Music and Math in Harmony
by Paul VanRaden
© 2002
People can enjoy beautiful music and soothing harmonies without understanding
math. Even songbirds, whales, and other animals make forms of music without
formal training in math. Harmony sounds better when the tones that blend
together follow some simple rules even if the artists are not aware of the
math. People can listen and practice and try for many years to create better
music, or they can learn the math and make better melodies and harmonies
today.
The human voice and some musical instruments can produce a continuous
range of frequencies between lower and upper limits. Other instruments require
a tuning system to choose a finite number of tones. Humans can sing using many
tones or in monotone, and they can paint colorful pictures or draw in black and
white. When choosing music, beauty is in the ear of the beholder.
Hearing and
Harmony
Ears sense sounds of different wave length in much the same way that eyes
sense lights of different wave length. Sounds move from vibrating surfaces to
your ears on air pressure waves traveling at 300 meters per second (700 miles
per hour), whereas light moves from vibrating molecules to your eyes on
electromagnetic waves traveling at 3 million meters per second (7 million miles
per hour). If the source vibrates slower or faster, the time and distance
between waves will increase or decrease, resulting in different tones of sound
or colors of light. The sounds of music range from the low tones of tubas and
cellos to the high tones of flutes and violins. The colors of a rainbow range
from low frequency red and orange light to high frequency blue and violet
light.
Harmony results when two or more tones with simple frequency ratios are
mixed. Some ears are tone-deaf and some eyes are color-blind, but most ears
are able to hear several tones at the same time and most eyes can see several
colors at the same time. When two tones with a simple frequency ratio are
played together, the result may sound better than either tone played
separately. Two colors with a simple frequency ratio also may blend into
another pleasing color (such as blue and yellow combining to form green). Other
pairs of colors may look bad when mixed, and too many bright colors mix into
brown or black. Similarly, certain pairs of tones can clash, and too many or a
poor choice of tones can mix into noise instead of a chorus or a symphony.
Better songs and pictures result when the artist mixes tones of sound or
colors of light that match. Artists can use their ears and eyes and trial and
error to find matching tones and colors, or they can use math.
Tones and
Tuning
A tone is a series of repeating air pressure waves of uniform size and
frequency. Most music is based on a reference tone with standard frequency such
as "middle C" with 264 waves per second or the "A" above it
with 440 waves per second. A reference tone allows standard tuning of
instruments and lets all other tones be set in proportion to the standard tone,
which can be assigned a relative frequency of 1 to keep the math simple. The
relative frequency of any other tone is its frequency divided by the standard.
For example, a series of "C" tones with 132, 264, 528, and 1056
waves per second may be assigned relative frequencies of .5, 1, 2, and 4,
respectively.
Two tones with relative frequency 1 and 2 are called doubles (formerly octaves).
They always sound good when heard together because the two tones blend into a
repeating wave pattern that is easy for ears to hear: large wave, small wave,
large wave, small wave. Large and small increases in pressure continue to
alternate with every other wave as long as both tones are played. The
pattern repeats because the less frequent waves of the lower tone add to every
other wave of the double. Similarly, tones 1, 2, and 4 add into a repeating
wave pattern: large, small, medium, small, large, small, medium, small. Every
4th wave is large, with medium waves halfway between the large waves, and small
waves halfway between the medium and large waves. This pattern of doubles, and
patterns including even more doubles played at the same time, all sound very
nice.
Long ago, doubles were called octaves. Doubles are so basic to music and
so easy that they are not usually considered part of harmony. The harder task
of choosing tones between the doubles and blending two or more tones together
is known as harmony. Ratios such as 1 : 1.5 or equivalently 2 : 3 are harmonic
because the two tones combine into a pattern of air pressure waves easy to
hear: large, small, small, large, small, small, etc. Harmony can be
understood by starting with the usual choices of tones available on
standard keyboards. But an easier method is to start at the beginning and use
simple math to choose musical scales.
Many tuning systems are possible, including linear, log, or reciprocal.
Linear tuning is easy to understand because the increase in frequency from each
tone to the next is constant. For example, the series 1, 1.25, 1.5, 1.75, and 2
has a constant increase of .25 between each tone. Log tuning is another common
system in which tones are equally spaced on the log of frequency rather than
frequency scale. This makes the ratios rather than increases constant from one
tone to the next. Reciprocal is a third system in which tones are obtained as
reciprocals of the linear tuning frequencies. Reciprocal tuning results in the
same tone ratios as linear tuning, but in the opposite order. Math allows
artists to choose different sets of tones and to make scales scalable.
Scalable
Scales
A scale is a list of the tones available from one tone up to its double.
Higher or lower tones outside the range 1: 2 can be obtained by multiplying or
dividing the original frequencies by 2, or 4, or 8, etc. The tones in the scale
can be set with simple formulas that allow artists to choose more or fewer
tones per double, resulting in scalable scales. Selecting the number of tones
in the scale is similar to buying boxes of 8, 16, or even 64 color
crayons. As you choose more tones or colors you get less change from one to the
next.
Musical scales require compromises between several goals. The tones
chosen should 1) be evenly spaced for simplicity, 2) have simple frequency
ratios to improve harmony, 3) have enuf tones to provide choice but not so many
that the ear cannot hear them all, and 4) have the same number of tones as
beats in a musical measure so that scales and measures are in step.
