Advanced dotting...

In my notation seminar, we were talking last Friday about "Crumb dots". These are George Crumb's idiosyncratic notation where dots are placed on either side of a note. Such as . ♩ . which is worth the length of a quarter note tied to a sixteenth note. I think the logic is that it's a double-dotted quarter note where the second dot instead subtracts instead of adds.  In any case, it's worth 1.25 quarter notes in length.

We also talked in the seminar earlier in the semester about medieval "dot groups."  These are not symbols found in actual medieval music, but things that are really useful for transcribing medieval music, where 9/8, 9/16, 9/4, 9/2, etc. are commonly implied meters.  There is no single note that can fill up a measure of 9/X where X is a power of two.  So we tend to use things such as a dotted quarter note tied to a dotted eighth note ( ♩. ⁀♪. ) for 9/16, and so on. But consider that that figure comprises two notes, the second of which is half the length of the first, and they are tied together.  That is the basic definition of a dotted note: a dotted half note is a half note tied to the note half the length of a half note, or a quarter.  So what ♩. ⁀♪. needs is a way of "dotting" a dotted quarter note, or  ( ♩. ) . ––note that this note has a different length than a double-dotted quarter note (which is worth 7/16, not 9/16) and this "dotted-dotted-quarter note" is worth more than a half note.  When it is used, it is usually written with two dots vertically aligned: ♩:

Formulas and Extensions to negative numbers

Working out the length of notes with multiple dots can be hard work.  In this screen capture from the "Reimagined" Battlestar Galactica, Kara "Starbuck" Thrace doesn't get it right even though the fate of humanity rests on her deciphering the secrets of a mysterious melody:

The third line should have two 32nd notes, not 16ths, and the fourth line should have triplet 64th notes.

So, let's get back to the basics. How long is a note with d dots? Let's see.  If a quarter note gets the beat then:

♩ = 1 beat
♩. = 1.5 beats
♩.. = 1.75 beats
♩... = 1.875 beats

and so on to

lim (d _⃗ ∞ ) = 2

So, what's the pattern?

1
1 + 1/2
1 + 3/4
1 + 7/8
1 + (n – 1)/n

Where n = 2^d

Given that, we can work out that –1 dots is:  1 + (2^–1– 1) / 2^–1 or 1 + (–½) / ½ or 1 – 1 or 0. So a note with negative 1 dots has no length.

For d = –2 we get –2 beats.  I don't know what that would mean.  I suppose play the note backwards beginning two beats before it should start?  (David Lewin has written about the uselessness of negative note lengths).

Just to round it all out:

d = –3, implies duration –6 beats.
d = –4, implies duration –14 beats.

d = –5, implies duration –30 beats.

or for any negative d, duration = – (2^|d| – 2).

[This section added 15 August 2015]:
Note also that the fraction of the preceding duration added by each dot is as follows:

One dot: 3:2
Two dots: 7:6
Three dots: 15:14
Four dots: 31:30

So the numerators follow the series of Mersenne numbers while the denominators are always one less.  The general formula this is:

(2^{numDots + 1}– 1) : (2^{numDots + 1}– 2)

Look what happens though if d (numDots) is zero:

(2¹– 1) : (2¹– 2)  = 1 : 0

Thus a note with no dots is 1:0 or infinitely longer than what preceded it in the dot series. One way of looking at this is that a plain note receives all of its duration not from its shape or number of flags, but from the fact that it has no dots; if it lacked no dots then it would have no duration. :-)  I told this theory to Noam Elkies and he said that I was dotty.

Anyhow, back to other weird numbers of dots.

Non-integer dots

Fractional dots create irrational durations, a more useful concept than negative durations. Conlon Nancarrow^[*]  has used these durations in some of his pieces, such as one where two canonic lines are set against each other with tempi in the ratio of √2 to 2.  After the first note of the piece, no two notes will ever coincide no matter how long the piece lasts.   There's no way to notate anything like this in normal notation, but with fractional dots, we can do this.  If n = 2^d and beats = 1 + (n – 1)/n then a note with 1/2 dot has n =  2½ or √2, so a half-dotted quarter lasts 1 + (√2 - 1)/√2 beats, or approximately 1.29 beats.  With this knowledge we can create many irrational lengths for notes just by giving fractional dots. (Though not all irrational lengths.  For instance Nancarrow's "Transcendental Etude" puts two canonic lines against each other with tempi in the ratio of e to π.  Fractional dots could not notate this).

