Scales! Wait. Come back.
I hated scales in middle school. Oh I was good at scales, even got a certificate for playing all four forms of every scale by memory on my wee clarinet. I was a good boy.
And still I hated them.
It just seemed like such a pointless, hoop-jumping thing to me. Memorize these note patterns, then play each one starting from every possible note, and you win. But aside from the value of rote memorization and the fingering practice…what was the point?
There were other unanswered questions in my head — like Why are there three different kinds of minor scales? Why is the melodic minor different going up and going down? Why does the harmonic minor make me sound like a snake charmer? Why does minor sound sad and major sound happy? Are these the only four ways to get from C to C?
It’s like the Steven Wright joke
Why is the alphabet in that order? Is it because of that song?
Scales are actually cool and interesting once you know what they’re for. Stick with me to the end of the post, then tell me if I’m wrong. (Spoiler: I’m not.)
Last time I split the octave into 12 equal parts to make the chromatic scale. The name hints at the purpose. By containing all 12 of the pitches we use in Western music, the chromatic scale is like a paint store with 12 available colors.
You don’t want every available color for a painting project. You choose a palette — a subset of colors, a theme. If you want this result:
…don’t use every color in the store. Use this palette:
Same for music. The chromatic scale has every available pitch in the Western tonal system. The distance from one to the next is a half step. Here are the 12 half steps of the chromatic scale (plus the next C):
When I write a piece, I don’t need every note. If I want a certain mood, I’ll choose a subset of pitches from the 12 available, and a different subset gets me a different mood.
Example: If I choose every other pitch, instead of 12 notes a half step apart, I’ll have 6 notes a whole step apart. That’s called a whole tone scale:
There’s something strange about that scale, right? It starts going off the rails at the 4th note and never gets normal again. Here’s why: All of the pitches are the same distance apart. No half steps. Because of that, your ear can’t easily track where you are. Aside from whatever note we happen to start on, none of the pitches stands out as “home,” so the whole tone scale creates music with an ethereal, directionless quality. If you were a child of the 70s, you know it well. Whenever there was a campy flashback in The Brady Bunch, or Scooby Doo was being hypnotized, there’s a good chance a harp or bells were playing music drawn from a whole tone scale:
One particular Debussy piano piece called Voiles (Sails) is the go-to for illustrating a whole tone scale. Feel that unanchored, homeless quality (0:47):
The fact that pretty much every theory teacher uses sitcom flashbacks and one Debussy piece as whole tone examples shows you just how rare it is. Beyond drifty sails and coming unstuck in time, a scale with all the pitches equally spaced just isn’t all that useful. If someone gave you driving directions like this — Go one block and left, one block and right, one block and right, one block and left, one block and right, and on and on across town, you’d have a helluva time finding your way home again. Every step is the same. It’s confusing.
If instead the directions were Go one block and left, three blocks and right, one block and left, then five blocks and you’re there — now the steps are different enough to landmark in your head. You need a scale that works the same way — a mix of steps of different sizes so your ear knows where you are relative to home.
Major and minor scales do just that, creating a pattern of pitches that are different distances apart. They do it by choosing only 7 of the 12 notes, leaving gaps where the other pitches were. The scale shows the ladder of steps between the notes if you arrange them in order. Here’s a C major scale:
You start on C but leave out C# and go to D — a whole step. It’s like going from red to orange and skipping reddish-orange. The major scale then goes another whole step (to E), then a half step (F), then three more whole steps (G-A-B), and finally a half step to the next C. If you start on C, the white keys on a piano fall naturally into the major scale without any adjustments. The black keys are the gaps, the five chromatic pitches that were left out of the major scale:
If you start on a note other than C, you’ll have to raise or lower notes along the way, using flats and sharps on the page and black keys on the piano to get the same pattern of whole steps and half steps.
Scales aren’t meant to be played in that order any more than the letters in a word are supposed to be spelled out in alphabetical order, or a set of colors in your paint theme is supposed to be painted left to right on the wall by wavelength. Hey, rainbows are great, but you don’t want a rainbow every time you paint. So when you paint your house, this is the scale:
…and this is the music:
The scale is a set of instructions to pick out the colors (pitches) that establish a certain palette (key) for the music you want to write.
But music goes beyond the visual analogy to something that makes its emotional power possible. This pattern also creates a kind of aural map. The sequence of half and whole steps sticks a pin in one of the notes, calling it home, then creates a series of landmarks to tell you where you are relative to that home at all times. By listening to music all your life, your ear has learned how to follow that map — to go away from home, have adventures, and return (or not). The next three JET posts will get into that.
