5.4.1 “How Come I Can’t Tune this #@*&!% Thing?”
5.4.2 Don’t Lose Your Equal Temperament
5.4.3 “It’s Too Low (or High) for My Voice”:
5.4.4 How Transposing Instruments Work
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If you write a piece of music in a single key, you'll likely have no
problem with musical unity. The arrangement of intervals in the
diatonic scale ensures a strong tonal centre. A nice, small
assortment of six related notes (scale degrees 2 through 7) all point
to the tonic note.
Moreover, assuming your song has words, you'll likely organize
the words into verses and choruses, each sung to the same musical
phrases. This reinforces musical unity.
But too much musical unity ain't necessarily a good thing. It can
To get some variety happening, you have the option of changing
keys partway through the song. And, since every key has its own
tonal centre, you preserve unity at the same time as you create
Several hundred years ago, when musicians and musical
theorists were experimenting with changing keys within a piece of
music, a nasty problem kept bedevilling them. Whenever they tried
to switch to a new key, their instruments sounded out of tune. Hellish
out of tune.
The problem was the dang Pythagorean comma. As discussed
in Chapter 4, if you tune an instrument using exact Pythagorean 3:2
frequency ratios, you end up with an octave that is slightly bigger
than it ought to be. About a quarter of a tone too big.
For example, Middle C has a frequency of 261.6 Hz. So the
frequency of the C above Middle C ought to be exactly double: 523.2
But if you use exact Pythagorean fifths, you end up with C above
Middle C having a frequency of 530.3 Hz. Noticeably too sharp.
If instead you tune in perfect 2:1 octaves, then the other notes
derived from simple frequency ratios such as 3:2, 4:2, and so on,
end up either too sharp or too flat.
What to do?
Musicians and theorists tried all sorts of variations on the theme of
just intonation, methods of tuning with simple whole-number
frequency ratios. Such tuning systems—and there are
many—enable the tuning of an instrument so that it's playable in one
key, or perhaps even several keys. But if you try to play in keys the
instrument isn't tuned for ... forget it. Doesn't work.
Eventually, one of the numerous tuning solutions attempted over
the centuries emerged the clear winner. The solution was to:
1. Stick to an exact 2:1 octave, despite the Pythagorean
2. Divide the octave into 12 exactly equal semitones.
This system is called equal temperament (from
Latin root meaning to mix or mingle).
To get the frequencies for each semitone:
• Start with the first note of the scale and multiply its frequency
by the 12th root of two.
• Take that frequency and multiply it by the 12th root of two,
which gives you the frequency for the next semitone up.
• Repeat until you get to the next octave.
The 12th root of 2 is the number 1.05946 (rounded off). So the
ratio of any semitone to the semitone below is 1.05946:1.
Table 29 shows the frequencies of all the notes from Middle C to
the octave above Middle C, with each successive frequency
multiplied by the 12th root of two:
TABLE 29 Equal Temperament Frequencies for
Tones from Middle C to C Above Middle C, and
Associated Simple Frequency Ratios
So, when you're playing in any given key, only the two octave
notes are in an exact 2:1 simple frequency relationship. Every other
note is slightly out of tune, compared with the simple frequency ratio
expected from the harmonic series.
For example, in Table 29 above:
• The frequency for the note G would be 392.4 Hz if it were
tuned in an exact 3:2 ratio with Middle C. But the equal
temperament frequency of G is 392.0 Hz (slightly flatter).
• The frequency for the note E would be 327.0 Hz if it were
tuned in an exact 5:4 ratio with Middle C. But the equal
temperament frequency of E is 329.6 Hz (slightly sharper).
• The frequency for the note C♯ would be 279.0 Hz if it were
tuned in an exact 16:15 ratio with Middle C. But the equal
temperament frequency of C♯ is 277.2 Hz (slightly flatter).
Similarly, in equal-temperament tuning, all of the other notes are
either slightly flat or slightly sharp, compared with their simple-frequency-ratio counterparts.
The equal temperament solution works. Your brain accepts the
small “pitch errors”—slight deviations from simple ratios—when
they're equally distributed over all 12 semitones. Since every
semitone interval is exactly equal, you can construct diatonic scales
using any of the 12 semitones as the tonic note, and the octave
notes will always have a frequency ratio of exactly 2:1. Equal
temperament makes something called modulation possible (coming
Consequently, equal temperament has been the norm for about
three centuries in Western music.
Equal temperament works only because the pitch errors are
small—so small that your forgiving brain processes them as though
they were simple frequency ratios.
When you try to tune a guitar or other stringed instrument using
harmonics from string to string, it doesn't quite work out because
you're not using equal temperament. That's why the best tuning
device is a digital tuner, with equally-tempered frequencies built into
the electronics that are accurate to many decimal places.
Get That Man a Digital Tuner
Some people think equal temperament is a Bad Thing because
every single note between the octave notes in any key is slightly
dissonant. Others think equal temperament is a Good Thing for
two main reasons:
1. It solves the dang tuning problem, already; and
2. Every single note between the octave notes in any key is
slightly dissonant—and therefore music played on equally-tempered instruments sounds more colourful and interesting
than it would if all the notes were exactly in tune.
Obviously, J. S. Bach agreed with the latter view. Nobody had a
keener ear. Bach would surely have been able to easily hear the
out-of-tuneness of equal temperament. Yet he famously
celebrated equal temperament by composing The Well-tempered
Clavier, a two-book masterwork of 24 preludes, one in each major
and minor key, and 24 fugues, one in each major and minor key.
One other tuning-related problem took even longer to solve: what
to do about a reference frequency. One note and its associated
frequency needs to serve as a standard to derive the frequencies for
all the other notes, using equal temperament.
