Morton Feldman’s Palais de Mari: a pitch analysis
by Frank Sani
The musical materials in this paper were prepared for a conference paper given on the 8th of May 2004 (TAGS day) at the University of Oxford, Faculty of Music. The script, however, was written exclusively for the Morton Feldman Page, and is published here for the first time.
Much to the disappointment of the analytical mind, a piece like Palais de Mari defies explanation. To be negative about it, therefore, would be tempting, especially after hours of investigation leading nowhere; however, it is more fruitful to take this as the triumph of a composer who allowed the irrational to penetrate a precisely notated score.
My investigation deals with the pitch content of the piece, in an effort to provide empirical evidence supporting the following view: Palais de Mari shows a catalogue of playful workmanship, making through-composing into a highly skilled flow of invention, where groups of pitches are inverted, transposed, reshaped, and where the introduction of new pitches from time to time is instinctively alternated with echoes of previous harmonies. This playful workmanship, Feldman’s inventiveness, generates a musical universe that challenges any tight-laced models of pitch analysis; as this paper will show, there is enough evidence to conclude, if anything, that new approaches are required to place Feldman’s late pieces through such a set of analytical parameters as to make the resulting commentary not only intelligible for the conscious mind but also true to the composer’s flexible approach.
Undoubtedly, it is not ethically correct for the analyst to offer pitch-set theories where the composer intended none; therefore, I took the position of offering more a catalogue of pitch content than a definitive theoretical explanation of it, hoping to provide a good background study for anyone intending to listen to this piece critically, and willing to penetrate its surface.
II. An introduction to Group Classes
Looking at the piece as whole, or listening to a performance of it from end to end, allows a glimpse at unmistakable similarities between certain parts, some of which are exact repetitions. That Feldman took an almost improvisational approach to his compositions, akin to Jackson Pollock’s drip paintings, must be kept in mind at all times: it is an irrational approach, where pitch notation serves as mere editing, where sound-shapes are pictorial and serve no tonal precept or structure. If anything, the use of pitch in late pieces like Palais de Mari may have been influenced by the colour variations observed by Feldman in the weft-and-warp of the Anatolian rugs he collected. With these premises, I will now proceed to give a full account of the pitch content of Palais de Mari.
From the point of view of pitch, the very first impression one gets from a performance of this piece is fundamentally correct: the opening four notes are repeated in exact or near-exact order (and octave placement) at a later stage; also, there are clear moments of reminiscence, where among the many pitch-groupings presented to the ear lies an intriguing web of subtle connections, weaving together most newly-sounded combinations with consequent re-iterations, in a stream of consciousness littered with intangible echoes operating within an extremely fluid time continuum. The main task, then, would be to identify what pitch-groups exist within Palais de Mari, with an eye to revealing their underlying connections.
Seeing no immediate and obvious structure to the pitch content of the piece, the only safeguard against inaccurate conclusions seemed basing any analysis on a careful observation of every single measure in the score. This would involve taking the piece apart into single units, beginning from the very first bar, and grouping together any similar chords or formations encountered on the way: each group was copied on large manuscript paper, beginning with the very first bar. Any exact repetitions of a group would be recorded under its first entry; non-identical repetitions would be notated vertically below the original form to show changes in duration, transpositions, changes in order of pitches within a group, and other mutations. Groups deemed to bear little resemblance to any others already catalogued would be entered in a new column; however, all groups were labelled using a combination of roman numeral and Greek letters, which allowed for any previously unseen connections to be highlighted using the Greek alphabet to link seemingly disparate sub-classes to a roman numeral (the latter representing a main branch of variants).
Often there had to be made highly subjective decisions with regard to whether combinations of pitches represented a new class or a variant of an existing class; this was true of one set in particular, group IVβ (see further down the essay for more details), which links together the majority of pitch-groups in the score on the basis of an immediately recognisable feature, namely an acciaccatura followed by a chord. Also, it appeared that group IVη was none other than IVβ in disguise; therefore, the two are grouped separately in the original classification but later united under the IVβ/η label, especially in respect of compiling statistics tables.
At this point it is necessary to highlight also how the tables presented in this essay imply a definition of ‘group’ that, in the context of Palais de Mari, incorporates all of the following: 1) a single pitch; 2) two or more pitches presented separately but still recognisable as a unit; 3) two or more pitches sounded together (at once or as an arpeggiato). Furthermore, certain groups present themselves as one-bar units first, and over several bars later; also, some multi-measure groups are composites made of previously encountered one-bar groups.
