Combinatorics of some musical objects

Excerpts from Appendix C (page 164) from Keith [1991].  All results assume a 12-tone equal-tempered scale.

Number of diatonic scale classes: 3
Number of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano:  7
Number of scales one note different from the Major scale: 9
Number of notes in the most common equal-tempered scale:  12
Number of common musical keys (C + 1-6 flats/sharps):  13
Number of 7-note diatonic scales (=7 * 3):  21
Number of elementary 2-fold polychords: 23
A k-fold polychord is an n-note chord sub-divided into k non-empty subchords, for k=1,  . . ., n.  For example, the 6-note chord <C, D, E, F#, G, A> can be subdivided into the 3-note 2-fold polychords, <C, E, G> and <D, F#, A>.
Number of 7-note chords:  66
Number of distinct interval sets (partitions of 12):  77
Number of 7-note triatonic scales (=7*35):  245
Number of notationally-distinct diatonic scales (=13 *21):  273
Number of distinct chord-types (= N(12) – 1):  351
Number of 7-note musical scales (=7*66):  462
Number of scales (=Number of n-note scales, summed over all n)  (=2^(12-1) = 2^11):   2048
Number of chords without rotational isomorphism (= 2^12 – 1):  4095
Number of notationally-distinct scales (=13 * 462):  6006
Number of non-syncopated 8-bar 1/4-note rhythmic patterns:  458,330
Number of non-syncopated 8-bar 1/8-note rhythmic patterns:  210,066,388,901

Reference:
Michael Keith [1991]:  From Polychords to Polya:  Adventures in Musical Combinatorics.  (Princeton, NJ:  Vinculum Press.)

Mr Sculthorpe, please call your office

I was thinking recently about concert performances I have attended where the composer was present, or rather, where I knew the composer to be present. Here is my list, as best I remember it:

Don Banks (1923-1980)
Sally Beamish (Hover, London, 21 February 2024)
Richard Rodney Bennett (1936-2012)
Pierre Boulez (1925-2016)
Ann Carr-Boyd (Britannia Fanfare, 2012)
Barry Conyngham
Palle Dahlstedt
Jasper Eaglesfield (London, 13 June 2024)
David Fanshawe (1942-2010, African Sanctus, Liverpool)
Rolf Hind
Archie John (East China Sea, London, 7 June 2024)
Robin Holloway
Keith Humble (1927-1995)
Gerard McBurney
James MacMillan (premiere of “Precious in the sight of the Lord”, Mass for the Fourth Centenary Celebration of the British Society of Jesus, 21 January 2023)
Olivier Messiaen (1908-1992)
Stephen Montague (Piano Concerto, BBC Proms, London)
Nico Muhly (premiere of Electric Violin Concerto, performed by Thomas Gould, and premiere of opera, Two Boys, at ENO, London)
Olli Mustonen (playing his own Sonata for Violin and Piano with Pekka Kuusisto, London 2013)
Tristan Murail (Reflections/Reflets, performed by the BBSO under Sakari Oramo, London 2013)
Nicola Murphy (Wavelength, Brisbane, 29 August 2024)
Loretta Notareschi
Jim Penberthy (1917-1999)
Behzad Ranjbaran (premiere of Violin Concerto, performed by Joshua Bell)
Johannes X. Schachtner
Peter Sculthorpe (1929-2014)
Larry Sitsky
Toru Takemitsu (1930-1996)
David Urquhart-Jones
James Wishart
Iannis Xenakis (1922-2001)

Of course, my presence at a performance does not constitute an endorsement of the music performed:  some of the music of these composers I like or appreciate very much, and some I think is unpleasant, boring or otherwise of low quality.   Although I generally prefer downtown and minimalist music, the music of the composers listed here also includes neo-romanticism (eg, Holloway, late Sculthorpe), abstract expressionism (eg, Penberthy, early Sculthorpe, Takemitsu), and uptown complexity (eg, Boulez, Muhly, Xenakis).  And, I have not included in this list jazz performers, who almost always play some of their own compositions.

For Peter Sculthorpe, one occasion (of several where we have both been present) was a performance in January 1975 near Patonga, Broken Bay, Sydney, of his profound and achingly-beautiful Sun Music III, in which I had the great good fortune, as second percussionist, to play the guiro. One has to wonder how the same person could compose the innovative Sun Music series of the 1960s and also the derivative, warmed-over, late-romantic tosh that Sculthorpe has written in recent years.  Bill Burroughs would have seen it as a clear case of spirit possession.

