What Is the Harmonic Series?

The harmonic series is an acoustical phenomenon consisting of
a sequence of pitches that is related to a lower pitch. A sound
having a frequency of, say, 100 vibrations per second (Hz) will
also cause sounds to resonate in simple ratios with those 100 Hz,
namely 200 Hz, 300 Hz, 400 Hz, and so on. These ratios, 2:1, 3:1,
4:1, are actually in an invariable harmonic relationship with the
note at 100 Hz, or fundamental note, as it’s called (see figure 1).
The sequence of successive intervals is always octave (2:1), fifth
(3:2), fourth (4:3), major third (5:4), minor third (6:5), two intervals
between a minor third and a whole tone (7:6 and 8:7), large
whole tone (9:8), small whole tone (10:9), and progressively
smaller intervals ad infinitum. The semitone is said to begin in
the sequence at 16:15—f°u r octaves above the fundamental—
but that interval can also be derived as the difference between a
5:4 pure major third and a 4:3 pure fourth (invert and multiply:
4/5 x 4/3 = 16/15).
Each note of the harmonic series is found at frequencies corresponding
to multiples of the fundamental frequency. Basically
any sound possesses all the notes in its harmonic series to varying
degrees, and that variability is the reason the timbres of
instruments playing the same pitch are different: their harmonics
have different strengths. It is also thought that the prevalence
of the octave, fifth, fourth, and major and minor thirds in
the lower part of the harmonic series contributed to the development
of our concept of harmony, in which those intervals
form the most common components of chords. Chords in the
Western (that is, European) music tradition, therefore, are not
merely a culturally evolved arrangement of musical sounds into
a system but a natural phenomenon based on the physical science
of acoustics.

What Are Pure Intervals?

Pure intervals occur when the speed of the vibrations of two or
more notes (that is, their frequencies) match the simple acoustical
ratios of the harmonic series. For the most commonly used harmonic
intervals, these ratios are as follows:
unison 1:1
octave 2:1
fifth 3:2
fourth 4:3
major third 5:4
minor third 6:5
For example, a note vibrating at a frequency of 400 times per
second (Hz) would be an octave above a note vibrating at 200 Hz,
and an octave below one vibrating at 800 Hz. It would be a fifth
below a note vibrating at 600 Hz because 600:400 is the same as
3:2. It would be a fourth above a note vibrating at 300 Hz and a
major third below a note vibrating at 500 Hz. Except for the octave,
these intervals are not exactly the same size as those in equal temperament,
but they are pure because they match the harmonic
At C E G#
Pure Major 3rd Pure Major 3rd Pure Major 3rd
Pure 8ve
A\> A\>

Three Pure Major Thirds Compared with an Octave.
That was mostly all right because Renaissance composers
weren’t using all the major thirds anyway. The most common
temperament of the Renaissance was something called quartercomma
meantone, which had eight acoustically pure thirds and
ratios and because, when sounded together, they are completely
beatless. That means that their sounds fit so perfectly together
that there is no interference between their respective vibrations:
Any variance from the exact ratio will cause a beat or pulsation—
waoo-waoo-waoo-waoo—and the faster the pulsation, the less
pure the interval must be. For example, two notes vibrating at 400
and 401 Hz, respectively, will pulse or beat once per second in
addition to sounding the two frequencies; notes sounding
together at 400 and 402 Hz will beat twice per second, and so on.
Those intervals are not pure unisons because the frequencies of
the notes are not exactly in a 1:1 ratio.

‘What Is Temperament?

