One Step at a Time: The power of the major scale
In the previous section, I claimed that the interval called the octave
Example 1:
Example 1:
unifies practically everything we commonly define as music. If you don't believe me, listen to this:
Example 2:
Example 2:
I'll bet my house that you are familiar with that musical structure, even if you don't know it by its formal name, the major scale. By "familiar," I mean that you have an intuitive, non-verbal grasp of what a major scale is, even if you've never known what the words mean. It's probably musical territory you've lived in most of your life; if you've ever sung the Christmas carol "Joy to the World," you've sung the descending major scale.
Example 3: Joy To The World
Example 3: Joy To The World
In fact, "My Country 'Tis of Thee," Beethoven's "Ode to Joy," " Frère Jacques," "Row, Row, Row Your Boat," "Happy Birthday to You," the first 30 seconds or so of the last movement of Beethoven's 5th Symphony, "Three Blind Mice," "God Bless America," etc., etc. - I could go on much further - all use the tones of the major scale, and those tones only, rearranged in various ways.
Of course, there's plenty of music that doesn't use the major scale. Many pieces use its close cousin, the minor scale,
Example 4:
Example 4:
and, in fact there many different sorts of scales in the world's music. Moreover, most pieces of music, including the classical repertory and practically any jazz standard, are far more complex than the simple tunes I've listed. But the major scale is more than sufficient to our present needs and so for the rest of this section I'll refer to it simply as the scale.
If the term interval in music is analogous to the term word in language, then the scale is the material that allows us to string intervals together into the musical equivalent of sentences, sentences that can be put together into coherent melodies like the ones I listed above. Here are some of the scale's crucial characteristics:
First, as Example 2 shows, the scale is bounded by the octave. Or, to put it another way, it divides the octave up into a series of smaller intervals.
Second, that series of intervals repeats cyclically, starting over again at each octave boundary.
Example 5:
If the term interval in music is analogous to the term word in language, then the scale is the material that allows us to string intervals together into the musical equivalent of sentences, sentences that can be put together into coherent melodies like the ones I listed above. Here are some of the scale's crucial characteristics:
First, as Example 2 shows, the scale is bounded by the octave. Or, to put it another way, it divides the octave up into a series of smaller intervals.
Second, that series of intervals repeats cyclically, starting over again at each octave boundary.
Example 5:
Third, for most people, the intervals that make up the scale have the cumulative psychological effect of making the scale's octave boundaries feel like a sort of home base, a natural resting point; in fact, all the tunes listed above (except for Beethoven's 5th, which is not really just a tune) end on that octave boundary.
Perhaps you're familiar with the song "Do, re, mi" from The Sound of Music. In that song, kids are taught the scale using a mnemonic code - do re mi fa sol la ti do - that has been around for centuries in one form or another. This method of learning the scale is called solfeggio.
Another method, the one I use, is simply to label the steps of the scale with the numbers 1 though 8, starting with the lowest step. (Simplistic though it sounds, I urge you to return to Example 2 and sing the numbers along with the scale. Then try it with Example 3.) Aha! you might well be thinking. So that's why that interval bounding the scale is called an octave!
Yes, indeed, that is why - and in fact intervals are named by how many scale steps they comprise: from 1 to 2 is called a second, as is the interval from 2 to 3, 3 to 4, etc; from 1 to 3 (or 2 to 4, etc.) is called a third, and so on. As we're about to see, it's more complicated than that, but the basic naming convention for intervals really is that simple.
Perhaps you're familiar with the song "Do, re, mi" from The Sound of Music. In that song, kids are taught the scale using a mnemonic code - do re mi fa sol la ti do - that has been around for centuries in one form or another. This method of learning the scale is called solfeggio.
Another method, the one I use, is simply to label the steps of the scale with the numbers 1 though 8, starting with the lowest step. (Simplistic though it sounds, I urge you to return to Example 2 and sing the numbers along with the scale. Then try it with Example 3.) Aha! you might well be thinking. So that's why that interval bounding the scale is called an octave!
Yes, indeed, that is why - and in fact intervals are named by how many scale steps they comprise: from 1 to 2 is called a second, as is the interval from 2 to 3, 3 to 4, etc; from 1 to 3 (or 2 to 4, etc.) is called a third, and so on. As we're about to see, it's more complicated than that, but the basic naming convention for intervals really is that simple.