POSTSCRIPT (2014-01-11):  New Zealander Alannah Currie, of The Thompson Twins, plays a guiro in their hit Hold Me Now (1983).  See video here.

Macho mathematicians

Pianist and writer Susan Tomes has just published a new book, Out of Silence, which the Guardian has excerpted here.  This story drew my attention:

Afterwards, my husband and I reminisced about our attempts to learn tennis when we were young. I told him that my sisters and I used to go down to the public tennis courts in Portobello. We had probably never seen a professional tennis match; we just knew that tennis was about hitting the ball to and fro across the net. We had a few lessons and became quite good at leisurely rallies, hitting the ball back and forth without any attempt at speed. Sometimes we could keep our rallies going for quite a long time, and I found this enjoyable.
Then our tennis teacher explained that we should now learn to play “properly”. It was only then that I realised we were meant to hit the ball in such a way that the other person could not hit it back. This came as an unpleasant surprise. As soon as we started “playing properly”, our points became extremely short. One person served, the other could not hit it back, and that was the end of the point. It seemed to me that there was skill in hitting the ball so that the other person could hit it back. If they could, the ball would flow, one got to move about and there was not much interruption to the rhythm of play. It struck me that hitting the ball deliberately out of the other person’s reach was unsportsmanlike. When I tell my husband all this, he laughs and says: “There speaks a true chamber musician.”

This story resonated strongly with me.  Earlier this year, I had a brief correspondence with mathematician Alexandre Borovik, who has been collecting accounts of childhood experiences of learning mathematics, both from mathematicians and from non-mathematicians.  After seeing a discussion on his blog about the roles of puzzles and games in teaching mathematics to children, I had written to him:

Part of my anger & frustration at school was that so much of this subject that I loved, mathematics, was wasted on what I thought was frivolous or immoral applications:   frivolous because of all those unrealistic puzzles, and immoral because of the emphasis on competition (Olympiads, chess, card games, gambling, etc).   I had (and retain) a profound dislike of competition, and I don’t see why one always had to demonstrate one’s abilities by beating other people, rather than by collaborating with them.  I believed that “playing music together”, rather than “playing sport against one another”, was a better metaphor for what I wanted to do in life, and as a mathematician.
Indeed, the macho competitiveness of much of pure mathematics struck me very strongly when I was an undergraduate student:  I switched then to mathematical statistics because the teachers and students in that discipline were much less competitive towards one another.  For a long time, I thought I was alone in this view, but I have since heard the same story from other people, including some prominent mathematicians.  I know one famous category theorist who switched from analysis as a graduate student because the people there were too competitive, while the category theory people were more co-operative.
Perhaps the emphasis on puzzles & tricks is fine for some mathematicians – eg, Paul Erdos seems to have been motivated by puzzles and eager to solve particular problems.  However, it is not fine for others — Alexander Grothendieck comes to mind as someone interested in abstract frameworks rather than puzzle-solving.  Perhaps the research discipline of pure mathematics needs people of both types.  If so, this is even more reason not to eliminate all the top-down thinkers by teaching only using puzzles at school.”

More on the two cultures of mathematics here.

The sources of silence

I listed here many of the teachers and thinkers whose influence I have felt.   In his wonderful new book on John Cage’s 4′ 33”, the indefatigable Kyle Gann says this (pages 71-72):

The meme that Cage was more of a music philosopher than a composer has become commonplace, most of all, it seems, among people who don’t like his music and are in need of a way to justify his celebrity.  Cage was not a philosopher in any sense that the philosophy profession would recognize, but he was very much a composer who drew inspiration for his music from philosophical ideas.  The list of artists, writers, and thinkers he names in justification of his musical trajectory is a long one:  Meister Eckhart, Huang-Po, Kwang-Tse, Erik Satie, Henry David Thoreau, Gertrude Stein, Arnold Schoenberg, John Cage Sr., Marcel Duchamp, Sri Ramakrishna, Daisetz Sukuki, Joseph Campbell, Ananda K. Coomaraswamy, Alan Watts, Antonin Artaud, Robert Rauschenberg, Morton Feldman, David Tudor, Norman O. Brown, Marshall McLuhan, Buckminster Fuller, Gita Sarabhai, and Christian Wolff, among others.”

I was reminded of James Pritchett’s intention, when writing his book on Cage’s music, to as much as possible read everything that John Cage had himself read, and in the order he had done so.
References:
Kyle Gann [2010]: No Such Thing as Silence.  John Cage’s 4′ 33”.  New Haven, CT, USA:  Yale University Press.
James Pritchett [1993]:  The Music of John Cage.  Cambridge, UK:  Cambridge University Press.

