The Daily Parker

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W is for Well-Tempering

Blogging A to ZToday's Blogging A-to-Z challenge entry examines the physics of music. Specifically, when a musician looks at a note on a page, what tone does she actually produce?

Most people today have passing familiarity with the piano, which has one key per note. This means the frequency of each note remains the same no matter what key a pianist plays in. If she hits the A above middle C, the piano strings vibrate at 440 Hertz (cycles per second). The A below middle C is 220 Hz, the A below that is 110 Hz, and so on. All of the notes in between have fixed frequencies as well.

This system, dating from about the beginning of the 20th century, is called equal tempering. It has some pretty interesting consequences, first among them that only the octaves are perfectly in tune. Every other interval is slightly out of tune—sometimes in two ways.

Equal tempering is a compromise, driven in part by the popularity of the piano, because they're so hard to tune. Other instruments don't have this limitation, so in some circumstances (i.e., string quartets), you might hear well-tempering instead.

In well-tempered tuning, some intervals actually do retain their proper mathematical relationships. But only some. Well-tempering is another compromise, resulting in different keys having wildly different tone colors. Bach promoted well-tempering with his two-volume set of preludes and fugues called The Well-Tempered Clavier. (Cue the irony that most people today have never heard it played on a well-tempered instrument. I found a good demonstration of the differences between equal- and well-tempering that's worth 12 minutes of your time.)

Before well-tempering, musicians used Pythagorean tempering (based on perfect 5ths) and meantone tempering (imperfect 5ths to get better 3rds).

For more about this topic, Nathan Nokes has a good video about the physics of these earlier tuning systems, with pure-tone examples. Notice how just and Pythagorean temperaments sound out of tune. Except they're not; they're just different.

U is for Unison

Blogging A to ZThe Blogging A-to-Z challenge entry today starts with a joke: what is the definition of a minor second? Two oboes playing in unison.

Sorry, oboes.

We already know what unison means: it's two voices sounding the same note. And earlier I mentioned that different instruments use different clefs. And we've covered key signatures. Now I'm going to tell you a dirt secret of the orchestra: unison sometimes looks like a bunch of completely different notes.

The problem comes from the way that certain brass and woodwind instruments work. Different instruments naturally play in different tunings. For example, if you play a natural scale on a French horn, you're actually playing (usually) an F or Bb scale. So we say that the horn is "in F" or "in B flat." Which leads to the part that string players and singers struggle to understand: music for the French horn is written as if C is F. Or Bb. In other words, horns fall into the class of transposing instruments, because they naturally transpose the notes written on the page.

To illustrate, here is how we write an F major scale played in unison—exactly the same notes—between a viola and a French horn in F:

The viola starts on F, but the horn starts on C. Except that the horn's written C is actually sounds like F.

It gets even crazier when you combine a bunch of instruments that transpose in different keys. Here's the opening of Brahms's Symphony #4 in e minor:

The flute, bassoon, and strings are in C, so when they play the note written as C they actually play a C. The clarinets are in A, so when they play what's written as C out comes an A. The key signature has to change, too; theirs is in C minor, because when they see a C the sound they make is actually E. Opposite issue for the French horns, which Brahms wanted in E: when they play the note written as C (as they do for four bars), they're actually playing G, which is also why their staff is in A minor.

It's all finally coming together, isn't it?

T is for Tritone

Blogging A to ZToday's Blogging A-to-Z challenge post sits right in the middle of everything.

The tritone is the interval between the perfect 4th and the perfect 5th. Depending on which direction you're going, it's either an augmented 4th or a diminished 5th. And it's always going somewhere.

In the C major scale, the natural tritone is between F and B (where it's an augmented 4th) or B and F (where it's a diminished 5th). B, remember, is the leading tone in the key of C, so it really, really wants to resolve to C. The F (subdominant) wants to resolve to E (the mediant). Therefore, if you have an augmented 4th, it wants to resolve to a minor 6th (which is also the tonic chord in first inverstion):

And if it's a diminished 5th, it resolves to the major 3rd:

Remember ethnic 6th chords? Tritones all the way down. Even in a basic I-V7-I progression, the V7 has the tonic chord's tritone in it:

The tritone is between the first and second lines from the top. Just as in the very first example on this post.

