Fiddly Details
A. Naming and identifying intervals
At various points in the book I have mentioned that the jump in pitch between any two notes is called an interval. The interval we have discussed most is the octave—the jump in pitch that corresponds to a doubling of the frequency of vibration of the note. All the other intervals also have names, some of which I have referred to already. The table opposite presents a list of interval names along with the size of the interval in semitones. You can also see three photos of a pianist playing two notes a fourth apart. In each case we start on any lower note and count up to the note which is five semitone steps higher (counting the black notes as well as the white ones). I have included three illustrations here in order to make it clear that it doesn’t matter which note you start on—the one five semitone steps up will always be a fourth higher, and similarly the one nine semitones up will always be a major sixth higher.
After years of musical training you get to recognize each of these intervals and you can write down any tune which comes into your head (start on any note, up a fifth, down a major third, etc.). But there is a way of identifying musical intervals which anyone can manage. All you have to do is learn the names of the intervals which occur at the beginning of several songs. I have made a list of suitable songs in the table below. The first two notes of the songs identified in this list give you the interval in its rising form (unless I have stated otherwise). Most tunes start with a rising interval, so examples are plentiful and you might want to substitute other songs for your own list. You might also want to collect twelve descending intervals from other songs.
So now, when you’re bored, sitting at an airport, you can identify the interval of whichever annoying “bing bong” noise they are using by matching it to one of these songs.
Three photos of a pianist playing two notes a fourth apart. It doesn’t matter which note you start on—the two notes are always separated by a gap of five semitones.
B. Using the decibel system
1. The decibel system is a method of comparing the difference in volume between two sounds. These differences are measured in the following way:
• If the difference between two sounds is 10 decibels then one sound is twice as loud as the other.
• If the difference is 20 decibels then one sound is 4 times as loud as the other.
• If the difference is 30 decibels then one sound is 8 times as loud as the other.
• If the difference is 40 decibels then one sound is 16 times as loud as the other.
• If the difference is 50 decibels then one sound is 32 times as loud as the other.
• If the difference is 60 decibels then one sound is 64 times as loud as the other.
• If the difference is 70 decibels then one sound is 128 times as loud as the other.
• If the difference is 80 decibels then one sound is 256 times as loud as the other.
• If the difference is 90 decibels then one sound is 512 times as loud as the other.
• If the difference is 100 decibels then one sound is 1,024 times as loud as the other.
• If the difference is 110 decibels then one sound is 2,048 times as loud as the other.
• If the difference is 120 decibels then one sound is 4,096 times as loud as the other.
2. When using rule 1 (above) it doesn’t matter which decibel number you start on. For example, the difference between 10 decibels and 20 decibels is 10 decibels so 20 decibels is twice as loud as 10 decibels. But the difference between 83 decibels and 93 decibels is also 10 decibels—so 93 decibels is twice as loud as 83 decibels. Similarly, 72 decibels is 16 times as loud as 32 decibels (because the difference between 72 and 32 is 40 decibels).
3. Although the decibel system should only be used to compare the relative loudness of two noises (as in rules 1 and 2 above), many people use decibels as a definite measure of the loudness of a single noise. In this case they are not saying “a large bee creates a noise of 20 decibels”; what they are actually saying is “a large bee creates a noise which is 20 decibels louder than the quietest noise we can hear.” The quietest noise we can hear is called the “threshold of hearing,” so we are really saying “a large bee creates a noise which is 20 decibels louder than the threshold of hearing.” When people appear to be using decibels in a non-comparative way (e.g., “The loudness of that motorbike is 90dB”) it’s just because they haven’t bothered to include the phrase “louder than the threshold of hearing”—it’s taken for granted.
C. Tuning an instrument to a pentatonic scale
If you are doing this as an experiment it would be convenient to use a guitar, as it’s the commonest six-string instrument. On a guitar, the strings are traditionally numbered 1 to 6 with 6 being the thickest one and the lowest note. Unfortunately, this is the opposite of the numbering system I used in chapter 8 . I tried reversing the numbers in the chapter to match the guitar system, but it spoiled the clarity of the chapter. So I’m going to give you instructions twice—once following the numbering system in chapter 8 , and once using the traditional guitar numbering for the strings. You can use either set of instructions—they both give exactly the same result.
If you tune a guitar to a pentatonic scale, you will change the difference between the thickest string and the thinnest from two octaves to one octave. If you make the thickest string give its usual note before you start, this will mean that the thinner strings will be very slack by the time you have finished. This doesn’t matter much if you are doing this out of curiosity and you are going to return the guitar to its normal tuning in a few minutes. If, on the other hand, you are doing this as a long-term plan or as a demonstration for students, I’d recommend that you tighten the thickest string to give a higher note before you start and/or change the thinner strings for thicker ones.
Re-tuning like this takes a while if you start with a normally tuned guitar, because the strings resent being de-tuned so much and take a long time to settle down to their new tighter or slacker tensions. I also suspect that the unbalanced tension of the strings will cause the neck to bend if you leave a guitar tuned like this for several days.
So, here we go. One finger of one hand will be used to pluck the strings individually, and one finger of the other hand (called the “non-plucking finger,” below) should gently rest on the string—as shown in the photo in chapter 8 .