The minor scale is a different pattern of whole steps and half steps spanning the octave. The result is a different emotional palette:
The minor scale results when you lower the third, sixth, and seventh pitches of the major. That’s worth remembering. No need to memorizing key signatures or build scales yourself, but we’ll certainly talk about the various shenanigans the third, sixth, and seventh notes of the scale get up to.
Three great scales from elsewhere
Step outside of Western music and you can really start to see how different scales color the music. I hinted at that last time with the Carnatic microtonal scales. (Don’t be skipping stuff, or all is lost.) Here are three more.
Arabic music has at least 90 different scales, or maqam, each of which provides a different mood. Here’s one, maqam nawa athar:
Which sounds like this:
Hear that? It’s just crammed with intervals that sound exotic to Western ears. If a major scale is a map across your hometown, this one is a map to the market in Marrakech.
Several of the pitches of the Indonesian pelog scale fall in the gaps between our notes, which further shows that all scale divisions are arbitrary and cultural. You can’t even notate pelog properly with Western notation. If you try, you end up with these apologetic little plus and minus signs to indicate “a little higher” or “a little lower” than the notation:
And why not? Remember the sixth overtone, the B-extra-flat? To Western listeners, these pitches sound out of tune because they don’t match our particular divisions, but they have no less claim. Here’s pelog:
And here’s a phenomenal Balinese gamelan playing music using a variant of pelog. If you haven’t heard Balinese gamelan before, strap in, this is awesome:
Want to evoke a Middle Eastern vibe? Use the Byzantine scale:
It starts with a half step, which is instantly exotic (neither major nor minor does that), and twice again you get that wide step and a half (the ties above). The composer Saint-Saëns knew he could throw you into the ancient Levant for his opera Samson and Delilah by using the Byzantine scale:
Other minors and the modes down the road. Now go outside and play.
Pitch: Slicing up infinity
The electromagnetic spectrum covers a huge span of wavelengths, from a trillionth of a meter to (hypothetically) the size of the universe. Most of this energy — gamma rays, X-rays, microwaves, ultraviolet, and so on — is invisible. But a tiny sliver is visible light:
Different wavelengths within that tiny window present as different colors. Newton first described the rainbow as having five colors — purple, blue, green, yellow, and red. He later added orange and indigo for a silly reason I’ll describe in a minute. Neither answer was wrong, and neither was right.
When I was six, my box of crayons had eight colors. By the time I was ten, there were 64. The crayonic scientists hadn’t discovered 56 more colors — they just split the spectrum into finer divisions, adding colors like blue-green and green-blue. (Some speak of a 120-color Crayola box, but the few photos that exist are suspiciously blurry.)
My first color computer monitor displayed 256 colors. The one I’m on now has the weirdly non-specific “millions of colors.” The possible colors in that spectrum are literally infinite: Just as you can endlessly divide a number line into smaller and smaller fractions, you can tweak wavelengths forever. So instead of being handed a specific natural palette, we divide the spectrum up arbitrarily to meet our needs: five colors in the rainbow, or seven; eight crayons or 64; 256 colors on the monitor, or millions; and 1500 choices in the Sherwin-Williams store. Until you mix a speck of mauve into the gallon of Carnival Orange and get 1,501.
Sound works the same as color. The range of frequencies we can hear is a sliver of the possible range of vibrations. Here’s the full range that the average human can hear, from 20Hz to 20,000Hz:
Like color, pitch is infinitely divisible, so we make choices about how finely to slice up that spectrum of sound. But one division is so basic to the physics of sound that it serves as the starting point for pretty much every music system on Earth: the octave. Give us an octave, Dorothy:
Doubling any vibration sends you up one octave, and the octave is the relationship between any note and its first overtone. It’s the strongest pitch relationship in music, and you can hear the family resemblance. In Western music, we designate that close relationship with a letter name. Notes vibrating at 110, 220, 440, and 880Hz are all “A,” for example.
So octaves are basic. But using octaves as the only musical steps would make for pretty poor music, like painting with 50 shades of grey. We want additional pitches to use, smaller divisions to give us more of the available colors. How small is the question.
There’s a slightly hilarious psych term for the amount something has to change before the average person will notice. You’d think it would be the “minimum discernment threshold” (MDT) or something similarly sciencey, but no: It’s the JND — the Just-Noticeable Difference. When my wife gets a haircut, it’s often a JND experiment for me.
The concept reaches even beyond haircuts to the perception of things like color and pitch.