S is for Syncopation

Blogging A to ZThe Blogging A-to-Z challenge will get a little funky today as we look at syncopation, which is nothing more than an unexpected rhythm.

Here's a simple example. Take this clunky melody:

Now let's syncopate it a little, by shifting some of the notes off the beat:

Instead of hitting 1, 2, 3, 4, 1, 2, 3, 4, now it hits 1, and, and, and, 2, and, and, 4. It's harder to dance to but more interesting.

More examples? How about Mozart's Symphony #40, third movement:

Or the Rolling Stones? Beethoven? Scott Joplin?

Syncopation is every——where!

R is for Rondo

Blogging A to ZToday's Blogging A-to-Z challenge post will take a look at common musical forms.

We've already seen some examples of common musical forms, even though I didn't call them out: the canon and the fugue. Both are imitative forms, though as you've seen the fugue is far more complex than the canon. "Row Row Row Your Boat" is a canon (but, of course, someone made a fugue out of it).

When we talk about other forms, we usually note large sections of music with letters. So a form of, say, A-B-A means that you have a theme first, then another theme, and then the first theme comes back.

The basic musical forms all show up in symphonies written between about 1760 and 1830, in the Classical and Early Romantic periods. Haydn codified the symphonic template, Mozart perfected it, and Beethoven took it up a notch.

The first movement of a Classical symphony usually uses sonata allegro form. It starts with an optional introduction, progresses to exposition, through development, then a recapitulation, and concludes with an optional coda. In short, A-B-A1, because the recapitulation usually doesn't strictly repeat the exposition.

The first movement of Mozart's Symphony #25 demonstrates this beautifully. The exposition starts immediately, stating the bold g-minor theme in several forms for the first 90 seconds or so. Development ensues, taking us around several related keys and themes (including the primary theme). The recap begins at 5:05, once he's brought it back to g minor. Note, also, that Mozart wrote this at 17.

Second and fourth movements often use rondo form, which is A-B-A, A-B-A-C-A, or A-B-A-C-A-B-A. The A just keeps coming around, you see. Here's Haydn's 49th symphony, second movement. It deviates from a typical classical symphony in that the second movement is fast, not slow as was common in the period. But it's a complete A-B-A1-C-A2 rondo, as well as an excellent example of the Sturm und Drang ("storm and stress") period in the late 1770s-early 1780s.

Third movements generally follow the minuet and trio form, which uses 3/4 time and a simple A-B-A or A-B-C-A form. Example: Brahms' Symphony #3, third movement. Even though Brahms wrote this decades after Mozart and Haydn had died, it still maintains the minuet and trio form—though with lusher orchestration and harmonies than either classical composer would have used.

Listen for these forms the next time you hear a symphony. They're in there.

Q is for Quaver

Blogging A to ZToday's Blogging A-to-Z challenge post explains how musicians keep time.

Through all the examples I've posted this month, you may have noticed that a note's stem has a relationship to how long the note sounds. They do. Starting with a whole note (open, with no stem), each change to the stem reduces the length of the note by half. This also works when you start with a whole rest, except a rest means "don't do anything here." Example:

(In the UK, those notes are called whole, half, quarter, quaver, semiquaver, hemisemiquaver, and hemisemidemiquaver. Don't ask me why.)

You can also add a dot after a note, which increases the length by 50%. The dotted quarter notes in this example are the same as three 8th notes:

On the same subject, let me draw your attention to the numbers (and sometimes letters) that follow the key signature in a staff. That guy, called the time signature, tells performers how to count.

The most common time signature is 4/4. That means there are 4 beats in each measure (the top number) and the quarter note gets one beat. It's so common that it can also be represented by the letter C, as in the top example above. Sometimes you'll see a C with a line through it. This is called cut time, or 2/2, in which there are two half-note beats per measure.

The second example above shows 6/8 time, in which there are six eighth notes to each measure. Typically musicians count 6/8 as two beats per measure, with the dotted quarter getting the beat, so it feels like a march.

The time signature 3/4 has the same number of eighth notes per measure, but since the quarter note gets the count, there are three quarter-note beats per measure. This is the time signature for waltzes.

Almost any combination of beats and notes can form a time signature. I've seen 7/4, 3/16, 1/1, 9/32...literally anything as long as the bottom number is divisible by 2 (or is 1). That said, the most common time signatures by far are 4/4, 6/8, 3/4, and 2/2.