Temperament is a way of tuning the notes of the scale using
intervals that have been modified (tempered) from their pure
forms. The usual reason for doing this is utility, which is to say,
to make a tuning system more useful in a wider variety of musical
situations. Pythagorean tuning is not a temperament because
its scale is constructed using notes in a chain of pure, untempered
fifths. Just intonation is not a temperament because it
calls for pure intervals of every type at all times. These systems
have advantages, but they also have drawbacks, particularly in
situations where a keyboard instrument is the model for the
musical system, and that’s where temperaments arose in the first
place.
ET is a temperament because its fifths are narrowed very
slightly from the acoustically pure ratio of 3:2, so that twelve ET
fifths in sequence arrive at a note that is seven octaves above the
starting note, creating, at the same time, a scale where all twelve
notes in the octave are equidistant.Its great advantage is that it is universally
usable: It can be used in every key with identical musical effect.
Its drawback is that its major third is much, much wider than an
acoustically pure major third, and sounds quite harsh. That’s
one of the main reasons that other temperaments were more
popular than ET throughout history: Musicians were not willing
to tolerate thirds that were so dissonant.
Typically temperaments tried to balance euphony with u t i l i t y—
to have the most common chords and keys sound good at the
expense of the least common chords and keys. To achieve that
they usually tempered some fifths more than in ET (that is,
made them narrower) while others were wider than in ET—
often pure, in fact. A temperament in which all of the tempered
fifths are the same size is called a regular temperament,
and one in which fifths are of different sizes is called an
irregular temperament. The most common regular
temperament, of course, is ET, where the notes of the scale are
created by tempering equally each of the twelve fifths. It is possible
to create equally tempered systems that use more (or less)
than twelve notes in the octave by tempering the “fifths” a different
amount, such that a certain number of them in sequence
eventually comes back to the starting note.

What Is a Comma?

Commas are discrepancies. Tuning discussions from earlier centuries
tend to toss around the word “comma” as if it were a single,
fixed interval, but there are actually different discrepancies
that theorists refer to when they use the word. There are two
main commas. The first is the Pythagorean comma (also called
the ditonic comma), which is the discrepancy between twelve
pure fifths and seven pure octaves. That’s the
comma used in ET, where each of the twelve fifths is narrowed by
one twelfth of the Pythagorean comma, creating an octave of
twelve equally spaced notes.
The other discrepancy often used in tuning discussions is the
syntonic comma, which is the discrepancy between four pure
fifths and two octaves plus a pure major third . This
is the comma referred to in quarter-comma meantone, for example,
where each of the fifths is narrowed by one-quarter of the
syntonic comma, so that its resulting thirds are acoustically pure.
Another interval often referred to as a comma in historical discussions
is technically a diesis, which is the difference between a
major semitone and a minor semitone.
This tiny interval can vary according to the amount
of tempering used in the fifths. Quarter-comma meantone, for
example, has a larger diesis than sixth-comma meantone: The
greater the tempering of the fifths (1/4 is greater than 1/6, so its
fifths are tempered more), the greater the difference between
major and minor semitones, and the larger the diesis.
Last, commas are historically used to describe very small intervals
in general, as in the individual units of octave division systems.

What Is an Octave Division?

Octave divisions are tuning systems created by dividing the
octave into a certain number of equal parts. ET is an octave division
system, because its twelve equally tempered fifths also happen
to divide the octave into twelve equal parts.You can see, however, that if you
were not tied to a modern keyboard with its twelve notes per
octave, it would be possible to divide the interval of the octave
mathematically into any number of equal parts and make a
musical system based on that division. The contemporary composer
Easley Blackwood has demonstrated that principle with
his Microtonal Etudes for divisions of thirteen up to twentyfour
notes to the octave. Those pieces are available on CD:
Cedille CDR 90000 018 (1994).
Some octave division systems were discussed in earlier centuries,
too, especially because of their relationship to established
temperaments. The 19-division system actually corresponds with
extended third-comma meantone, since the intervals created in
the division correspond with those of third-comma meantone.
The same is true for 31-division and quarter-comma meantone,
43-division and fifth-comma meantone, and 55-division and
sixth-comma meantone. Theorists (including Isaac Newton)
also discussed the 53-division of the octave because of its close
approximation to Just intonation. Historically, theorists often referred to the
individual units of octave division systems as commas.

from Ross W. Duffin’s book  ” How equal temperament ruined Harmony ”