That audible range from 20Hz to 20KHz is about 10 octaves. (For comparison, the piano keyboard is just over 7 octaves.) We could split the full range of human hearing into a maximum of about 1,400 just-noticeably-different pitches — 140 pitches per octave. But just as octaves are too big to be our only interval, these are way too small to make musical sense. Going from one to the next sounds more like drifting very slightly out of tune than anything musically useful.
So over time, instead of using 140 pitches in the octave that are barely noticeable in their differences, music cultures have divided up the octave into fewer, more useful and noticeable pitch differences.
South Indian Carnatic music uses the term śruti (or shruti) for the smallest meaningful interval of pitch. Some Carnatic theorists have put that number as high as 66 pitches in the octave, but Carnatic practice actually uses 22 — still a lot of small steps to Western ears. The result is some amazingly nuanced, slippery melodic contours.
Listen to the tiny pitch differences in this performance. The string drone establishes a central pitch, like the Western tonic. The singer’s voice dances in the amazingly tight Carnatic spaces around that center. Just before the midpoint of the clip, she locks in with the drone, then dances away again, returning home once more at the end (58sec):
Western music settled on 12 pitches, equally spaced about twice as far apart as the Carnatic śruti:
That’s the arbitrary paint store that provides pitches for nearly all Western music — folk tunes, opera, rock, pop, everything. It’s called a chromatic scale, which is perfect — chroma is Greek for color, and scala is Italian for ladder or steps. It isn’t music any more than a paint store is a painting. It’s the resource we draw on to create music.
As the Carnatic example shows, the octave can be divided up any number of other ways. There are big gaps between those notes in the chromatic scale, and there could be notes in those spaces, or in some of those spaces but not others. The composer Harry Partch built his own instruments to bust out of the constraints of the Western pitch ladder. Try this on for a minute:
Several more recent inventions like the fluid piano and tonal plexus have also poked around in those pitch gaps. (If you love this kind of boundary-pushing, find 17 minutes for this stunning soundscape by Wendy Carlos.)
Instead of finding more pitches in the cracks of the chromatic scale, we usually go the other way, taking pitches away from the 12 and playing around with what remains. You can get a scale with steps of various sizes that way. Even the 22-pitch Carnatic system doesn’t treat all 22 as equal players in a given piece. There’s a subset that provides the melodic backbone, and the rest serve as ornaments and variants.
The Western major scale works the same way, using seven of the 12 pitches as the backbone of melody and harmony — that’s your do re mi fa sol la ti. The other five pitches are “non-diatonic,” meaning outside of the key, and are used as ornaments and variants — delightful, delicious ornaments and variants.
The seven-note major scale, by the way, is why Newton added orange and indigo to make seven colors in the rainbow. He was enamored of the Pythagorean idea of a “harmony of the universe” (musica universalis) and thought the rainbow ought to reflect the pitches of the musical scale, as well as the known objects in the solar system at the time and even the days of the week.
Never mind that the scale and the week are also arbitrary and the astronomical count turned out incorrect — it’s a thing we do, grabbing on to patterns and relationships to make sense of things. But the divisions themselves are neither right nor wrong — just useful.
More on the pitch/color connection:
The man who heard his paintbox hiss (the synaesthesia of Kandinsky)
Music for Measure: on the 300th Anniversary of Newton’s ‘Opticks’
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Overtones: Every note is a chord
Listen to this:
That’s a low C, a single note played on a piano. At least that’s how I’d have described it when I was 13. Within a year I was hearing that very differently, thanks to two great musicians.
One was my older brother Ron, who started playing piano when he was seven and I was three. I grew up in his shadow, musically and literally, laying by his feet while he played Chopin and Ravel, Scott Joplin and West Side Story (which he can still shred). He led me to Elton John and Genesis (until Genesis betrayed all that was good) and Jesus Christ Superstar. When I said The Rite of Spring was just noise, he ordered me to put the headphones on and listen again. And then I couldn’t stop.
When I was 14, Ron introduced me to my second huge musical influence by bringing home a set of LPs from the library. It wasn’t music, but a lecture series about music: Leonard Bernstein’s 1973 Norton Lectures at Harvard.
What Carl Sagan did for me with science, Bernstein did with music. I’d never heard anybody talk about music like Bernstein did in those lectures. He went below the surface in a way that was completely new to me. It was the start of my interest in music theory, which turned into a major, my first career, and a lifelong fascination.
The topic Bernstein covers first is the most stunning of all — that a single pitch contains an infinite number of pitches.