Sing the scale a few times and ask yourself this question: are the intervals that make it up all the same? If you have not been prejudiced by previous musical training, I think it's very natural to answer "yes." The steps of the scale seem so familiar and so natural, they all seem to work so nicely together. Besides, didn't I just say that the scale is made up of a sequence of seconds?
Well, it turns out that not all seconds are equal and, in fact, the remarkable ability that the scale has to establish musical order depends on a built-in asymmetry, an asymmetry that is just as fundamental to the way music behaves as the octave itself - and just as mysterious.
Well, it turns out that not all seconds are equal and, in fact, the remarkable ability that the scale has to establish musical order depends on a built-in asymmetry, an asymmetry that is just as fundamental to the way music behaves as the octave itself - and just as mysterious.
At this point, a visual aid would be very useful. Luckily, we have just the thing we need: the piano keyboard.
If you've never had any formal musical training, if you never took piano lessons, you probably have only a vague idea about what a piano keyboard looks like. I imagine you know there are small black keys and larger white ones, you might know that once the white keys, plastic nowadays, were made from ivory (as in "tickling the ivories"), and you might even know that there are 88 piano keys in all. But you probably never had any occasion to consider why the keys of a piano are arranged the way they are. So, let's take look:
Example 6: The standard 88-key piano keyboard:
If you've never had any formal musical training, if you never took piano lessons, you probably have only a vague idea about what a piano keyboard looks like. I imagine you know there are small black keys and larger white ones, you might know that once the white keys, plastic nowadays, were made from ivory (as in "tickling the ivories"), and you might even know that there are 88 piano keys in all. But you probably never had any occasion to consider why the keys of a piano are arranged the way they are. So, let's take look:
Example 6: The standard 88-key piano keyboard:
There are 88 keys, producing constantly ascending pitches from left to right, starting here
Example 7:
Example 7:
and ending all the way up here.
Example 7a:
Example 7a:
You'll notice that there is a repeating pattern of white and black keys. No matter what key you start on, the pattern repeats in groups of 12, leaving 4 orphan keys to finish the 88. However, if you had ever taken piano lessons for as little as a day or two, you would, for reasons that will become obvious very shortly, choose this pattern as your reference.
Example 8:
Example 8:
It will not amaze you, I imagine, to learn that the 12-key pattern repeats every octave. That is, each key in the pattern is an octave away from the corresponding key in the adjacent pattern(s). Knowing that, you might naturally ask:
To answer the first question, click on the link below. Doing so will open a new window containing a piano keyboard that you can play with mouse clicks. You might want to bookmark that window for use again in subsequent sections.
Click here for the virtual keyboard.
(I picked this one from among the several free online keyboards I could find because it's relatively uncluttered, easy to use, and produces a decent sound. If you know of better ones, please let me know.)
Putting aside, just for the moment, the question of why the keys are labeled the way they are, click your mouse on any one of the keys labeled C - C3 for example - and proceed from left to right, clicking the white keys only. Voila! The major scale!
This is one of the first things any beginning piano student learns, and it is a powerful lesson, one that tends to stay with folks even if they only took lessons for a week: If, and only if, you start on one of the the keys labeled C, you can play the major scale using white keys only, without having to touch any of those obviously exotic and possibly dangerous black ones.
What about the labeling of the white keys with the letters A though G?
For most of us (including me), the labeling of the keys with the letters A through G is just an arbitrary reference system. Most people, even trained musicians, cannot hear a note struck on the piano and identify its letter. But most people do hear octave equivalence and, reflecting this, keys an octave apart all have the same letter. However, about one in every thousand people can perceive pitches in something like the way most of us perceive color; strike a G on the piano, and these one-in-a-thousand folks will say "Ah, that's a G!" - and they'll say that no matter which octave version of G is played.
It seems almost certain - although there's no way of proving it - that most if not all the great composers had or have this ability, which is called perfect pitch or absolute pitch. The terms are misleading: like all phenomena involving the musical capacities of the brain, this one is full of uncertainty; even within the population with this talent, there are big variations of "perfectness." (For example, my brother has perfect pitch, especially with regard to the piano, but has always been less sure of it where voices are concerned.) Nobody yet understands why some folks have the this ability. It does not seem to be a skill one can acquire through practice. (The Wikipedia article on absolute pitch is, for once, informative and thorough without being mind-numbingly verbose.)
