CHAPTER 4:
  How Scales and Intervals REALLY
  Work
  _______________________________
  
  4.1 Scales: Brain-averse, Brain-
  friendly

 
There is geometry in the ringing of strings.

                         —MONTY PYTHAGOR

PAGE INDEX
  

4.1.1 What’s a Scale?

4.1.2 Chalk Marks on a Cello Fingerboard

4.1.3 Brain-averse: Why Random Scales Suck

4.1.4 In Search of an Organizing Principle that Will Yield a Brain-friendly Scale

4.1.5 Using Simple Frequency Ratios to Build a Brain-friendly Scale

4.1.6 That Familiar “Do-Re-Mi” Scale

4.1.7 More Brain-averse: Equal-interval Scales

4.1.8 Brain-friendly: A Naturally-selected Specialization for Simple Frequency Ratios

4.1.9 Filling In the Last Gaps: The Chromatic Scale

4.1.10 E Unum Pluribus ... Many Scales Out of One

~ • ~ • ~ • ~

4.1.1

WHAT’S A SCALE?

Whether speaking or singing, humans automatically and effortlessly use discrete pitches, with only the occasional slide. By contrast, our primate cousins, such as gibbons and chimpanzees, either vocalize in pitch glides or without distinct pitches—just grunts and pants.

     Discrete pitches in speech and music serve to organize sound in such a way that the brain can recognize patterns and make sense of them. Once you have more than one discrete tone, you can have a scale of some sort.

     Humans undoubtedly turned discrete tones into songs long before anybody recognized the existence of musical scales. At some point, it must have become clear that the tunes people remembered tended to use the same sets of notes: scales.

     A tune or melody is a coherent or distinctive succession of tone pairs called intervals. The notes of a tune (melody) move up and down in pitch, stepping or skipping from note to note, using the same notes time after time, like stepping up and down the same staircase.

     That means the notes themselves must come from some set of related notes of different pitches. This set of notes is called a scale. But how does the brain recognize a set of pitches as a scale?

4.1.2

CHALK MARKS ON A CELLO FINGERBOARD

Imagine you have a cello. (Maybe you do have a cello.) As you know, the fingerboard of a cello has no frets. Which makes the cello an ideal instrument for this thought experiment.

     •   Take a piece of imaginary white chalk and make some horizontal marks at random places along the fingerboard of your imaginary cello. Say, oh, maybe eight chalk marks.

     •   Remove the excess imaginary chalk from your fingers by wiping your hands on your black pants or dark skirt. Nobody will be able to see the chalk marks on your clothing because, even though your clothing is real, the chalk is imaginary.

     •   Now, pick a string, any string. Press your finger on the string over the chalk mark nearest the narrow end of the fingerboard (the end nearest the tuning pegs). Pluck (or bow) that string. Then move to the next chalk mark. Pluck the string. Then the next chalk mark, and so on, until you’ve played all eight notes.

     Technically, that’s a scale.

     Good thing that was a thought experiment. Because the scale you created and played on your imaginary cello sucks. Your brain just does not recognize it as a meaningful scale. How come?

4.1.3

BRAIN-AVERSE: WHY RANDOM SCALES  SOUND BAD

If you create a random scale, a scale comprised of notes having no natural, physical relationship with each other (the way you did using random chalk marks on the cello fingerboard), then try to play a tune using that scale, your brain interprets the sound as chaos, not music.

     Studies of both children and adults indicate your brain is hardwired at birth to reject random scales. Infants prefer non-random scales, as do adults. The frequencies of the notes comprising a scale have to have some kind of internal order—ordered relationships with each other—or your brain interprets the sound as noise.

     But not just any ordered relationships.

     Particular ordered relationships that your brain recognizes: “brain-friendly” ordered relationships of tones, as opposed to “brain-averse” chaotic non-relationships.

4.1.4

IN SEARCH OF AN ORGANIZING PRINCIPLE THAT WILL YIELD A BRAIN-FRIENDLY SCALE

Recall what happens when you pluck a guitar string that you’ve cut in halves, thirds, quarters, fifths, and so on, by damping the string over various frets. You get a whole series of soft overtones—overtones that sound different from the fundamental.

     As the guitar-string-damping experiment reveals, each overtone not only sounds different, it also sounds good. Brain-friendly. So it would be a reasonable guess that a brain-friendly scale might have something to do with the relationships of overtones to each other.

     Hmmm. Maybe relationships among overtones hold the secret that will yield a useful scale, a group of tones in a brain-friendly ordered relationship.