The flexibility of the cataloguing and labelling of groups meant that certain groups could be seen as different or similar depending on how one chose to link them; for this reason, the statistics of group recurrence presented later in this essay exclude multi-bar or multi-unit groups wherever these can be broken down into one-bar or one-unit groups, with the latter ones being those upon which the figures are based.
Having thus introduced the criteria behind the compilation, labelling, and classification of the existing or supposed groups in Palais de Mari, I will now present them outside the text proper via the following list of hyperlinks (NB – whenever different groups were originally grouped together in one page, these may appear in the list either as separate entries, in which case there will be several hyperlinks referring to the same document, or as one entry, joined by hyphens into a string; as a consequence of this, the numerical order of I to XXXII may not be strictly observed through the list):
Links to Group Tables (listed in Group Class order)
IV (overview table, IVα to IVν), with separate tables for variants of groups IVα, IVβ, IVδ, IVε-IVζ-IVη-IVι-IVκ, and a separate document for IVλ-IVμ-IVν
XIIIα-XIIIβ-XIV-XV-IXγ (this link includes the following unlabelled composites: XII+IXα; XII+X; XII+XIIIα)
NB – Some of the documents referred to in the above list contain information on further links within different groups; it is therefore advisable to read each document with care in order to better cross-reference between groups.
Arabic numerals are used to show how single components change position within certain groups, as can be seen in the table for Group Class I, where pitches three and four swap places.
Notice also that whenever two or more bars in a row, reading from left to right, present the same time signature, the latter is written in brackets or omitted after the first bar; the same applies to clefs.
III. An interpretation of Group Classes
In the previous section, the entire piece was organised into groups, and each and every one of these was catalogued and labelled; also, links were established between groups from time to time, as suitable.
The major preoccupation with the group classes system was with simplification of the existing material: having reduced eight pages of seemingly disconnected pitch formations to thirty-two basic sets, it was easier to see the connections between different parts of the score, especially where those were not immediately intelligible (due to factors such as inexact transposition and displacement across the score).
This new wealth of material is in a way self-explanatory, and the juxtapositions presented in the scanned documents hyperlinks tables offer as clear a visual guide as possible to what we may cautiously describe as the organising principles or building blocks, which underpin the harmonic content of Palais de Mari. Of course, trying to further explain these findings would be desirable, in an attempt to find a centre, a core around which all pitch content develops symmetrically (or asymmetrically).
An immediate difficulty can be found in the relative value of the group classes presented, since we saw how the idea of ‘group’ changes through the piece to incorporate not only one-bar units but also larger ones; relative is also any deduction of a nucleus or core, around which Feldman might have centred the entire edifice, where such conclusions are based on the subjectively chosen group system. In other words, different analytical approaches to a group theory could bring a number of solutions. Some groups in Palais de Mari are obviously interconnected, but others are not; therefore, identifying a core group class must be pursued with a degree of scepticism, as quite often the multitude of mutations undergone by a musical idea (like the acciaccatura-and-chord motif) turns most attempts to unifying these under one tree into an unnatural straight-jacketing of Feldman’s intuitive creativity.
Moreover, the presented model of organisation is only one of several, and leaves the way open for further investigation; for instance, the group class tables are not based on interval analysis, which would imply re-structuring the grouping system according to subtler criteria of similarity, for instance the interval class formed between top and bottom note of each chord or motif, or even between each note within a chord (or motif).
It is interesting to point out one more difficulty with the group-class theory: depending on the criteria adopted to ascribe any set of pitches to any particular group-class, the exercise of labelling by group-class number each bar of the entire score in the printed order could result in a range of different results, where more inclusive criteria would generate a lower number of group-classes and, in turn, a higher recurrence of same group-classes. In other words, the more differentiated the numbering of the sets of pitches in the score, the lesser the chances to show (through rows of numbers alone) any underlying symmetry or periodicity in the repetition of same groups.
This was very much the situation I was in when numbering each bar (excluding empty ones) of Palais de Mari according to my own group-class numerals: even excluding eight classes from the process (namely all the composite groups), I was left with twenty-five numbers, most of which followed one another without any obvious mathematical consequentiality, symmetry, or periodicity.