- What does this 12-key pattern have to do with the 8-tone scale we were just discussing?
- Why are the black and white keys grouped the way they are? Or to put it another way, why are there missing black keys?
- What's the distinction between black and white keys anyway?
To answer the first question, click on the link below. Doing so will open a new window containing a piano keyboard that you can play with mouse clicks. You might want to bookmark that window for use again in subsequent sections.
Click here for the virtual keyboard.
(I picked this one from among the several free online keyboards I could find because it's relatively uncluttered, easy to use, and produces a decent sound. If you know of better ones, please let me know.)
Putting aside, just for the moment, the question of why the keys are labeled the way they are, click your mouse on any one of the keys labeled C - C3 for example - and proceed from left to right, clicking the white keys only. Voila! The major scale!
This is one of the first things any beginning piano student learns, and it is a powerful lesson, one that tends to stay with folks even if they only took lessons for a week: If, and only if, you start on one of the the keys labeled C, you can play the major scale using white keys only, without having to touch any of those obviously exotic and possibly dangerous black ones.
What about the labeling of the white keys with the letters A though G?
For most of us (including me), the labeling of the keys with the letters A through G is just an arbitrary reference system. Most people, even trained musicians, cannot hear a note struck on the piano and identify its letter. But most people do hear octave equivalence and, reflecting this, keys an octave apart all have the same letter. However, about one in every thousand people can perceive pitches in something like the way most of us perceive color; strike a G on the piano, and these one-in-a-thousand folks will say "Ah, that's a G!" - and they'll say that no matter which octave version of G is played.
It seems almost certain - although there's no way of proving it - that most if not all the great composers had or have this ability, which is called perfect pitch or absolute pitch. The terms are misleading: like all phenomena involving the musical capacities of the brain, this one is full of uncertainty; even within the population with this talent, there are big variations of "perfectness." (For example, my brother has perfect pitch, especially with regard to the piano, but has always been less sure of it where voices are concerned.) Nobody yet understands why some folks have the this ability. It does not seem to be a skill one can acquire through practice. (The Wikipedia article on absolute pitch is, for once, informative and thorough without being mind-numbingly verbose.)
What about those black keys? On this keyboard, they are not named. In fact, the black keys don't have names of their own; they are named with reference to their adjacent white keys. For example, take the black key between C and D. It can be called either C-sharp (written C#) or D-flat (Db); the choice of which is appropriate involves issues of music theory that we need not worry about.
This naming convention reflects that fact that for reasons having to do with the long and complicated history of musical tuning systems, the black keys once were indeed exotic and problematic. Today they're equal members of the musical community.
To demonstrate: For the moment, let's call the distance between piano keys separated by a black key whole steps, and the distance between the keys without intervening black keys half steps. Look at Example 8 and notice the pattern of whole and half steps in the major scale:
Step name :1 2 3 4 5 6 7 8
Step size: whole whole half whole whole whole half
If you start on any key, black or white, and follow this pattern of whole and half steps, you'll create a major scale. I highly encourage you to try out a few cases on the virtual keyboard.
There are 12 keys within the octave, and so 12 possible major scales, each with a unique combination of white and black keys, each as valid as any other. If you were to study piano formally, much of your time early on would be spent learning those patterns until they were second nature to your mind and to your fingers.
Over the past century or two, after many previous centuries of dispute and experimentation, the named keys have become associated with fixed pitch frequencies. For example, A3 on our virtual keyboard - and on all correctly tuned piano keyboards the world over - has a frequency of 440 Hz. That's the pitch the oboe plays as reference when an orchestra tunes up. You can download a free iPhone app that does nothing but play A 440. And, as it turns out, once we've agreed on A 440, the pitches of all the keys on the keyboard are determined as well.
Why is A named A? Wouldn't it make more sense for C to be called A? Beats me. Why are traffic lights red and green rather than blue and yellow? Why do we drive on the right while the British drive on the left? Why do mathematicians use x and y axes instead of, say, a and b?