Deaf and blind musicology

Looking through some old scores, I come across the following note written by one Edouard Lindenberg, and copyrighted 1951:

Schumann said of Mendelssohn that his first name, Felix (happy) suited him admirably.  Mendelssohn was, in fact, of a carefree disposition, full of gaiety and optimism, and he was spared material cares.  Sorrow almost always passed him by – he never experienced any really severe shocks of any kind. Is it on this account that his music never attains the highest summits?  Or was it perhaps that he was too universally gifted – for he spoke several languages, read Greek fluently and had translated Terence; he was, moreover, one of Hegel’s best pupils and his talent as a draughtsman and painter in watercolours was very superior to that of an ordinary amateur.”

What an amazing person Lindenberg must have been!  He was clearly deaf, because even a short acquaintance with Mendelssohn’s music would tell you that the composer had experienced profound sorrows and emotions, and had expressed these in his music.  Listen to his last quartet, written after the death of his sister, Fanny, for example, or the two violin concertos.  Or listen to the opening orchestral number of the oratorio Elijah, which, again and again and again, seems about to resolve but has its resolution postponed, thereby expressing  human anguish better than any other composer before or since.

But my amazement at this man Lindenberg is even stronger.  He must also have been blind as well as deaf, since his concert note (concerning the Hebrides overture) then quotes from a letter Mendelssohn wrote from Paris on 21 January 1832.

I cannot have the Hebrides played here because I don’t consider the work finished yet.  The central section ff in D major is very stupid, and the whole development smells more of counterpoint than of seagulls and fish – whereas it should be the other way round.  And I am too fond of it to be played as is stands.”

But just two weeks later Mendelssohn wrote the following letter, immediately after hearing of the unexpected death from tuberculosis of his violin teacher and close friend, Eduard Rietz (17 October 1802 – 22 January 1832).  Surely, unless he was blind, Lindenberg must have also seen this, the very next letter in the published edition of Mendelssohn’s letters:

You will, I am sure, excuse my writing you only a few words to-day:  it is but yesterday that I heard of my irreparable loss.  Many hopes, and a pleasant bright period of my life have departed with him, and I never again can feel so happy.  I must now set about forming new plans, and building fresh castles in the air; the former ones are irrevocably gone, for he was interwoven [page-break] with them all.  As I shall never be able to think of my boyish days, nor of the ensuing ones, without connecting him with them, so I had hoped, till now, that it might be the same with those to come. I must endeavour to inure myself to this, but the  fact that I can recall no one thing without being reminded of him, that I shall never hear music, or write it, without thinking of  him, doubles the sorrow of such a separation.  The former days are now indeed departed, but it is not these alone that I lose, but also the man I so sincerely loved.  Had I never had any, or had I lost all cause for loving him, I must without a cause have loved him all the same. He loved me too, and the knowledge that there was such a man in the world – one on whom I could rely, who lived to love me, and whose wishes and aims were identical with my own – this is all over: it is the hardest blow that has yet befallen me, and never shall I forget it.

This was the celebration of my birthday.  When I was listening to Baillot on Tuesday, and said to Hiller that I only knew one violinist who could play the music I loved for me, L______ was standing beside me, and knew what had happened, but did not give me the letter. He was not aware indeed that yesterday was my birthday, but he broke it to me by degrees yesterday morning, and then I recalled previous anniversaries, and took a review of the past, as every one should on his birthday; I remembered how invariably on this day he arrived with some special gift which he had long [page-break] thought of, and which was always as pleasing, and agreeable, and welcome as himself.  My day was very sad; I could neither do anything, not think of anything, but the one subject.

To-day I have compelled myself to work, and succeeded.  My overture in A minor is finished.  I think of writing some pieces here, which will be well remunerated.

I beg you will tell me every particular about him, and every detail, no matter how trifling; it will be a comfort to me to hear of him once more.  The octet parts, so neatly copied by him, are lying before me at this moment, and remind me of him.   I hope shortly to recover my usual spirits, and to be able to write to you cheerfully and more at length.  A new chapter in my life has begun, but as yet there is no title.
— Your Felix. ”

[Mendelssohn-Bartholdy 1864/1870, pp. 327-329, letter from Paris (to Fanny?), dated 1832-02-04]

To imagine that a privileged person does not suffer normal human sorrows in the same way that the rest of us do is a peculiar form of irrationality, contrary to all human experience.  To further imagine that such a person is not capable of profound artistic expression despite the evidence of own’s own senses is just perverse.  But Mendelssohn seems to have attracted the perverse among his critics, from the explicit anti-semitism of Richard Wagner to the anti-Victorianism (and possible antisemitism) of George Bernard Shaw.