And here, for your listening enjoyment, is a piece that flouts all the rules and somehow works—as humor:

P is for Pachelbel

Blogging A to ZThis morning, my Blogging A-to-Z challenge post will discuss a composer whose music I absolutely loathe because of its insipid, simplistic, earwormy pabulum, Johann Pachelbel (1653-1706).

You have, no doubt, heard his Canon in D, which, thanks to its inclusion in an otherwise forgettable film 51 years ago, continues to besmirch weddings and other cultural events with its demonstration of what happens when you strip music down to the essentials and add nothing back. In a way, the Canon in D resembles a lot of modern music by providing nothing more than a repeating theme of such simplicity that only a performance by 3rd graders on recorder could do it justice.

So why did I include this composer in a series on music theory? Because in that simplicity is just about all popular music of the last half-century.

Here are the first few bars of the piece:

It is I-V-vi-I6-IV-I-IV-V, repeated endlessly, until someone in the audience starts yelling "Please, for the love of Dog, make it stop!" Notice that the string parts are also boring, and (because this is a canon) repetitive.

When I say it has infected music in the last 50 years, I mean it's like a staph infection that can shut down an entire hospital. Here, to make the point better than I can, is Rob Paravonian:

O is for Ornaments

Blogging A to ZThis morning's Blogging A-to-Z challenge entry will take a quick turn and possibly trill your heart with a brief overview of ornaments.

You got a glimpse of two of the most common Baroque ornaments on Saturday as the Bach snippet I posted contained a grace note and a mordent:

The grace note tells the performer to add the note within the duration of the main note. For example, the grace note in the first measure would be played out as shown in the second measure:

A mordent tells the performer to do a little flip up from the written note (or up if it's got a line through it). This example shows a mordent, then its written-out equivalent, followed by a rising mordent and its equivalent:

The turn comes up frequently as well, and does exactly what it says on the tin:

Let's not forget the trill:

If only there were a single piece of music that put all of them together brilliantly...like maybe Bach's Sinfonia #5 in Eb major, BWV 791? Why, yes:

N is for Notation

Blogging A to ZToday's Blogging A-to-Z challenge post will, like yesterday's, take us back in time.

Almost every day I've shown samples of music using modern notation. Any contemporary musician should have no trouble reading them.

Almost a thousand years ago, in 1025, the monk Guido d'Arezzo decided to record music on paper in a way that would enable people to read it even if they'd never encountered it before. He used blocks on lines with stems indicating how the notes were connected, and it looked like this:

The innovation here is that the note heads convey absolute pitch, and the stems convey timing, just as they do today. Someone who has never seen this before could (if they understood the language) produce the music intended by the composer.

During the early 17th century, as instrumental music came to dominate the scene, musical notation became much more complex. By the 1620s and 1630s, scores looked almost the same as they do today.

Specialized music uses specialized notation schemes, however. And non-Western music, which may use entirely different tonalities, sometimes has entirely different ways of notation.

M is for Modes

Blogging A to ZToday's Blogging A-to-Z challenge entry goes back in time a little bit. Before there were keys, there were modes: the original scales used in Ancient Greece that still pertain today.

Our C-major scale roughly corresponds with the Ionian mode. (I say "roughly" because while the fifth, octave, and probably fourth notes would have sounded the same back then as they do now, the rest of them probably would have sounded slightly out of tune to modern ears. This is a topic for next week.)

If you start on D, you get the Dorian mode:

It's not quite D minor, because instead of a flat 6th and natural 7th as in a minor key, we have a natural 6th and flat 7th. You can hear Dorian mode in Deep Purple's "Smoke on the Water."

Next comes Phrygian mode, starting on E:

If you start on F, you get Lydian; G, Mixolydian; A, Aeolian (corresponding closely to natural minor); and B, Locrian.

I mentioned Deep Purple because everything old is new again. A lot of modern popular music has reached back to modes, mainly (I think) because they sound cool. Lydian mode has a tritone instead of a perfect 4th; Locrian just sounds...off.

In fact, while Ionian, Dorian, Phrygian, and Aeolian pop up a lot, there aren't many examples of the others. But they're out there.

Homework assignment: Find modal pop songs (LMGTFY). Then listen to them.