Music is math, which is truth, which is beauty, which is here at SIAO (for a moment, at least)

by Michael S. Kaplan, published on 2007/10/19 10:31 -04:00, original URI:

One of the problems in being a generalist in so many different areas is that I am interested in so many different things that I had no way of becoming an expert in all of them.

Luckily I have friends like Melanie (the technical editor and musical expert from Garden Path Menus) who are experts in areas like music (from Medieval times to now) and with her permission I am quoting an email she sent yesterday after a great conversation she was in with a few people about music, math, and more:

...I wrote up a little bit about the history of music notation, complete with pictures. J  

The dates on the chart in the link are not quite right (it depends on what country you’re documenting), but they’re close enough for most folks.

The left column of squiggles is what Hildegard von Bingen still used in the 12th century. In the 6th century, when these neumes began to be used, they were not drawn on ledger lines, so there was no way to know how great the interval was between two notes—accommodating text might require that a note identical in pitch to a neighboring note be placed higher or lower on the page. These were called “unheightened neumes.”

By the 9th century, composers started to draw ledger lines and the pitch intervals between notes began to be consistent (“heightened neumes”). For the first time, someone could accurately melodically replicate a song they had not heard, although the length of notes was still up to the individual performers. They narrowed the number of ledger lines to four in about the 10th century (notes in a sequential scale on a row—on the line and between the lines—added up nicely to the eight notes of a modern scale. Music was still in five-note modes at the time, but because finding the drone note was an obvious way to pair voices, the octave became an important concept to the evolution of early harmony and polyphony. More on that later).

The block notes in the center column of the table began to be popular in the late 11th century. The block notes very much imitate the earlier neumes, but the duration of a note began to be relevant. The duration was influenced partly by text and musical sensibility, but composers could add dots and little lines (called epicemas) to note shapes indicating that a note should be lengthened and by how much. The primary motivation for developing these larger note shapes was so that groups could participate (rather than soloists, for whom the earlier music was intended). They were large enough to be seen from a distance. (A whole choir sang from one large book.) Block notes are still in use by people who perform Gregorian chant. The earlier neumes are still used by Hildegardians and by scholars working on the Gregorian chants that are old enough to have been written in them originally. The neumes provide perfectly singable information, and are considerably more musically informative than modern notation regarding nuance of inflection, duration, intensity, and intention.

The link’s chart is missing a stage in development, so here’s a link to what happened next.

Next came white note mensuration. Although it was developed in the early days of the 15th century, it was fast replaced (within century and a half, or so) by modern notation (in the right-hand column of the earlier link’s table). White note mensuration was not adopted by all countries—the Italians, always innovators, were the primary adopters. In white notes, the black diamond was a count of one beat, the open diamond with a stem was two beats, the dot meant to add half again (so a dotted open diamond was three counts, a dotted black diamond was a beat and a half), and so forth. The number of ledger lines increased to five because relatively little music took place in less than an octave. (More singers had training than ever before, and keyboard instruments began to increase in popularity and offered considerably greater range than previous instruments.) For the first time, duration of a note was prescribed by the composer. How fast or slow the piece was performed was largely up to a conductor, a role that was another new invention (someone who interpreted the music and led the group in a formal way—previously there was a leader to keep groups together, but leaders had no special training and just waved a hand around to show when to change notes). Some of the symbols (like the C with a slash through it on the top lines near the left edge) told the leader how fast to go (a C meant—and still means—common time, which is heartbeat or walking speed, or 60 beats per minute; a C with a slash meant half that speed, or twice as slow, etc.), and instruments and singers could finally begin to make complex music together because the rules of music were starting to converge on a standard.

The end of the 16th century saw the establishment of major innovations that would change music composition forever: key signatures (how many sharps and flats), measure lines (separating a series of notes into discrete chunks), clefs (marking a range of notes. Some clefs already existed, but they were considered movable. The 16th century created a standard and pitch range for each), and time signatures (how many beats per measure throughout the piece—a major innovation in allowing larger groups and diverse instruments to perform together).

The rules of music were pretty stable by the end of the 16th century, when modern notation was in common usage. This is where the story I told of Michael Praetorius and Syntagma Musicum fits in, the documentation of modern music notation and performance that J.S. Bach finished, and the standards that we still use today. They got it so right that nothing has changed in 300+ years.

What we do with these notation standards has changed considerably, in large part, thanks to Beethoven. But that’s a story for another day. So is the tangent about rhythm and rhythmic patterns and its merry companion, counterpoint. In brief, rhythm wasn’t part of Western music until white note mensuration made standard note lengths. With rhythm came harmony (as opposed to polyphony), and in a hop, skip, and jump, music began to sound modern. Consider the Italian Renaissance the beginning of modern music, with rhythm, harmony, notation standards, and pitch standards. Harmony plus rhythm bred counterpoint further north. But I digress.

Here’s a little running parallel that I can’t resist. Polyphony—multiple parallel lines of melody—began to develop in the very late 12th century and hit its height in the 14th century (or so, depending on what country you are talking about). Harmony (in the modern sense, where chords follow an established formula and some musical lines are not at melodic, but are provided purely to fill in harmonies) evolved around the late 15th century, right around the development of modern notation. The pianoforte had been invented and allowed faster evolution than ever before because, for the first time, a performer could play loudly and softly on a single instrument and specific emotions could be defined by a composer. Emotion hadn’t really come into it before. It was all about the math....

Do you see what I mean?

I could read a dozen books and not come out of the experience understanding as much as I did from the above.

The people you get to feel smarter just for knowing them are very cool....


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# Gene on 21 Oct 2007 5:03 PM:

Wow... that's abso-frickin-lutely fascinating, even though my knowledge of music barely extends to playing Guitar Hero... I'm glad I have all my audio plugins disabled, though.

# Maleeha Khan on 24 Nov 2007 2:32 PM:

i m glad to visit on this site its a gud site.  KEEP IT UP!HEY DUDE

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