     Time to bring back the overtone series and have a look at overtone frequency relationships (Table 8 below). Frequency relationships among the first few overtones, the strongest ones, are of greatest interest. They’re the ones you can hear by damping a guitar string at various fret positions.

TABLE 8  Fundamental and First 15 Overtones of the “Middle C” Overtone Series

     •   Start with the ratio of the first overtone to the fundamental frequency, which is 523.2 Hz : 261.6 Hz, which boils down to a simple ratio of 2:1. This simple ratio comes from the first two numbers of the middle column.

     •   Next, the ratio of the second overtone to the first overtone. It’s 784.8 Hz : 523.2 Hz, a ratio of 3:2. (middle column, second and third numbers).

     •   Keep doing this for the first few overtones, and you end up with a list of simple ratios of frequencies, like this (Table 9):

TABLE 9  Simple Ratios of Frequencies

2:1

3:2

4:3

5:4

6:5

etc.

       Next step: try out simple ratios of frequencies as an organizing principle to build a scale.

4.1.5

USING SIMPLE FREQUENCY RATIOS TO BUILD A BRAIN-FRIENDLY SCALE

Any organizing principle worth its salt should work universally. That is, you should be able to pick any old frequency as a starting point for scale building.

     •   Start building the scale with the tone Middle C, the first tone in Table 8 above, with a frequency of 261.6 Hz.

     •   Next, in accordance with the organizing principle of simple frequency ratios, add a second note, derived from the simplest possible ratio, 2:1. What you get is a two-note “scale.”

     •   This scale clearly has its limitations. But you have to start somewhere (Figure 5).

FIGURE 5  Scale of “Middle C” and “C Above

Middle C”

     •   Next, add a tone derived from the next simplest ratio of frequencies, 3:2. (The simplest possible frequency ratio that can identify a relationship between two tones is 2:1.) For reasons that will become clear in a little while, you can label the 3:2 tone G.

     •   Notice that when you add G to the scale, the relationship between G and the C above Middle C also happens to be a simple ratio of frequencies, 4:3.

     •   Now you’ve got a scale of three notes. It sounds good, too. The organizing principle looks promising (Figure 6).

FIGURE 6  C - G - C Scale

     •   Notice the big gap between Middle C and G. Fortunately, a tone derived from the simple frequency ratio 5:4 fits beautifully, right between Middle C and G. Call it E.

     •   When you add E to the scale, the relationship between E and G turns out to be a simple ratio of frequencies, namely, 6:5. Amazing.

     •   The scale grows to four notes (Figure 7 below). Sounds great, too. These four notes correspond to the words “say, can you see,” in the American national anthem, music composed by John Stafford Smith, a London, England, church organist.

FIGURE 7  C - E - G - C Scale

     •   Next, have a look at the big gap between G and C above Middle C. It so happens that yet another tone derived from a simple frequency ratio, 5:3, fits right in there. This tone happens to be the lovely and talented Concert A (also commonly called A-440). More about lovely, talented Concert A later on.

     •   The scale grows to five notes (Figure 8):

FIGURE 8  C - E - G - A - C Scale

     •   You can fill in another big gap, the one between E and G, using another note derived from the simple frequency ratio, 4:3. The note F relates to Middle C by the this simple ratio.

     •   When you insert F into the scale, it relates to Concert A by the simple ratio of frequencies, 5:4.

     •   Now the scale has grown to six notes. So far, so good (Figure 9).

FIGURE 9  C - E - F - G - A - C Scale

     •   Only a couple of big gaps remain, one between Middle C and E, and another between Concert A and C above Middle C. The simplest frequency ratio available to fit between C and E is 9:8, which yields the note D.

     •   You can use the same 9:8 frequency ratio to stick a tone between Concert A and C above Middle C. Call it B.

     •   When you insert these two notes (D and B), you notice a few things:

                -    The scale has no more big gaps between notes;

                -    The smallest gap between notes has a ratio of frequencies of 16:15—not exactly simple;

                -    The order of the letter-names of the notes makes some sense, though not plain, common horse sense. The alphabet starts at C, stops at G, then starts again at A.

     •   Now you’ve got an 8-note scale. Which includes the 8th note. Which is the same as the first note, but higher in pitch (Figure 10):

FIGURE 10  C - D - E - F - G - A - B - C Scale

     This scale definitely sounds brain-friendly. Looks like the organizing principle of tones derived from simple frequency ratios has worked. (Whew!)