The following tables show the results of my score-labelling (according to group-class number), with the first page also listing the aforementioned omissions. It is clear to see that there is no unequivocally obvious cycle of repetitions, even after labelling twice any bars containing repeat signs and splitting up all composite groups into their original parts. Also, in spite of reducing the numerals employed to twenty-five, the abundance of sub-classes (which, as we saw, are signalled by two or more Greek letters referring to a same roman numeral) makes the visual immediacy of the tables less effective; in turn, this means that existing patterns of repetition, if any, are lost to the eye.
The hyperlinks below show said tables, to be used in conjunction with the group-class ones:
IV. From Group Classes to Pitch Occurrence Tables
Having shown how the pitch content of Palais de Mari can be broken down into a number of recognisable groups, and having thus demonstrated, by implication, how such groups form an intuitive web of cross-references through non-periodical and often non-identical reiterations, we may ask the following question: if any, which groups are dominant and why?
Invariably, by listening to a performance of the piece and by looking at the previous three tables (at the end of part III) we can both hear and see intuitively that certain formations recur more frequently, or even that they appear to be stronger in character or identifiable as dominant; the following stage, then, consists of corroborating this intuition with some degree of factual evidence.
First, we may compile a list of all the groups, in class order, and notate the total number of occurrences according to the aforementioned three tables; the results are seen here:
Group Classes and number of occurrences (listed by class order).
NB – The excluded group-classes are the same composites as listed in the tables at the end of part III.
Having collected plain statistics of the kind shown in the above hyperlink, we are only one step away from the results of the following table, which puts the data in descending order of occurrence:
NB – All groups which appear within repeat signs in the score have been counted twice in this table for the sake of verisimilitude with regard to the aural experience of the piece (thrice in the case of Group II in bar 9). Please note that some groups are not included here, for which see NB to the previous hyperlinked table.
It is possible, looking at this table, to see how sub-class IVb/h dominates above all other combinations, and that if we added up all sub-classes of Group IV (that is α, β/η, γ, δ, and so on, minus exclusions), we would obtain one hundred and forty iterations, approximately between half and a third of the sum total (353).
The difficulty in taking the next step, namely assessing in what direction the dominance of one group class pushes the aural plane of Palais de Mari, lies not only in the subjective criteria that allowed the above group class tables to be compiled, but also in the potentially misleading simplification to which numerical statistics can lead if taken at face value: figures alone tell half the tale.
It is appropriate, at this point, to retrace our steps and look again at the composition of Group Class IV in terms of pitch content; in order to do this, one must revisit firstly the overview table for said group and secondly all other tables showing variants of each sub-class – for which refer back to the list of group class tables in part II.
Most important of all is to look at the overview table for Group Class IV; there, we are reminded that:
<![if !supportLists]>1) <![endif]>only the second chord of IVα is connected with IVβ;
<![if !supportLists]>2) <![endif]>only IVη is an exact transposition of IVβ, whereas for all the other Group Class IV sub-classes the connection with IVβ is the acciaccatura (whose feature is to always precede a two-note chord and to be pitched nearly always between those two notes).
Focussing on Group Class IV, it is understood that the criteria applied while compiling tables for this group class seek to compromise between strictness and flexibility, by allowing on the one hand both exact and inexact transpositions to unite under one roman numeral, while on the other labelling under different class numbers combinations that generally did not share its characteristic (of the acciaccatura followed by a two-note chord, hence the exclusion, for instance, of the groupings seen in Group Class XVIII).
The system of grouping is complex, and a review of an existing system may be preferable to creating a new one: given a set of flexible criteria, every new system where the two processes of measure-by-measure cataloguing and cross-referenced classification take place simultaneously is open to the possibility of overlooking certain connections, due mainly to the enormous scale of the process and to the partly intuitive nature of such work. Inevitably, chains of connections are formed, sometimes leading to a break from the intended criteria, as was the case in my system where sub-classes XXIVβ/γ/δ/ε, all of which feature the acciaccatura followed by a two-note chord, became linked to the arpeggiato chord in XXIVα instead of occupying their rightful place within Group Class IV. I suspect that this was allowed to happen because the unchanged top and bottom pitches were seen as a more immediately recognisable connection between XXIVβ and XXIVα at the very moment of the classification process where XXIVβ first appeared, two bars after XXIVα – although the presence of IVδ, appearing as the intermediary chord (on page 3, system 4, bar 3), should in theory have shown me the right path.