Why is A named A? Wouldn't it make more sense for C to be called A? Beats me. Why are traffic lights red and green rather than blue and yellow? Why do we drive on the right while the British drive on the left? Why do mathematicians use x and y axes instead of, say, a and b?
To tell the truth, if you're not a practicing musician or planning to become one, I can't see how knowing about the names of keys will contribute to your listening skills or your musical pleasure. But you now know what it means to be "in" a key; if you start on C and sing "My Country 'Tis of Thee," you are using the C major scale and are said to be singing the tune in C major. Mozart heard his 41st Symphony in C, so its official name is Mozart Symphony No. 41 in C Major.
But, to get back to the scale, the relationship among the notes of the scale rather than their absolute properties is crucial to understanding why music affects many of us the way it does.
Go back to the our piano keyboard and this time, starting anywhere you like and going in either direction for as long as you like - but for more than an octave - click as regularly as you can on adjacent keys, both black and white. Or listen to example 7 again.
All the keys are the same musical distance apart; that is, adjacent keys on the keyboard always form the same interval. (It's natural to imagine that equal musical intervals are separated by a constant frequency difference, like marks on a ruler, but in fact the brain measures equality by the ratio of frequencies, not arithmetical difference. Of course, your brain supplies you with absolutely no conscious knowledge of this fact, any more than it tells you what frequencies light waves are.) When adjacent piano keys are played simultaneously, they always produce the same pungent sound.
Example 9:
But, to get back to the scale, the relationship among the notes of the scale rather than their absolute properties is crucial to understanding why music affects many of us the way it does.
Go back to the our piano keyboard and this time, starting anywhere you like and going in either direction for as long as you like - but for more than an octave - click as regularly as you can on adjacent keys, both black and white. Or listen to example 7 again.
All the keys are the same musical distance apart; that is, adjacent keys on the keyboard always form the same interval. (It's natural to imagine that equal musical intervals are separated by a constant frequency difference, like marks on a ruler, but in fact the brain measures equality by the ratio of frequencies, not arithmetical difference. Of course, your brain supplies you with absolutely no conscious knowledge of this fact, any more than it tells you what frequencies light waves are.) When adjacent piano keys are played simultaneously, they always produce the same pungent sound.
Example 9:
In fact, that's the smallest interval there is in standard music theory; it is called the minor second. In standard music notation, there is no way to write down two notes closer together in pitch. That doesn't mean two pitches couldn't be closer together; there as many possible pitches between these two as there are points on a line - that is, an infinite number. But intervals smaller than a minor second are generally not recognized in our culture's musical system.
Playing a series of ascending or descending minor seconds, as in Example 7, obliterates the musical order created by the major scale. We're left with an undifferentiated series of notes in which none has the quality of being a home base.
The interval formed by every other piano key - that is, minor second plus minor second - is called a major second. Here's what a series of major seconds sounds like:
Example 10:
Example 10:
That certainly sounds different, but major seconds alone also create a structureless series of undifferentiated notes.
Only by mixing major and minor seconds (or, as they're also called, whole steps and half steps) in a particular sequence can we create the major scale. If you go back to the virtual keyboard and play the scale starting on C again, you'll see that steps 3 and 4 as well as steps 7 and 8 are separated by half steps; all other steps are a whole step apart. So that explains the missing black keys: without that gap, it would not be possible to play a major scale using white keys only.
Only by mixing major and minor seconds (or, as they're also called, whole steps and half steps) in a particular sequence can we create the major scale. If you go back to the virtual keyboard and play the scale starting on C again, you'll see that steps 3 and 4 as well as steps 7 and 8 are separated by half steps; all other steps are a whole step apart. So that explains the missing black keys: without that gap, it would not be possible to play a major scale using white keys only.
Even though I know intellectually that the major scale consists of a series of whole steps and half steps, and even though I am aware of the musical riches that can be unlocked because of those strategically placed half-steps, I really don't experience the scale in terms of a series of major and minor seconds.
It's kind of like this: when I see a face that I recognize, I don't say to myself: "Hmm, there are two eyes, a nose, and a mouth; it's 3/4 of an inch from the nose to the upper lip, the eyes are blue, so that must be ..." Instead, I immediately recognize whoever it is as all at once. It's the same with the scale: its power to organize things, to create musical logic, is what registers, not the details of its construction.