Goal 1) is met if the frequency of each tone is a constant ratio of the
previous tone and the number of tones is the same within each doubling of
frequency. Goal 2) is met if the tones are spaced evenly on the frequency scale
instead of the log scale. Goal 3) is met if scales have fewer tones per double
(2-6) for beginners and more tones (7-24) for more advanced musicians.
Goal 4) is met if scales have 3, 4, 6, or 8 tones per double because most music
is based on measures with those beats and few songs are written with 5 or 7
beats per measure or higher primes. Then, a scale could begin and end on the
same beat of the next measure. All goals cannot be met at the same time,
but modern electrical instruments can make more choices easily available.
Scalable scales can be produced using linear tuning, log tuning,
reciprocal tuning, or other tuning systems. If w represents the number of tones
desired per double, then the following formulas can be used to set the tones.
With linear tuning, tone n gets a frequency of (1 + n/w). With log tuning,
the formula 2 to the power (n / w) is used instead. With reciprocal
tuning, the formula is 2 / (1 + n/w). With two tones per double (w=2), the tone
chosen between the doubles is 1.5 with linear tuning, 1.414 with log tuning,
and 1.333 with reciprocal tuning. More tones are obtained between the doubles
by setting w equal to 3, 4, 6, 8, 12, or other numbers.
Fewer tones result in fewer wrong notes. More tones too closely spaced
result in poor harmony because the wave series takes too long to repeat
and the ear cannot pick up the pattern. The worst harmony results from tones
with ratio 1 : 1.03 because higher ratios have shorter repeat lengths and
ratios closer to 1 begin to sound like the same tone (no harmony). Similarly,
3% above and below ratios with nice harmony such as 1 : 1.5 are ratios with bad
harmony such as 1 : 1.55. Because ratios with best and worst harmony are so
close, singing can be difficult. If you try to sing the best tone and miss it
by 3% you sing the worst tone instead.
All tones have nice harmonic ratios in scales having 4 or fewer tones per
double (w=4) if tuning is linear or reciprocal. Then, a beginner or child
can play any or all of the available tones and never hit a wrong note. Simple
math can produce better music, but scalable scales are not yet available
because of the key pattern on the standard keyboard.
Keys and
Keyboards
The standard keyboard was invented hundreds of years ago and is the
foundation of current music. The 7 repeating letters A, B, C, D, E, F, and G
were engraved onto organ keyboards in Europe as early as the year 1090. The
earliest keyboards had just the front row of keys with 7 instead of 12 tones
per double. In about1350, “new” keys for sharps and flats were added in an
uneven pattern in a back row. This arrangement allowed organists trained on the
“old” keyboard without sharps and flats to continue playing the same old songs.
The standard keyboard survived for several hundred years because the tones were
reasonable, because all Christian churches used it, and because a standard used
for so long by so many was hard to change.
The standard scale probably developed by trial and error rather than by
math. The uneven spacing of the 5 black keys gives better harmony to the 7
white keys than the previous evenly spaced keys. The standard musical scale
does not start at tone A but instead at C. It includes relative frequencies 1,
1.125, 1.25, 1.333, 1.5, 1.667, 1.875, and 2. These tones happen to be very
close to tones produced by the log tuning system with 12 tones per double: 1,
1.122, 1.26, 1.335, 1.495, 1.682, 1.888, and 2. The early organ builders
probably didn’t use logarithms to determine pipe length and diameter but
instead probably used their ears to find simple frequency ratios with nice
harmony.
Linear tuning with 8 tones per double produces many of the standard tones
exactly: 1, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875, and 2. Linear tuning
with 3 tones per double produces the remaining standard tones 1.333 and 1.667.
The linear math is easier to understand than standard theory and also can
produce better harmony. For example, the nice sounding chord 1 : 1.25 : 1.5 :
1.75 : 2 (equivalently 4 : 5 : 6 : 7 : 8) isn’t available on the standard C
scale. Singing also may be simpler with linear scales because frequency
increases are even and predictable.
With a keyboard, one person can play several notes at the same time. With
most other musical instruments such as brass and woodwinds, each musician plays
only one tone at a time. Some exceptions are harmonicas and bowed instruments
which let the musician play two adjacent tones, and the guitar with its
multiple strings (commonly 6 or 8) that can all sound at the same time. Such
multi-tone instruments are tuned with harmonic tones side by side and have
fewer notes per doubling of frequency than the piano. Thus, the increase in
frequency from one guitar string to the next is greater than the increase in
frequency from one standard piano key to the next. Similarly, the
keys of electric keyboards could be rearranged so that the tones beside each
other sound good when played together.
Scalable keyboards with evenly spaced keys let math be applied more
easily. Standard keyboards can’t easily be tuned to have fewer or more tones
per double because then each double would contain a different pattern of black
and white keys. Some basic principles of scalable keyboards are covered in
another document.
Notes and
Notation
Music notation is based on the standard keyboard and does not support the
new tuning systems. Standard notation is difficult if all 12 notes are used.
Five sharps or flats and five naturals are needed to record a simple 12-note
scale. New notation is needed.
Before music and musical instruments were invented, people began to talk. Most of the things they said weren’t worth repeating, but some of their more important ideas and feelings were put into poems and songs. Words were easier to remember if they are repeated with a distinct tempo, pattern, rhyme, and melody. Music without words is also useful to express feelings and simply to entertain, but perhaps the main use of melodies and music is to bring words to life. Some lessons last longer when sung in pleasing songs or when accompanied by well tuned tones.