4.1.6

THAT FAMILIAR “DO-RE-MI” SCALE

If you’re European, you’ll recognize the scale in Figure 10 as the “do-re-mi” scale, using the solmization system, which designates notes using syllables instead of letter names:

do re mi fa so la ti do

     Or, if you’re going down the scale:

do ti la so fa mi re do

     Or, as the von Trapp family sang in The Sound of Music:

Doe, a deer, a female deer

Ray, a drop of golden sun

Me, a name I call myself

Fah, a long long way to run . . .

     (They could sing better than they could spell.)

     You get the same scale when you play the white notes on the piano starting at C. Any old C. You don’t have to start with Middle C (Figure 11):

FIGURE 11  The “Do-Re-Mi” Scale in All Its Glory

     Since the scale has eight notes (including the first and last notes), the pitch gap between the first note and the eighth note is called an octave.

     In music, the pitch gap between any two notes is called an interval. Think of an interval as a relationship between two pitches. You can play the two pitches successively—usually the lower one first—or simultaneously.

     So, that makes the pitch relationship between the first note and the eighth note an interval of an octave.

     That last note of the scale sounds exactly like the first note, and yet ... well ... “higher” in pitch. The same, but somehow different. The terminology, familiar to everybody who plays music, goes like this: the last note of the scale is an “octave higher” than the first note.

     As you can see in Figure 11 above, the eight notes of the do-re-mi scale are not evenly spaced. Still, when you play this scale, it sounds agreeable whether you play it from bottom to top or top to bottom. It sounds as though the notes are proceeding smoothly up and down the pitch “staircase.” As though all the notes are the same distance apart. Even though they obviously are not.

     How come? What if all the pitches were actually the same distance apart?

4.1.7

MORE BRAIN-AVERSE: EQUAL-INTERVAL SCALES

So far, you’ve tried two different organizing principles to construct an agreeable-sounding scale:

     •   A scale of random notes—the experiment with the chalk marks on the cello fingerboard. Result? A brain-averse scale. Chaotic and completely “unmusical.”

     •   A scale of notes related to each other by simple ratios of frequencies. Result? A brain-friendly scale. Clearly “musical,” beautiful- sounding. A scale consisting of a distinctive but uneven order of tones.

     Now, just for good measure, try a third organizing principle: a completely regular, evenly-spaced order of tones.

     Start at Middle C and divide the octave into seven equal intervals, for a total of eight notes (Figure 12 below). The lowest note is Middle C and the highest note is C above Middle C. All eight notes are spaced the same distance apart, frequency-wise (37.4 Hz between each note).

     No point in naming notes 2 through 7 because this scale is only theoretical.

     And a good thing, too. Because, like the random scale of chalk-and-cello fame, this scale also sucks (Figure 12):

FIGURE 12  The “Eight-Note, Seven-Equal-Interval” Scale

     Table 10 below shows the frequencies for the eight notes of this scale, compared with the “do-re-mi” scale frequencies. As you can see, they’re all different, by roughly five to 24 Hz, except for the first and last “C” notes.

TABLE 10  “Eight-Note, Seven-Equal-Interval” vs “Do-Re-Mi” Scale Note Frequencies

4.1.8

BRAIN-FRIENDLY: A NATURALLY-SELECTED SPECIALIZATION FOR SIMPLE FREQUENCY RATIOS

     Substantial research findings show that, if you try to create music using scales that have no tones in relationships of simple frequency ratios, your brain stops recognizing “musical” sound and hears chaos. Like the static you get when you move your analog radio dial between stations.

     Infants respond to changes in pairs of tones only if the tones are related by small-integer, simple frequency ratios—the tones that emerge from the harmonic series. Tones not related by simple frequency ratios simply do not elicit responses from babies. This strongly indicates that the human brain has a naturally-selected specialization for simple frequency ratios—that these preferences are not cultural constructs.

     Not only that, infants remember scale tones when the intervals of the scale are of unequal size, compared with scales having intervals of equal size. This is consistent with the unequal-interval scales that emerge from the harmonic series. As you’ll see in Chapter 5, most scales used commonly worldwide have only 5 to 7 different tones (i.e., not including the second octave note), which are unequally spaced.

     Infants also have difficulty resolving tones that are close together. Tones spaced close together are not related by simple frequency ratios.

     To summarize, your brain can make sense of, and prefers, scales of non-random but unequally-spaced tones—pitches related to each other in simple multiples or simple fractions of a fundamental frequency.

4.1.9

FILLING IN THE LAST GAPS: THE CHROMATIC SCALE

Look at all those (comparatively) wide intervals between some of the notes (Figure 13 below). Between C and D. Between D and E. Between F and G. Between G and A. Between A and B. Five intervals.

