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