic filling as one would expect, but this is very limited in its use.

Much, or all of this commentary may be very familiar to those who regularly play these sonatas, but in the world of education revisiting the familiar often helps us see where we have ourselves become lax with acquaintance. Some commentators have described the design as "text book sonata form", however one responds to that observation the rhythmic work is far from commonplace. Even within the expectations of sonata form the use of dissonance (in the sense of chords being distant from the tonic) the recapitulation provides additional and surprising tension before settling down to the tonic. Playing through the sonata and considering the rhythmic play has refreshed my thinking about composition and for this reason alone I am happy to share the observations in this blog.

Mozart's early sonata No. 7 is full of rhythmic diversity with a large number of different rhythmic cells in the exposition. These are considerably reduced in the development and reinstated, as one might expect, in the recapitulation.

it would be strange if Beethoven's approach to rhythm had not developed in the Mozart in the intervening period. Like Mozart's sonata Beethoven's exposition is full of rhythmic play, the essential difference is that each of Mozart’s rhythmic cells has its own identity while Beethoven's rhythms create the sensation of evolving from each other. One would expect the repetition of rhythms (Mozart usually repeats his twice or three times), in Beethoven’s hands these are fewer and successive phrases undergo small degrees of variation.

The opening four bars are rhythmically complex, the first pair of notes being followed by their diminution, the second phrase takes this second (shorter) rhythm and extends it with a flourish (trill and demisemiquavers). This moves the musical argument on to the “proper” start of the music, the same intention is heard again at bar 27.

The rhythm of bars 5 and 6, and their exact repetition 7 and 8 have a simpler outline with bar ten developing the pattern in the second half. Bar 6 is a repeat of bar 2 so we can hear 5 and 6 as a simplification of the opening bars.  The increased momentum of bars 9 to 11 leads to the demisemiquaver arpeggios.  Looking at the pitch structure for a moment each first beat of four demisemiquavers form octaves over 5 octaves, and the momentum is marked by a rising scale on the beat (A flat, G, A flat to A flat one octave above).

The rhythmic interest is not in the running demisemiquavers but in the left hand where the pulse is repeated twice and then developed by chords on the second and third beats.  From bar 20 the interest is in the left hand / right hand interplay of dotted quaver semiquavers figures. This gentle (legato) section has the effect of slowing the music down at the half way point of the exposition with the trills acting as a further restraint before the demisemiquaver turn of bar 27 moves on to the second half of the exposition. Bar 31 sees another close variation on the opening figure before going through an extended semiquavers passage taking us back to the development. The presence of the opening bar is also felt at bar 33 and in augmentation in the left hand at 36/37.

Several commentators have noted the simplicity of the design of the development, from the rhythmic viewpoint bars 1 and 2 form the main argument taking us from bar 40 to 55 with eight repetitions. After the first four bars (40 – 44) the left hand rhythm forms syncopated phrases from the second semiquaver beat, this has been prepared for the listener in bars 33 to 35.

56 marks the start of the recapitulation. Bar 55 has the left hand provide the demisemiquaver lift into the main argument, this time the pulse heightened with continuous 32^nd notes in the left hand while the right has 2 versions of the opening rhythm before left and right hands exchange the role. Bar 63 follows the pulse pattern of bar 5 onwards with slight variations increasing the momentum. The argument is familiar even if small changes take place, however bar 100 introduces a new rhythm with a minim tied over to a quaver, quaver minim. This interjection might be taken as a final breath before moving to the close but it may also be heard as a preparation for the much simpler rhythms of the Allegro molto.

The Mozart sonata was composed in 1777 and the Beethoven in 1821, the 44 years difference mark considerable changes in the use of key structure, tonal qualities of the piano and many other features.  The developments in rhythm are of equal interest, of course it can be argued that the rhythms of this particular Mozart sonata are notable for their “small unit” approach. It is, in my opinion, worthwhile exploring these works through a filter to reveal their structures; though our use of rhythm has evolved to explore repetition and variation in both simpler and more complex forms there is much to learn from these works about creating momentum that can be of use in music today.

Frank Bridge Piano Sonata commentary

Now in PDF format so tables and illustrations are easier to read.

Bitonality and Polytonality

Polytonality and bitonality

This link takes you to the full article on the subject, please let us know if you have any feedback or any questions arising from the topic.

KH/NE

This post introduces Nurtan's thoughts on polytonality in the form of a series of articles forming a guide to the topic.  I found it an informative read and felt like I was being invited to travel across the world with him in his exploration of the topic.