Each string can produce four notes easily:
- the natural note of the open string (which we will call “open”)
- the “octave above” pinged note—created by pinging the string with your non-plucking finger on the middle of the string (on a guitar, the middle of the string is directly above the twelfth fret)
- the “double octave above” note—created by pinging the string with your non-plucking finger one quarter of its length from either end—directly above the fifth fret
- the “new” note—created by pinging the string with your non-plucking finger one third of its length from either end—directly above the seventh fret.
If we call the thickest string number 1
First of all we tune the thickest string (string 1) to a higher than normal note (to stop the thinner strings being too slack when we have finished).
Then the “octave above” note of string 1 should match the “open” note of string 6.
The “new” note of string 1 should match the “octave above” note of string 4.
The “new” note of string 4 should match the “double octave above” note of string 2.
The “new” note of string 2 should match the “octave above” note of string 5.
The “new” note of string 5 should match the “double octave above” note of string 3.
All done—instant Oriental sound.
If we call the thickest string number 6 (normal guitar numbering)
First of all we tune the thickest string (string 6) to a higher than normal note (to stop the thinner strings being too slack when we have finished).
Then the “octave above” note of string 6 should match the “open” note of string 1.
The “new” note of string 6 should match the “octave above” note of string 3.
The “new” note of string 3 should match the “double octave above” note of string 5.
The “new” note of string 5 should match the “octave above” note of string 2.
The “new” note of string 2 should match the “double octave above” note of string 4.
All done—instant Oriental sound.
D. Calculating equal temperament
As I said in chapter 8 , Galilei and Chu Tsai-Yu found that calculating the equal temperament system is pretty easy once you have presented the problem clearly and logically:
1. A note an octave above another must have twice the frequency of the lower one. (This is the same as saying that if you use two identical strings one must be half the length of the other—the frequency of the note produced by a string goes up as the string gets shorter and half the length gives double the frequency.)
2. The octave must be divided up into twelve steps.
3. All the twelve steps must be equal. (If you take any two notes one step apart, then the frequency ratio between them must always be the same.)
Let’s look at an example to make the calculation clearer. In this example we will make every string 90 percent as long as its longer neighbor—and all strings are made of the same material and are under the same amount of tension.
- Let’s make our longest string 24 inches long.
- String number 2 is 90 percent of the length of string 1 (i.e., 21.6 inches long).
- String number 3 is 90 percent of the length of string 2 (i.e., 19.4 inches long).
- String number 4 is 90 percent of the length of string 3 (i.e., 17.5 inches long).
- Continue until you reach string 13.
This example shows how the percentage shortening system works, but unfortunately we picked the wrong percentage. There is too much of a jump between strings. We want the thirteenth string to be half as long as the original string so that it will produce a note an octave above it. However, if you make every string 90 percent of the length of the previous one, your thirteenth string will be far shorter than you want it to be. So what percentage should we use for shortening our strings?
This is where Galilei and Chu Tsai-Yu come in handy. They calculated* exactly the correct percentage to make string 13 half the length of string 1. And the answer is… 94.38744 percent. Or, to put it another way, you need to remove 5.61256 percent of the length of any string to find the length of its shorter neighbor.
So now let’s do the calculation using this correct percentage:
- String 1 is 24 inches long.
- String 2 is 94.38744 percent as long as string 1—i.e., 22.65 inches.
- String 3 is 94.38744 percent as long as string 2—i.e., 21.38 inches.
- String 4 is 94.38744 percent as long as string 3—i.e., 20.18 inches.
- String 5 is 94.38744 percent as long as string 4—i.e., 19.05 inches.
- String 6 is 94.38744 percent as long as string 5—i.e., 17.98 inches.
- String 7 is 94.38744 percent as long as string 6—i.e., 16.97 inches.
- String 8 is 94.38744 percent as long as string 7—i.e., 16.02 inches.
- String 9 is 94.38744 percent as long as string 8—i.e., 15.12 inches.
- String 10 is 94.38744 percent as long as string 9—i.e., 14.27 inches.
- String 11 is 94.38744 percent as long as string 10—i.e., 13.47 inches.
- String 12 is 94.38744 percent as long as string 11—i.e., 12.71 inches.
- String 13 is 94.38744 percent as long as string 12—i.e., 12.00 inches.
Now the thirteenth string is half as long as the first string, which is what we wanted. Also, the relative length of any string compared to its neighbor is always the same—so it doesn’t matter which string you start your tune on, the same sequence of up and down jumps will give you the same tune (but the whole tune will be higher or lower in pitch).
E. The notes of the major keys
In the following list, “
” means “flat” and “#” means “sharp.”
A major: A, B, C#, D, E, F#, G#
B major: B, C, D, E, F, G, A
B major: B, C#, D#, E, F#, G#, A#
C major: C, D, E, F, G, A, B
D major: Db, E, F, G, A, B, C
D major: D, E, F#, G, A, B, C#
E major: E, F, G, A, B, C, D
E major: E, F#, G#, A, B, C#, D#
F major: F, G, A, B, C, D, E
F# major: F#, G#, A#, B, C#, D#, E# (or Gmajor: G, A, B, C, D, E, F)
A major: A, B, C, D, E, F, G
Note: F# is the same as G—so we could have either key in this case. In every other case where the key could have one of two names (e.g., D and C#), we generally choose the one with the least sharps or flats in the key signature. D involves five flats but its alternative, C#, would have seven sharps—so we generally use D. There is no such clear choice in the case of F#/G because F# has six sharps and G has six flats.