But before trying to figure those out (an exercise for the reader...), there are still lots of rational number lengths that cannot be notated with fractional (i.e. rational number) dots.  Is there a way to get any other rational length instead of just the ones based on powers of 2 that are standard? Can we, for instance, notate a triplet only with partial or negative or partial negative dots? Yes, we can, by making the number of dots a logarithm!  For instance, to get a note that is 5/3 the basic length, we set the number of dots to log₂(3). That's a triplet half note. To get a triplet quarter, we subtract two dots from that, or (log₂(3) – 2) dots. Or for a quintuplet, use (log₂(5) – log₂(3) – 1) dots.

Trying to come up with a way of notating this makes me understand why someone decided that a triplet mark was simpler. But since it is not too hard to prove that any nested tuplet can be written as a single tuplet, we find that any rational duration can be notated as a single note with different numbers of, possibly irrational, dots.  Something for the new complexity school to pursue next.  Or hopefully not.

[*] an earlier version of this post referred to "Colon Nancarrow" who is neither a piece of punctuation nor a part of the digestive tract.  He is also not the same as Loren Nancarrow, a really great weatherman in San Diego, that I've always wanted an excuse to make a shoutout to in a Conlon Nancarrow post.

Changing Musical Time Over the Past 1000 Years

One of the fundamental changes in the West’s conception of music over the previous millennium was the idea that music’s rhythm and meter could be precisely measured not in terms of seconds or minutes, nor in borrowed notions such as poetic feet, but in purely musical terminology like quarter notes or dotted eighth notes.^[2] This new way of thinking is intimately connected to the rise of musical notation, and in particular, the use of notation to show relationships among two or more coordinated parts. Pitch needed its own vocabulary even in the era of unwritten music—-organs needed to be built; lyres tuned—-but the necessity of specific terminology for musical rhythm arose only when it was written down. And while the notation of pitch became largely standardized within the first few centuries of the last millennium to the point where undergraduate music students can (more or less) read pitch from thirteenth-century manuscripts, rhythmic notation remained unsettled much longer.

Seemingly from the first moment when music became measured (musica mensurata), it began to slow down. The earliest measured music, that of Leonin and Perotin, the twelfth and early-thirteenth century composers of Notre Dame, used the figure of the “long” as a basic value, to be paired up and combined into a maxima or duplex long, or to be divided into three breves. By the late thirteenth century, music had slowed to such an extent that the breve became the basic note value, and the semibreve, worth either one-half or one-third the value of the breve, sprang into use.^[2] This word, semibreve, is still the most common name in the British world for the note that in the States is called the whole note. Because the process of slowing continued unabated, this note is now the longest value in frequent use. By the fourteenth century, the tempo of music had slowed sufficiently that a note even faster than the semibreve needed to be introduced. This note came to be called the minima or minim--still the British word for half note—-signifying that this was to be the minimum, that is, the final, indivisible smallest possible note. (The debates of the music theorists of the Trecento are echoed in the discussions of modern physics that postulate that time itself may move in discrete and indivisible minima, each lasting about 1/200,000,000,­000,000,­000,000,­000,000,­000,000,­000,000,­000,000th of a second.) But the force of the slowing trend could not be stopped: it was less than fifty years later that the first “semiminim” would appear. Its name embodies a contradiction, “half of the shortest possible note,” yet though it was an extremely fast note for its time, it eventually became our most basic beat, the quarter note.