After centuries of hair-pulling and fang-gnashing, everybody
agreed in 1939 that the note A above Middle C would always be
tuned to exactly 440 Hz, and would therefore serve as the reference
pitch for setting all the other pitches. (Then World War II started.)
This tuning pitch is called Concert A or A-440.
A Free Emergency Digital Tuner
When you're lost in Juarez in the rain and you don't have a digital
tuner with you but you must tune your guitar, what can you do?
Why, just reach in your pocket and whip out your trusty cell
phone. Or wander around until you find a pay phone. Get a dial
tone, and you've got your reference note. The dial tone is F.
Specifically, it's the F on the first fret of the low E-string of your
guitar, the F that's one and a half octaves below Middle C.
It happens to everybody. You swagger into the Wrong Ranch Saloon
and start singin' a tune and everything's goin' along fine until you get
to the lowest notes (or the highest notes), and you can't hit them.
People turn and laugh at you. Especially the dusty cowpoke, the
one who out-drew Billy Joe. You laugh along with them, vainly
attempting to hide your humiliation. Tears stream down your face.
It’s no use. They know, they know. Yes, they know you started
singing in a key that did not match your vocal register for that
it's too late. Marshal Puma senses trouble brewing and allows as how you might
live to see tomorrow if you get outta Dodge tonight. So you stumble out of the
saloon into the dusty main street. Sadie and Ellie Sue offer you a fresh horse
and away you go to join Ex-Marshal McDillon in exile.
If only you had thought to start over, in a different key.
Transposition refers to moving a whole group of notes (such as
the entire melody of a song) up or down in pitch.
• If you play guitar, you can do this easily without even
changing chord fingering. All you do is move your capo up or
down the fretboard.
• On the piano, it's not so easy. You have to change the way
you finger the melody and chords for every dang key you play
You can use tables 24, 25, 26, and 27 for transposing. They
show you, row by row, the scales for each key. If you're singing in
the key of C major and you want to know what notes you'd be
singing if the tune were transposed to E major, just go to Table 24
and compare the C Major row with the E major row. For instance, if
you want to transpose the notes C, D and E in the key of C major to
the key of E major, the equivalent notes would be E, F♯, and G♯.
It's that simple.
One important thing to keep in mind at all times with respect to
key changes and transposing: There's no such thing as a “high
key” or a “low key.” A key is just an interval order with respect to
a key note or tonic note. The key of E major is neither “higher” nor
“lower” than the key or C major or any other key.
The way a songwriter or composer has arranged the intervals of
a particular melody determines which key you will be able to sing it
in, without the tune being too high or too low for your voice.
You can sing some songs easily in the key of C major, but not in
the key of G major. You can sing other songs easily in the key of G
major, but not in C major. The determining factor is not the key. It's
how the melody itself is structured. The key of C major is not
inherently “higher” or “lower” than the key of G major.
That goes for all the keys, major and minor.
If you happen to read music notation, the idea of a “transposing
instrument” will make more sense than if you don't happen to read
Most musicians don't read music notation, which is why this book
has no music notation. Still, even if you don't read music, you might
find a brief description of the meaning of “transposing instrument”
mildly entertaining. George Martin, the classically-trained producer of
the Beatles, once tried to explain the workings of transposing
instruments to John Lennon, who did not read a note of music.
Lennon thought it was all pretty daft.
A true transposing instrument (as opposed to an octave
transposing instrument—more on the distinction in a minute) is a
wind instrument (aerophone) for which the musical notes on the
page differ from the notes the instrument makes. You see a note on
the page, you finger the instrument to play that note, and a different
note comes out of your instrument.
What's going on?
Any given musical instrument is constructed so that it can handle
only a certain range of pitches. The guitar, for instance, only has a
certain number of frets, limiting the upper and lower range of the
This applies to wind instruments, like any other. So it's common
to have “families” of wind instruments—families of clarinets, flutes,
and saxophones, for instance—of varying sizes. The smaller-sized
instruments handle higher pitches, the larger ones, lower pitches.
For instance, each of the four common sizes in the saxophone
family—soprano, alto, tenor, and baritone—is good for a certain
range of pitches, from a high-pitched range (soprano sax) to a low-pitched range (baritone sax).
All saxophones use the same fingering for a particular written
note. So, if you learn to play, say, alto sax, and you decide to switch
to another sax in the same family, you don't have to learn a whole
different way of fingering.
Problem is, because each instrument is built for a different pitch
range, when you finger the alto sax to play, say, the written note C,
the note you actually hear coming out of your horn is E♭, 9
semitones below C. On the tenor sax, when you finger the
instrument to play C, the note that comes out is B♭, more than an
octave below the C written on the page.
Therefore, composers and orchestrators must notate the music
so that it accounts for the difference between the notes that come
out of the transposing instrument and the notes on the page.
Suppose the composer wants the sound coming out of the alto
saxophone to be in the key of C. The composer needs to notate the
music nine semitones higher (an interval of a major sixth) on the
page—in the key of A. The alto sax player sees an A on the page,
fingers the horn to play A, and out comes the sound of the note C,
nine semitones lower—as the composer intended.
So, written music for the alto sax must be transposed up by an
interval of a major sixth (all notes!), in order to sound the way the
This all seems pretty odd, but it makes a lot of sense for wind
players who read music. They don't have to cope with learning new
fingerings for each instrument in a family. Instead, it's up to the
composer or orchestrator to ensure that the music is transposed on
the page properly for the intended instrument and the intended
Some instruments are “octave transposing” instruments. The
guitar, for instance. Notated music for the guitar is written an octave
higher than it sounds when you play the music. When you play the
note Middle C from the page, you still hear the note C, but it's the C
an octave below Middle C.
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