This isn’t an abstraction. These other pitches are tangible, and that matters. It’s one of the hidden keys to the ability of music to communicate emotions. But for reasons I’ve never understood, most of the current top-selling music theory textbooks never talk about it at all.
Actually, I do know why they leave it out: it isn’t necessary. It’s merely wonderful.
Don’t Skip the Miracle
Sound results from vibrations. The more regular the vibrating body is in structure — tuning fork, metal string, column of air — the less like noise the sound will be, and the more distinctively “pitched” and musical it sounds. The faster the vibration, the higher the pitch. As you go from left to right on the piano keyboard, you go from low to high, the strings get shorter and therefore vibrate faster, and the pitch gets higher.
The low C at the top of the post resulted when a felt hammer hit a string of a certain length in my piano. The string vibrated about 100 times a second (100 Hertz, or 100 Hz) along its entire length to produce that pitch.
If you double the speed of vibration to 200 Hz, the pitch goes up one octave to the next higher C. A string half as long will do this.
To go up another octave to the next C, just double the vibration again to 400 times a second. A string a fourth as long as the original one will vibrate four times faster, and you’ve got the next C.
And on it goes, with Cs at 800, 1600, 3200. You can hear the family resemblance between notes that are an octave apart. They’re like higher and lower versions of each other because of this simple ratio. If men and boys sing the same melody at the same time, they will tend to be an octave apart. But it doesn’t sound like they’re singing two different parts — just the same melody doubled at the octave. That’s how close the family resemblance is.
Play the low C again.
That string is vibrating at about 100 Hz along its entire length to produce the pitch C. Here’s a schematic of the string in vibration along its entire length:
Now here’s where it gets strange: at the same time it’s vibrating along its full length, the two halves of that string are vibrating independently. And because they’re half as long, they’re going twice as fast (200 Hz). Don’t ask me how this works — magic, I assume. We know that doubling the vibration sends the pitch up an octave. So in addition to the low C, you are also hearing the C an octave higher at the same time.
Again, I’m not saying you are “hearing it,” in quotes, like it’s some hypothetical thing. When you play a low C, you are also actually hearing the C an octave higher. It’s a small but audible fraction of the sound. The note you’re playing is called the fundamental. The higher C is the first overtone. But because the two Cs have such a strong family resemblance, it’s hard to hear it. They are too similar.
Fortunately the string is also vibrating in thirds, at 300 Hz. That produces a G, the second overtone, which is usually pretty clear. Here’s the pitch for G, so you can get it in your ear:
The string is also vibrating in fourths (400 Hz), which is four times as fast as the low C and twice as fast as the second C, so that produces yet another C, an octave higher still:
And also in fifths (500 Hz), which produces an E. This one will be fainter, but because it’s not another C, you can usually hear it. Here’s the pitch to listen for:
And also in sixths (600 Hz), which is twice as fast as the G, so that’s another, higher G:
The divisions of the string continue infinitely in smaller and smaller intervals, all vibrating simultaneously and producing an infinite number of pitches. But after about the fifth overtone, the vibrations are such a tiny sliver of the sound that you usually can’t hear them.
No worries, we’ve already achieved the magic. When you play a C, you actually hear all of these pitches:
I don’t mean to startle you, but that’s a C major chord. When you play a C, you don’t just hear a C, you hear the pitches of a C major chord — C, E, and G. It’s not only C that does this, of course — play any pitch, and you are actually hearing not just that pitch, but a major chord based on that pitch. It’s more subtle than that recording, but you can hear it, and we will shortly.
We could do the same with any pitch: A, F#, Bb, anything. Because of the physics of vibration, each of them produces its own major chord.
Why do overtones matter? Because the emotional fuel of music is the interaction between pitches sounding simultaneously — harmony. And when you play any two pitches, it isn’t just those two pitches interacting – there’s a whole cloud of other notes above them interacting as well. (More on this when we get to consonance and dissonance.)
Now it’s time to hear the overtones, not as bells outlined on my synthesizer, but coming out of an actual single pitch. Turn the volume up a little. A lot of things affect how strongly overtones pop out when you play a note, including humidity and whether the piano is in tune. My piano is not, but the day was humid, so we’ll see what happens. I’ll play that low C one more time, and you listen for the second and fourth overtones (G and E) blooming out of the sound:
Oh man, this was a good day for overtones. The G blooms out almost immediately. The E and the next G become audible around 4 seconds. And right around 5 seconds…
Oh my. (Whispery teacher voice.) Boys and girls, we have a very special guest today. Around the 5-second mark, you begin to hear an overtone that is usually not audible — the sixth overtone, which results from the string vibrating in sevenths. This one’s an oddball to Western ears because it doesn’t correspond to an actual pitch we use. It lands in the crack between A and B-flat on the piano. But there it is, ringing out all the way to the end like it has no shame:
(Now go back and play the C again. Did you hear the B-extra-flat, especially after the others have died out a bit? It can help if you whistle or hum the pitch you’re looking for.)