The process of slowing itself slowed slightly during the fifteenth, sixteenth, and seventeenth centuries. Faster notes continued to be introduced, but the fundamental tempos of music changed little. There are exceptions, and in some of these cases the slowness of the fundamental beat and the speed of the shortest notes produced dramatic effects: tuplet 128th notes appear in Beethoven’s third piano concerto, 256th notes in a concerto by Vivaldi (F. IV. n. 5), and most exceptionally, 1024th notes (incorrectly notated as 2048th notes) in the little known “Toccata Grande Cromatica” from The Sylviad by Anthony Phillip Heinrich (ca. 1825), a piece that in its own continual slowing of the fundamental beat is a microcosm of all of the notated music history of the West.^[3] (See Example 1).


Example 1: Anthony Phillip Heinrich, “Toccata Grande Cromatica,” excerpt, with 512th and 1024th notes (incorrectly notated as 1024th and 2048th notes) at the end of the example.

Between Perotin and Heinrich, there exists a range of time that is almost incomprehensible to our sense of how music unfolds. Within a single maxima, there are 1024 128th notes, and, as we have seen, even longer and shorter notes have occasionally been used. Example 2 shows (on a logarithmic scale) the amount of time various notes would take if they were all played at the same adagio tempo of one quarter note per second. Obviously it is absurd to use the entire range of note values in a single piece at a single tempo mark. In fact the pieces using the shortest notes tend to have the slowest tempos, and the contrary is true for pieces with the longest notes. But Figure 1 demonstrates the enormous pull towards expanding the range of rhythmic resources that composers have felt over the centuries.


Figure 1: Lengths of different note values at M.M. qtr. = 60.
(with actual CD frequency and Ring lengths for reference)

In contemporary art music, composers have played with all extremes of lengths, but the most significant innovations in notation have come, as they did in the Middle Ages and Renaissance, in the notation of the most fleeting notes.^[4] One of the most interesting case studies in the continuation of the medieval tradition of shorter notes comes in the American composer George Crumb’s string quartet Black Angels (1970), where, in the excerpt shown in Example 2, he uses a time signature of 7/128 in measure 2--written with the denominator as a note--along with the almost as equally unusual 7/64 with each note receiving an equal accent, thus providing the shortest non-compound meter ever published.


Example 2: Unusually short beat values in George Crumb’s Black Angels.

Other unusual and brief meters appear occasionally in modern music. Karlheinz Stockhausen’s 1956 composition Zeitmaße uses meters that are nearly as short as Crumb’s, such as 2/32. Composers allied with the New Complexity school, including Brian Ferneyhough and Thomas Adès, have used meters such as 5/6 and 2/10 that allow “tuplet” values (in these cases, triplets and quintuplets) to be fundamental and independent units. Finally, though multiple simultaneous meters were used in the works of both Bach and Mozart, the metrical experiments of Conlon Nancarrow use multiple meters in ways that particularly stretch the definition of meter. His “Transcendental” Etude contrasts two canonic lines whose rhythms are in the ratio of the transcendental irrational numbers e to π.

I am particularly interested in these unusual ways of specifying beat and meter in today’s music because the same spirit of experimentation was at work throughout the Italian ars nova. The feeling that the resources of the past were insufficient to express the creative metrical impulses of the present dominated musical thought in the late Trecento and early Quattrocento. Over the next few blog posts I hope to show how.

Endnotes

^[1] This post is adapted from part of an article "Changing Musical Time at
the Beginning of the Renaissance (and Today)" to be published by L. Olschki in an as-yet-unannounced surprise Festschrift in 2011.

^[2] On attempts to measure the slowing of medieval musical time in terms of clock time, see Marco Gozzi, “New Light on Italian Trecento Notation, part 1”, Recercare 13 (2001), pp. 5–78.

^[3] The source for these (and many other) extremes of musical notation is Donald Byrd’s excellent on-line resource. I thank him for many stimulating conversations on this topic. The score of Example 1 has been republished recently as Anthony Phillip Heinrich, The Sylviad, or, Minstrelsy of Nature in the Wilds of North America: opus 3, intr. J. Bunker Clark, Greenleaf, Wis. 1996. The term chromatic in the title of the piece refers not to notes out of the key, but to the amount of ink (or color) needed to express the short notes in the work.

^[4] On recent uses of extremely long durations in recent music see Alex Rehding, “The Discovery of Slowness in Music,” forthcoming.