It is clear that unresolved contradictions of this kind should be reviewed where possible. On the other hand, it seems that the group-class system presented here does offer a reasonably congruent tool with which the pitch combinations of Palais de Mari may be better understood. Let us now go back to the question posed earlier, and which we were attempting to answer, namely how the dominance of Group Class IV, and of sub-class β/η in particular, shapes the pitch content of Palais de Mari; for this reason we must return to the Group Class IV overview/variants tables and observe, by analysing the different sub-classes in terms of pitch content (including transpositions), whether any one pitch or group of pitches within the class emerges as dominant.
NB – Some of the Group IV sub-classes appear in other group tables; therefore, the results to follow will cross-reference with any such tables wherever necessary, for instance where Group IV sub-classes appear in two- or three-bar composites (like XXX).
All pitches are counted separately and treated enharmonically (bear in mind that some of the enharmonic equivalents in the column headings are not actually in the score, as for example F-flat). All repetitions are included.
From the above table we can deduce the following one, where the most frequently occurring pitch (for the whole of Group Class IV) is first to the left:
Composite of all Group Class IV pitch occurrences.
The table is a composite of all the Group Class IV pitch occurrences, including all transpositions, so it would be incorrect to say that because the first three notes (F, Eb, G) make up the original form of IVδ, this sub-class is the strongest. Conversely IVβ/η, as we saw in one of our previous tables, is the group with the highest number of occurrences, yet the pitches of its original form (G, D, C) only appear in third, fourth, and fifth place in the above table. A re-count could be done to exclude successive identical repetitions of each sub-class, and the new sum total for each pitch compared with the two “Group Classes and number of occurrences” tables seen before.
As it is, the connection between pitch occurrences and harmonic outlook of the piece, that is between how the piece is made and it sounds, is tenuous, even as we delved into the core of the most recurring Group Class. The above analysis would only be directly relevant if each Group Class and sub-class had never been transposed or re-shaped, so that there would be a more easily detectable connection between the pitch framework and the aural plane created by it. Perhaps the way forward in linking the sound of the piece with its pitch framework is by eluding the Group Class classification of the materials, as frequent transpositions within group classes and sub-classes complicate matters. If this approach were taken we could then look at the pitch content alone and derive statistics accordingly.
Based on the last premise, I counted all pitches in the score, page by page, regardless of their grouping, and marked the results in an enharmonic table, which is available through the following hyperlink:
Global table of pitch occurrences
The following table presents the sum totals seen in the previous table in decreasing scale, so that the most recurring pitch is at the top.
Table of pitch occurrences (in order).
It becomes apparent that the group system and the pitch content are two areas that require further investigation, as the above table makes clear: it is not enough even to catalogue every note played in the score to find a kind of focus point for the harmonic range Feldman presented. To say that D#/Eb, the highest occurring pitch, is present in the first and last bar of the piece is not enough: there is no central sequence to reinforce a sense of palindromic journey around a D#/Eb pivot, nor is there any particularly poignant singling out of this pitch. It is not even sufficient to say that the second highest scorer in the previously drawn “Table of comparison for pitch occurrence within each Group IV sub-class” is also D#/Eb: it is coincidental, as the Group Class IV materials cover a large portion of the work anyway. Also, the top three pitches (Eb, D, C#) do not form part of a high-scoring class or sub-class, but rather of XXIVζ, which according to our records only occurs once in the entire piece (though Group Class XXIV reaches a total of 36, yet even this is much lower than the 140 of Group Class IV). Whatever the meaning of the higher occurrence of this particular pitch, it cannot be gained through statistics, nor does it help to guide us towards an enhanced or more focussed listening experience of Palais de Mari.
At this point, we can draw a line under group tables and one under pitch tables, and ask the following question: what other criterion of analysis can we apply to the tonal content of Palais de Mari so that our results embrace the acoustic dimension of the piece, and do not stray into statistical abstraction? In other words, what form of pitch analysis would remain close to the composer’s choice of materials, or help unveil in part the composer’s original intention behind this piece? If there is no immediately intelligible pitch-set theory, intended or coincidental, behind the way the piece is shaped, then what other concern pushed Feldman?
The answer I propose is the following: resonance. It is clear that if we are to treat the pedal markings as something more than a special effect, then we must look at the way pitches are layered in time through prolonged use of the sustaining pedal. This seems closer to the real Feldman, who preferred composing at the piano in order to hear the departing landscape of echoes generated by the sustaining pedal, and at the same time to overlap different tonal colours through time. For this reason, the next session takes up the challenge of drawing together a number of pitches into “Superchords”.