PRACTICAL GUIDE TO POLYTONALITY

PREFACE:

The series of discussion papers on polytonality, bitonality and polymodality featured in this blog are not scholarly papers or extensive instructions on how to compose in a particular style. There will only be a few bookish references and technical detail will be kept to the minimum. The musical examples cited range from well know pieces to fragments of our own compositions.

This raises the question of the utility of this effort. That is easy to answer. Polytonality had a long history in different cultures and probably used extensively prior to music of early renaissance. We hear the traces of this in the folk music of Greece, Turkey, Armenia, Ukrainian and Russian steppes, Northern Persia and India. These folk songs have been transmitted by oral tradition and reflect to a large extent the ease of achieving expressive, emotional content by their use.

By definition, polytonality uses more than one key simultaneously. This is fairly easy to achieve through writing each voice in a different key. However, the selection of the melodies and the choice of keys can get complicated. In addition if the polytonal passages are too complicated, the ear will pick a key and unintentionally hear only that, the music may sound highly ornamented or off key. The polytonal music with three or more tonal centres will be discussed towards the end of this series.

If an entire composition of a section or a passage uses two tonal centres it is called bitonal. depending upon the mode of each key bitonal scales provide a very wide variety of musical language. In general, bitonal sections might be unimodal (for example, each is major, or minor etc.) but there is no restriction on this aspect. Also, the tempered scale of the Western music is not a requirement for the bitonal-bimodal arrangements.

I think the music of the Roman liturgy; Gregorian chant and medieval music are probably the exception rather than the universal in Western musical practice – especially in secular music. The huge influence of the Byzantine Empire on the culture of Eastern Mediterranean regions is undeniable. The examples of the use of polytonality in generating simple polyphony still survive in Armenian, Greek, Kurdish and Turkish multi-instrument folk music. This seems to be continuous from the antiquity through the Greek, Byzantine, Armenian, Ottoman and Islamic heritages.

From the early Baroque through common practice until late in 19^th century bitonality was more or less dormant, Composers seeking more a colourful palette started to weaken the rules of common practise tonality; which in turn led to several schools of thought in the harmonic language of the Western music. French Romantic Music of Debussy, Saint-Saȅns, et al. introduced ambiguous excursion (Debussy's Prélude à l'Après-midi d'un Faune starts and ends in E major, but it is all-over the place in between). Arnold Schoenberg introduced serial composition in which the tonality is deliberately destroyed.

One result of this activity is that composers start using bitonality as a means to write expressive and emotionally charged music, Charles Ives: Psalm 67 (1902), Frank Bridge: The Sea Idyll*, Capriccio No1, & No 2 (1908), Béla Bartok: 14 Bagatelles (1908), Igor Stravinski: Petrushka (1911) firmly established the use of bitonality. Later on Darius Milhaud: Saudades do Brasil (1920), Benjamin Britten in his  operas  Billy Budd and Peter Grimes among other composers completely integrated the technique into the  musical vocabulary.

Several scholars and theorists object to the concept of polytonality and present cogent arguments to give the technique other names and attributes. Since the technique survived over 100 years under this name, we present the rose is a rose argument for keeping the name as it is for practical purposes.

* The Sea Idyll was the first piece analysed by us when we started these cross-Atlantic discussions and will be included as this blog progresses. Ken

Before we post another part of the bitonality/polytonality discussion, I would like to share one of the many e-mails sent between us as we explore different possibilities and aspects of extending our composing techniques and understanding of new music (shall we say anything from Debussy to the present day).  I was encouraged to do this by Nurtan as it touches on, and perhaps extends on an aspect of his thoughts and arguments that were covered by Bernstein in his famous lectures.
For those who have heard and want to be reminded of the content of these lectures they are of course available on the net:
http://www.openculture.com/2012/03/leonard_bernsteins_masterful_lectures_on_music.html

to Nurtan
subject: some thoughts after a hospital visit

Yesterday I visited my mother in a hospital in North Wales, the location is a stronghold of the Welsh language and the majority of staff are fluent Welsh speakers, and for many it is their first language.

During my visit some of the medical staff were moving about the ward and I picked up on the term osteoporosis, a condition that affects my mother, and probably the reason why I tuned in to another person's speech.

I gave it little thought until I was driving back to my home and then I realised that I was working with words from three different languages.  There was no effort in exchanging between Welsh and English and knowing the Greek term and its roots made all three languages blend.  I recall Arthur Koestler making the same sorts of observations about the ease with which he could follow conversations in different tongues, so I assume it is a common enough feature.

