Tuning Theory 2: Temperament ("Microtonal" Theory)

Using western music's "meantone temperament" as a guide, I try to cover the basics of the regular temperament paradigm.

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the term temperament refers to the
tuning of musical intervals

away from just intonation
which may be done for various reasons

throughout history there have been many
systems of temperament

these include among others
equal temperament

where one interval, usually the octave,
is divided into equally spaced pieces

well temperament where
the chromatic scale was made

unevenly spaced so as to favor
the purity of certain keys

without completely sacrificing others as
in Bach's "Well-tempered clavier"

and meantone temperament which was similar
to Pythagorean tuning

in that it used stacked fifths to generate
its intervals

but instead the fifths were flatted

so that they stacked up
to produce pure thirds

in this video however I will be
introducing you to what is arguably

today's most relevant and thoroughly to
find system of tempering

the regular temperament paradigm

the fundamental concepts
of regular temperament

are the generators and the mappings

generators are intervals used
in combination with each other

to create all other intervals
in a regularly tempered tuning

the mapping is what ratios of just intonation

we say those intervals are approximating

You may notice that this method is
reminiscent of meantone temperament

in fact meantone temperament
fits perfectly

under the regular temperament paradigm

where the generating intervals are
the fifth and the octave

and the are mapped to 2/3 and 2/1
respectively

then combinations of the two are
used to arrive at other intervals

like the major third
which is mapped to 5/4

or the minor third which is mapped to 6/5

a more mathematically robust definition
of the mapping

specifies how many of each generator
are required to reach

each prime number of just intonation
taken into account

thereby defining how to arrive

at any just intonation ratio
in terms of those generators

a useful convention is to refer to
the larger generator

in a temperament with only two
generators as the period

so then in the meantone 2/1
is mapped to one period - the octave

3/1 is mapped to one period
plus one generator - a perfect twelve

and 5/1 is mapped to four generators
a major third of two octaves

an interesting repercussion
of mappings like this one

is that commas, small ratios close to 1/1,

cease to exist in the system and are
equated with 1/1 or the unison.

this is called "tempering
out that comma"

in most mathematical contexts it would be
gibberish to equate

any non 1/1 ratio with 1/1,
but in tuning theory

it is one of the most revolutionary
concepts in recent history

in meantone temperament
the comma 81/80

also known as the Syntonic comma,
is tempered out

we can show this starting with how 5/1
is mapped to 4 generators

that means
3/2 to the fourth equals 5/1

expanding 3/2 to the fourth
we get 81/16 equals 5/1

finding a common denominator
gives us 81/16 equals 80/16

then we can multiply both sides by 16
and get 81 equals 80

then finally dividing both sides by 80
gives us 81/80 equals 1/1

the meaning behind this seemingly
meaningless statement

is that in a tuning that tempers out 81/80

all ratios that differ by that comma

will be represented by the same interval

this has two important implications

one is that chord changes that
would drift by 81/80 in just intonation

now return to the same note you started on

eliminating chromatic drift
for that comma

the second important implication
is that we are giving away to represent

one dimension of just intonation
the dimension of the prime 5

in terms of two other primes -
2 and 3

in doing so we are approximating
a three-dimensional structure

with only two dimensions
therefore lowering the complexity

it is only an approximation however

as you may notice
if we treat a pure 81/16

as our 80/16 or 5/1

by definition it'll be out of tune
with respect to 5/1

here is the 1-6-2-5 progression
using pure fifths

and using 81/64 as our 80/64

but not to worry
as history already dictates

if we temper all 3/2's in the system,

that is to say
regularly temper them,

then stacking four generators
arrives at a much pure 5/1

here is that same chord progression in
several different meantone tunings

that form a compromise between the
ratios of 3

and ratios of 5

in addition
we could slightly temper 2/1 - the octave

to reach a compromise and purity between

all three primes we're representing

doing so can result in a remarkably good
approximation of just intonation

for such a simple two-dimensional
structure

here's the 1-6-2-5 chord progression

tuned to an optimized meantone tuning

as we'll find out in the next video

the use of generators
to define a tuning's intervals

has additional benefits
not related to harmonic purity

or chromatic drift