References:

Edouard Lindenberg [1951]: F. Mendelssohn  Fingal’s Cave (Hebrides Overture) Op. 26. Paris, France:  Heugel & Companie.

Felix Mendelssohn-Bartholdy [1864/1870]: Letters from Italy and Switzerland. Translated by Grace, Lady Wallace.  Fifth Edition. London, UK:   Longman, Green, Reader, and Dyer, 1870.  Includes preface to the First Edition, dated 1864-04-22.

Vale Richard Meale

Australian composer Richard Meale (1932-2009) has just died at the age of 77.   He was perhaps Australia’s best expressionist, especially in moving early works such as Homage to Garcia Lorca, and Clouds Now And Then.   In his later years, like so many 20th-century Australian  modernist composers, he turned to writing late-romantic tosh, as if the only function of composers was to support the film industry.

In honour of his memory, I repeat the profound Basho haiku which he quoted on the score of Clouds Now And Then:

Clouds now and then,
Giving men relief
From moon-viewing.

SOME LINKS:
The SMH obituary is here.  An account of his funeral is here and reminiscences by composers David Worrall and Ross Edwards are here.    Andrew Ford’s eulogy is here.  A review of a memorial concert held for him in February 2010 is here.

POSTCRIPT (Added 2013-02-23):  Meale visited UCLA in 1960 on a travel grant from the Ford Foundation, and there studied and played traditional Japanese and Balinese gamelan music.   I wonder if one of his teachers was Colin McPhee, who taught ethnomusicology there from 1958.  (And how cool was that:  to be teaching ethnomusicology at UCLA in the late 50s!)

POSTSCRIPT 2 (Added 2014-08-08):  Here also is a tribute to Richard Meale’s nephew, Tony Meale.
 

The Zen of Sunday-painting

In his famous account of learning the piano as an adult, Guardian editor Alan Rusbridger refers to a book by psychiatrist, Marion Milner, a pseudonym of Joanna Field.  Milner was the sister of Nobel-physicist Patrick Blackett, and great-neice of Edmund Blackett, the architect of colonial Sydney.   Her book is an account of her attempts to paint and draw, and to learn to paint and draw, as an amateur artist.

I am not enchanted by her artwork, and I find her Freudian accounts of artistic creativity and its barriers both implausible and untrue to life.  I believe Alfred Gell’s anthropological account of art to be far more compelling – that artworks are tokens or indexes of intentionality, perceived by their viewers or auditors as objects created with specific intentions by goal-directed entities (the artist, or a community, or some spiritual being).  These perceived intentions include much else beside the expression of feelings.

But Milner’s book is replete with some wonderful insights, many of which express a Zen sensibility.     Herewith a sample:
Continue reading ‘The Zen of Sunday-painting’

Recent listening 3: Eastern European flavours

Following my post about Czech Tropicala-post-punk duo DVA, I thought I’d write about some other East European-flavoured music which I listen to a lot.
The first is the album Mnemosyne, by Ljova and the Kontraband, which I have mentioned previously for their setting of  poem by Joe Stickney.  Their music is a mix of East-European gypsy, klezmer and jazz, played by virtuoso performers on viola, accordian, bass and percussion (plus guests).   This mix could only happen in New York City, which is where they are based.
A genuine EE gypsy sound is the painful blues of Dona Dumitru Siminica (1926-1979), who sings this gypsy Romanian parallel to Rembetika in a falsetto voice, with accordian, cymbalon and bass and sometimes violin behind him.  The re-issue I am listening to comprises tracks originally recorded in Bucharest in the early 1960s.
The third album brings the accordian into the 21st-century:  arrangements of various fast-moving electronica tracks for a trio of virtuoso Polish accordianistas, Motion Trio. After this album, no one can ever say again that a gentleman is someone who knows how to play the accordian but refrains from doing so!
Play Station
Ljova and the Kontraband [2008]:  Mnemosyne.  Kapustnik Records, New York City.
Dona Dumitru Siminica [2006]:  Sounds from a Bygone Age, Volume 3. Asphalt Tango Records GmbH, Berlin.  CD-ATR 1106.
Motion Trio [2005]: Play-Station. Asphalt Tango Records GmbH, Berlin. CD-ATR 0705.