I took to considering your experiment with major and minor tonalities and the way we hear sounds in combination.  Anthony Burgess wrote in one of his early novels about a character who could hear all 4 parts of a fugue in his head. 

Now I can recreate the sound of a Bach fugue in my head after playing it, sometimes only part of the fugue, and then it does a strange looping back to the beginning, which is rather frustrating!  The question is am I really hearing all the parts?  I don't know for certain as it isn't exactly like listening to a CD, though there are times when the music is particularly vivid.

Similarly, am I hearing two or more distinct languages at once, or is it a rapid alternation between words and their meaning?  When I think about it it isn't just a matter of vocabulary, Welsh has a cadence and stress to the speech quite unlike English.

If our brains can cope with this degree of information, coping with two or more keys in a bitonal piece shouldn't be beyond our listening skills, nor discriminating between various combinations of bitonal chords.  However, I have a certain sense of unease about hearing two tonic chords at once, why is this?  When I am hearing two languages does one take precedence over the other, if (let us say) Welsh was spoken first or more regularly than English would I gravitate to that language?  I am not suggesting that because we might start playing the piano with pieces in C that we would gravitate to that key over all others (don't even start thinking about writing pieces in C and C sharp for lesson one, we might never extract ourselves from that pathway)!  Has equal tuning ensured that we have no particular "home" key in our minds?

How much saturation could we cope with before we decide that there is no sense to the music presented? Does a massive amount of information become a global event that can be contrasted with another? In musical terms we could be talking about Ligeti's "Atmospheres" or Nielsen's "Commotio" or Nancarrow's player piano studies.

If we were discussing rhythm the same issues apply, a change from a crotchet pulse to a triplet crotchet to quaver can be heard as three events or as an acceleration.  We could grade each note in a sequence with a minutely different duration and get a similar effect, but would our ears mark the difference?  Of course one could compose with a legion of different tempi and have three orchestras to perform them, but would we manage to discriminate the material or combine it into a gesture?

Perhaps the question seems trivial, but if the composer has a responsibility to present his audience with coherence rather than chaos these fine lines need some consideration, and I know of no rule book which says where the boundary lies.

Have I taken the analogy too far?

As always I look forward to your opinions,

Ken

Symmetric scales

Click on the link above to read the PDF on symmetrical scales, how to build them and find scores  to illustrate their use. Tables in manuscript form are also provided at the end of the PDF.

A sample from the charts is shown.

Processes of transformation

Transformation has been used in one form or another throughout the history of music, but the levels of complexity and design have rapidly increased since the late romantic period, notably in the works of Liszt, Berlioz and Wagner. The use of transformation in the 20th century is notable in works by Peter Maxwell Davies. I refer to him in particular as his transformations work on many levels; parody, distortion of dance rhythms, pitch designs and alterations to  plainsong to name a few. The “cantus” melody discussed in a previous blog on "Stone Litany" will keep its melodic and rhythmic outline in certain sections but is treated to “compression” to create proportionally related phrases, it is also subject to retrograde formations to develop complex contrapuntal passages. In alternate sections the material is fragmented.  In later works the process of transformation is refined, for example progressive changes in the phrases may alter a theme into a related version of itself.  In the chart this would come under rearrangement by structure. These techniques will be examined in greater detail in later blogs.

Transformation is at the heart of the music of many British composers. Having discussed Bax at some length with Nurtan and presented several blogs on his symphonic works, the question of narrative music became a major issue for us. Bax makes use of the technique in his symphonies to enable his themes to “develop” the motives which identify his characters (in his case we believe based on real people).  In the later symphonies his treatment of themes is often subtle, incorporating pitch, rhythm and timbre to provide the continuous changes which make the sense of narrative so powerful.

In the table the generation of the related material, A1 or B, is not a musical terminus, it could be continued indefinitely. Maxwell Davies recognised and discussed in his Dartington Summer School composition classes the problem of terminating a process.  Though he didn’t discuss the question with reference to Eastern philosophy it comes down to the Tao again, it is a matter of balancing artistic and technical forces on the musical material.

There is an excellent (free) lecture on Jonathan Harvey's fourth string quartet on iTunes - University of London/ Arditti quartet, where the composer's use of interlocking themes is discussed in some detail and provides food for thought as to how the process of transformation is still developing as a technique.