Don’t be sad and scared if you can’t hear the overtones right away. You will eventually, and once you do, you’ll get quicker at it. It’s like those Magic Eye stereograms from the 90s where you [tippy title=”make your eyes go wonky”]Start with the image an inch from your nose. Gradually pull it back without adjusting your eyes.[/tippy] and boom, you see a space shuttle or the Mona Lisa or whatever:
Your assignment: Get to an actual piano. Thump that low C and hold it down. As the fundamental pitch decays, the overtones will bloom out of the sound, especially the G and E. If you’re lucky, you’ll even spot the exotic B-extra-flat.
Extra credit: Nine minutes of Leonard Bernstein demonstrating the same thing in a slightly different way. These are from the actual Harvard lectures I heard on LPs at 14.
Can you hear the overtones? Tell me in the comments.
(Next post: Pitch.)
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A blog about how music does that
- August 26, 2015
- By Dale McGowan
- In Intro
- 0
On September 1st, I’ll start a new blog here about music — not what it does to your brain, or how it can make your children brilliant or magically shrink your tumors, but the seldom-discussed fact that another person can make you feel intense and specific emotions, even take you through a changing emotional landscape, just by arranging sounds in a certain way.
I know how music does that. It’s the most interesting thing I know. I used to have a job teaching it, and after several years away, I’m ready to do it again.
I’ll start by walking through just enough music theory, then apply it to TV and film scores, classical, rock, and video game music, the strange startup music in my Nissan LEAF, the saddest piece ever written, the universal teasing song, and the interval British people use to say good morning when they don’t really want to talk to you.
While you’re waiting, you can learn more about how the blog will work, subscribe by email in the sidebar, and Like the blog’s Facebook page.
My Big Fat Christian Wedding
An ever-less-frequent series of posts while I’m working on a book about the secular/religious mixed marriage.
Deep into the wedding chapter now, including examples of the three general wedding flavors for secular/religious mixed couples: Sacred, Secular, and Swirl.
Mine was Type I, a traditional religious wedding. No one there would have guessed there was an atheist in the room, much less that he was the one in tux and tails. The setting was a beautiful, historic Lutheran church in San Francisco that we’d chosen not because it was Lutheran but because it was beautiful and historic and in San Francisco, Becca’s hometown.
We upped the religious ante with not one but two ministers–a Methodist friend of the family, and a Southern Baptist uncle of Becca’s whose contribution included a rafter-rattling reference to Matthew 21:21, the assurance that faith can move mountains. The readings were all Christian, ranging from the indispensable “love is patient, love is kind” from First Corinthians to a popular excerpt from The Prophet by the Christian mystic poet Khalil Gibran.
Ten years later, I’d have probably wanted to include some secular poetry or meditations and maybe nudged the scriptures a little — a nice humanistic bit from Ecclesiastes, say, instead of a verse on the telekinetic properties of faith.
But here’s the thing: At that point in my life, even though I was no less secular in my point of view, that played a much smaller part of my identity than it would later on. At 28, I was defined by music. My degrees were in music, and I was about to begin a 15-year career as a conductor and professor of music. I’d have been more offended by lame music than by all the Psalms in the KJV. It could rain little pillows embroidered with Proverbs for all I cared. The music was mine.
Included was Ravel’s Jeux d’eau and the “Menuet” from Sonatine, both played by my insanely talented brother Ron, plus Bach’s secular cantata “Sheep May Safely Graze,” a warhorse gone awesome in an arrangement for strings and two recorders. Becca entered to Bach’s “Air on the G String,” one of the most perfect things ever written, played by the San Francisco Conservatory String Quartet. We lit the unity candle to a prelude I wrote myself, also played by Ron, and we left to the ridiculously exuberant Widor Toccata for organ played by the organist of San Francisco’s Grace (Episcopal) Cathedral.
For those of you keeping score, we had now achieved the Protestant quadfecta: a Lutheran church, an Episcopal organist, and Methodist and Baptist ministers. I was awash in Christian ritual, text, and symbols — and I didn’t care a bit. I’d made my own heaven musically.
Are you in a secular/religious mixed marriage? This is the last week to submit your wedding story for possible inclusion in my book. Do it here!