FIGURE 13  The “Do-Re-Mi” Scale

     Those five intervals look suspiciously like they’re exactly twice as wide as the two smaller intervals, the ones between E and F, and B and C. If the five bigger intervals are exactly twice as wide as the two smaller ones, and if you were to insert a tone into each of those wide gaps, you’d have:

     •   A 12-equal-interval scale (a total of 13 tones, including the first and last ones, which are the same note, an octave apart);

     •   A scale composed of close-together tones.

     Precisely the recipe for non-musicality. So, would such a scale actually sound chaotic?

     The answer is yes, it would sound chaotic. Not at all musical.

     However, that does not mean such a scale would have no musical value. As you’ll see shortly, the 12-equal-interval scale serves a valuable purpose as a pool of tones you can dip into and use in the construction of many different, truly musical scales. You can also use the same 12-equal-interval scale as a pool you can dip into for colourful extra notes when writing a song.

     (While most equal-interval scales are inherently chaotic and unmusical, a few are actually palatable. Chapter 5 discusses an example of a musical-sounding equal-interval scale—an exception to the rule.)

     For now, go ahead and fill in the five wide gaps in the above scale (Figure 13) with five new notes.

     But before you do that, you’ll need names for the new notes. Problem is, there’s no letter of the alphabet between C and D, or D and E, or F and G, etc. What to do?

     •   Suppose you start at C, you’re going up to D, and you want to stick a note in between. Since you’re going “up” in pitch, call the in-between note a sharp note (symbolized ♯).

     •   If you’re going “down” in pitch, from D down to C, call the same in-between note a flat note (symbolized ♭).

     (This nomenclature will become a lot clearer shortly.)

     So . . . here’s what you get when you fill in the last five gaps of the “do-re-mi” scale (Figure 14):

FIGURE 14  The “Do-Re-Mi” Scale With the Gaps Filled In

     On the piano, if you start with the note Middle C (or any other C), you’ll notice that the in-between notes correspond to the black keys.

     •   The smallest interval in the above scale, the interval between any two adjacent notes, is called a semitone or half-step (for example, between C and C♯, or between E and F). So, an interval of an octave is comprised of 12 semitone intervals.

     •   The next smallest interval, the distance covered by two semitones, is called a tone, or a whole tone, or a step.

     And the name of the above 13-note (12-semitone) scale is the chromatic scale. The five new notes added to the do-re-mi scale are called chromatic notes.

4.1.10

E UNUM PLURIBUS ... MANY SCALES OUT OF ONE

To play the chromatic scale, just start at any C, then play every note ... C, C♯, D, etc., all the way up to the next C. When you do this, you play 13 notes, but only 12 intervals of one semitone each. (Remember, an interval is not a note. It’s the pitch distance between two notes.)

     So, the chromatic scale does sound chaotic—not naturally musical. However, you can grab notes from the chromatic scale to craft numerous naturally musical scales. These agreeable-sounding scales contain only eight or fewer notes, selected from the chromatic scale. Chapter 5 discusses some of them.

     For now, though, a bit more about the “do-re-mi” scale.

     Its common name is the major scale. It consists of eight notes, spaced by seven intervals of tones and semitones in this order:

tone, tone, semitone, tone, tone, tone, semitone

     This type of scale is called a diatonic scale. “Dia” comes from the Greek word for “through” or “by.” And “tonic” refers to the tonal anchor of the scale—the first note of the scale—called the tonic note. So a “diatonic” scale’s notes are related to each other “through” the first, or “tonic” note of the scale.

     More on this in Chapter 5, which discusses tonal music in detail.

~ • ~ • ~ • ~

~ • ~ • ~ • ~

 TABLE OF
 CONTENTS

 PART I

 The Big Picture
   Introduction

   1. W-5 of Music
   2. Pop Music
       Industry
  
 PART II
 Essential
 Building Blocks
 of Music
   3. Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6. Chords/
        Progressions

   7. Pulse/Meter/
        Tempo/Rhythm
   8. Phrase/Form
   9. Melody
 10. Lyrics
 11. Repertoire/
        Performance

 PART IV
 Making a
 Living In Music
 12. Business of
        Music
 
 Appendixes

 Notes

 References

 Index
  

   Top

 TABLE OF
 CONTENTS

 PART I

 The Big Picture
   Introduction

   1. W-5 of Music
   2. Pop Music
       Industry
  
 PART II
 Essential
 Building Blocks
 of Music
   3. Tones/Overtones
   4. Scales/Intervals
   5. Keys/Modes
 
 PART III
 How to Create
 Emotionally
 Powerful Music
 and Lyrics
   6.