The Tao of Musical Intentions

Having one clear musical intention is the point at which the blank sheet of paper, which troubles so many composers, gives way to a succession of possibilities. The blog will refer to many famous composers associated with particular intentions as an overlay to the main mind map. The musical intentions have been placed into two fields, one technical and one expressive, following the plan used in previous blogs to identify composing errors and the elements that may form a composition.

This table follows on from the latter in expanding the question "Is there a single musical intention to fulfill"? The list of intentions provided may be expanded but they form a sufficiently broad group to satisfy my requirements.

One of the issues that arise from using the table is that once the main intention has been selected it cannot develop in isolation, other considerations have to be heeded, but the table can be used to generate a hierarchy of choices.  Let us take an example, silence; one intention could be to create a work which balances sound and silence.  Once that decision is made the secondary considerations start flowing. Silence comes as a section of the parameter 'duration' and as Cage has much to say about this feature we can imagine his train of thought working on Imaginary Landscape No 4.

In this work one player selects the radio signal and will as a consequence deal with definite and indefinite pitch content and possibly rhythm and pulse.  The second player deals with amplitude and timbre in the sense of tone colour. Cage decides to use traditional notation to provide durations to events. As in other works chance elements are used to select values and create events, of these 64 events (I Ching has 64 events in the design of its hexagrams) half are of silence, with the further possibility that the radio may also tune into silence. The above considerations cover a large part of the ‘technical’ elements on the chart.

It is not outside the realms of possibility that Cage also considered the opposite side of the chart, the ‘poetic’ or aesthetic intentions. There is clearly the possibility of playfulness and drama arising from the choices. The audience may form relationships of their own between the events heard, it would be difficult not to. The following extract is from “Cage Talk” available to read as a PDF:

http://www.rednoise.org/pdal/uploads/CageTalk._Dialogues_with___about_John_Cage.pdf

I include this reference to Imaginary Landscape 1 as it relates to the audience response in creating “their own script.”

He created a piece that was played by manipulating two turntables.10 The role of the person who played it was to increase or decrease the volume or to shift the speed. He wrote it up as a score. What was intriguing for us was that it had no pulse, and we had never dealt with that before. Since the piece was called Imaginary Landscape, there was no reason we shouldn’t learn to work with it. We had to listen. It was played live, but because of the turntables it had to be piped in from our radio studio, which was attached to the theater, with the musicians in the studio. It had an extraordinary eerie quality. The piece and the sound absolutely intrigued people, and the audience began to build into it their own script.

As Cage wrote and spoke a great deal about his music we are fortunate enough to have a composing intention:

“I had a goal, that of erasing all will and the very idea of success.”

This philosophical outlook is Zen driven and describes the decreasing role of the composer over the outcomes of the performance.  However one could argue that Imaginary Landscape No 4 does not fully meet this intention, though it is a concern that develops throughout his life. As for “success” there are several recordings of the work on You Tube, the first link is set as a concert hall performance:

https://www.youtube.com/watch?v=SSSn0odpHKE

this unfortunately suffers from distant microphone recording, there is little chance to hear the events and we are given a vague mass of sounds.

https://www.youtube.com/watch?v=A0BNsBlzQII

Has closer microphone work and sounds remarkable like the Beatles Revolution No. 9. The performance starts about 1.50’ in to the video

https://www.youtube.com/watch?v=iRa0j_EIth0

No video of performance but clear sound.

It is possible to relate the composing intentions to particular composers as has been done with Cage.  Here is a short list, readers might enjoy expanding it to cover 21^st century music, or indeed any period.  Nurtan and I have shared some thoughts on the matter and several of his suggestions are included:

Stravinsky covers many different intentions; dance/drama is high among them so rhythm and pulse is a natural technical pairing.

Berg or Schoenberg for drama / set structure, variation

Ravel for timbre and texture / playfulness, and this is shared by his admirer Gershwin.

Berio meaning of sounds/language (e.g. Sinfonia) while many of the sequenzas concern themselves with particular technical matters.

Harry Parch for tuning and colour.

Minimalist composers for repetition and hypnotic states, but let us take Glass for repetition paired with drama for “Satyagraha”.

Dream music can be represented by George Crumb where timbre and texture is a natural pairing from the technical side, the use of quotation as in Dream Images being a potent force.

One could continue but for the self imposed 1K limit!

You may have asked yourselves why Tao of musical intentions instead of Zen.  The reason is that balance is the key element of the Tao, and it seems that finding the correct balance between the forces that come into play in composing is essential.