In the piano, once a key is struck, control is relinquished over the string vibrations, which eventually fade away and leave behind but the memory of that sound. The sustaining pedal helps lengthening this process further, yet it cannot escape from its end.
However, if we could prolong the life-span of each sound manifold, or even indefinitely, we would increase the level of interaction between successive new sounds, so that more complex chordal structures could be achieved through time.
In Palais de Mari, the sustaining pedal is depressed almost throughout; if we grouped the sounds within each depressing and releasing of the pedal into large units, we would count thirteen in total. Such units I have named Superchords, and I have illustrated in the following table:
For a breakdown of these chords through the score, that is to say for a note-by-note build-up to the Superchords shown in the above table, click on the following hyperlinks:
Superchord Sequences nos. 1, 2, and 3
Superchord Sequences nos. 4, 6, and 8 (first part)
Superchord Sequences nos. 8 (second part), 9, and 10
Superchord Sequences nos. 11, 12, and 13
NB – The first and last notes for each sequence may not match the bar-numbers given in the Superchords Complete Table, due to the omission of re-iterations of same notes, that is of repetitions of same pitches (unless appearing in the score at different octaves). Notice also that the horizontal spacing on the sequences is not a measure of time or an indication of duration, and that all pitches are treated enharmonically.
Please bear in mind that Superchord 5 and Superchord 7 are each a single chord sounded at once, therefore there are no Superchord Sequences for them.
Looking at the Superchords, I realised that none matched, or at least not exactly; once again we find that where it was hoped a larger pitch-grouping exercise, and one following Feldman’s pedalling, would bring the analysis closer to establishing a sense of harmonic purpose, the near contrary was a more acceptable explanation, and that is to say that from the point of view of pitch Feldman did not at any level, not even in the distribution of layering through the sustaining pedal, seem to follow anything but his own intuition.
Yet the Superchords idea does not end here: I wanted to hear at least one Superchord Sequence come to life. The result can be experienced through the following audio (*wav) file, produced through CuBase software:
Superchord Sequence 1 (duration: 3’07”: 3Mb download)
Each new pitch from the sequence was played approximately every 6.23 seconds; this was calculated by timing the equivalent of each Superchord bar-range in Marianne Schröder’s recording (1990, Hat Art, cd 6035) and then dividing each total by the number of pitches present in each Superchord, as shown in the following table:
Superchords total number of pitches and timings
The situation presented in the sound sample is of course completely separate from that of the score, and the mathematical timings only serve the purpose of giving each new note in the sequence equal amount of presence through the Superchord build-up.
Also, the sound-source chosen is not in any way similar to that of the instrument for which Palais de Mari was intended, and the interest in the sound file is only with regard to extending Feldman’s experience of the fading echoes within the piano (upon depressing the sustaining pedal) through a longer time-span.
Analysing Feldman’s works with the intention of finding a hidden pitch- or duration- organising principle is like looking for a needle in a hay-stack, or going round in circles. In this particular case I decided that in order to test more fully ideas regarding pitch content and its organisation I should not explore any other aspect of Palais de Mari.
I hope to have demonstrated, if anything, that Feldman’s intuitive grouping of pitches is an important feature of the piece, and also that my group classification system is a useful way (but by no means the only one) to shed light on the organisation of harmonic content in Palais de Mari.
I am aware, however, that there are other issues at play in late Feldman pieces generally, such as the practice of irregular repetition, developed by the composer though observation of the Anatolian rugs he collected; there may well be a case for using this connection to find a thread within pieces like Palais de Mari, but it is not in the scope of this paper to do so.
Perhaps the only thing left for me is to suggest that the true motif in the entire piece is not so much related to pitch-combinations as it is to evading them: the acciaccatura (or appoggiatura, in certain cases) hangs between beats in most places, and from this grace-note all of Group Class IV (which occupies between a third and half of the piece’s materials) is woven.
Besides this, I found no incontrovertible evidence to suggest a single interpretation or key in analysing the pitch content of Palais de Mari, and in spite of the different approaches taken I feel new avenues must be explored in order to find more answers. Meanwhile, I hope this paper will help listening to Palais de Mari with an enhanced awareness of its tonal content, and complement the experience more than adequately.
Copyright © October 2004 Francesco Sani