To borrow an image from physics: it's as if that combination of whole and half steps creates an invisible magnetic field, a field that makes the notes within it line up in orderly patterns like iron filings in the presence of a bar magnet. And, to continue the metaphor, that musical magnetic field can sometimes strongly repel notes that don't belong to the scale.
For example, lets return to "Joy to the World" and try to introduce a note outside the scale we're using.
Example 11:
It's kind of like this: when I see a face that I recognize, I don't say to myself: "Hmm, there are two eyes, a nose, and a mouth; it's 3/4 of an inch from the nose to the upper lip, the eyes are blue, so that must be ..." Instead, I immediately recognize whoever it is as all at once. It's the same with the scale: its power to organize things, to create musical logic, is what registers, not the details of its construction.
To borrow an image from physics: it's as if that combination of whole and half steps creates an invisible magnetic field, a field that makes the notes within it line up in orderly patterns like iron filings in the presence of a bar magnet. And, to continue the metaphor, that musical magnetic field can sometimes strongly repel notes that don't belong to the scale.
For example, lets return to "Joy to the World" and try to introduce a note outside the scale we're using.
Example 11:
That last note is clearly a "wrong note." It happens to be a half-step above where you expected it, and therefore outside the scale.
Does this mean that when you're in a particular key - that is, when you're using the notes of a particular major scale - that the five other notes (the black keys in C major) are for ever unavailable to you? The answer is no: a complex piece like a Mozart Symphony or a Chopin Nocturne will typically use up all 12 notes fairly quickly. We're a long way off from seeing how that's done, but without that capability, no music would ever get more complicated than "Rudolph the Red-Nosed Reindeer."
Does this mean that when you're in a particular key - that is, when you're using the notes of a particular major scale - that the five other notes (the black keys in C major) are for ever unavailable to you? The answer is no: a complex piece like a Mozart Symphony or a Chopin Nocturne will typically use up all 12 notes fairly quickly. We're a long way off from seeing how that's done, but without that capability, no music would ever get more complicated than "Rudolph the Red-Nosed Reindeer."
What does music sound like without the organizing power of scales? What would happen if a composer were to write music in which all notes are equal, in which there is no natural resting point, as in Example 7? The result would sound like this:
Ex 12:
Ex 12:
In this remarkable piece, Anton Webern's 3rd Bagatelle for String Quartet, it was Webern's explicit intention to do away completely with the organizing force of the major or minor scale. It so happens that I love this and its five companion pieces - my wife Gretta and I heard them performed in Philadelphia just the other night at a concert that made them sound positively old-fashioned - but I fully sympathize with anyone who doesn't.
In fact, one can make a convincing argument that music like this, music that abandons the rules implied by the major scale, isn't really music anymore. Surely, that's what Mozart or Bach would have thought. The impact of pieces like this was so strong that such music is still called "modern" - often with a strong pejorative tone - even though Webern's Bagatelles were composed in 1911! (In retrospect, this sort of music didn't appear out of nowhere; by the first decade of the 20th century, traditional scalar music had been pushed to its limits by composers like Wagner, Brahms, Debussy, Mahler, Mussorgsky and others. Nevertheless, one of the great puzzles of recent cultural history is why music that repelled so many people gained such a stranglehold on musical life for most of the 20th century.)
In fact, one can make a convincing argument that music like this, music that abandons the rules implied by the major scale, isn't really music anymore. Surely, that's what Mozart or Bach would have thought. The impact of pieces like this was so strong that such music is still called "modern" - often with a strong pejorative tone - even though Webern's Bagatelles were composed in 1911! (In retrospect, this sort of music didn't appear out of nowhere; by the first decade of the 20th century, traditional scalar music had been pushed to its limits by composers like Wagner, Brahms, Debussy, Mahler, Mussorgsky and others. Nevertheless, one of the great puzzles of recent cultural history is why music that repelled so many people gained such a stranglehold on musical life for most of the 20th century.)
Why does the scale organize things the way it does? As you may be figuring out by now, like so many questions involving the brain's relationship to sound, nobody knows. Nobody is remotely close to knowing. But, understood or not, the power of the major scale is undeniable, as we shall see in many different ways in future installments.