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