WEBVTT

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the term temperament refers to the
tuning of musical intervals

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away from just intonation 
which may be done for various reasons

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throughout history there have been many
systems of temperament

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these include among others 
equal temperament

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

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well temperament where 
the chromatic scale was made

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unevenly spaced so as to favor 
the purity of certain keys

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

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and meantone temperament which was similar
to Pythagorean tuning

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in that it used stacked fifths to generate
its intervals

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but instead the fifths were flatted


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so that they stacked up 
to produce pure thirds

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

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

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the regular temperament paradigm


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the fundamental concepts 
of regular temperament

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are the generators and the mappings

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generators are intervals used 
in combination with each other

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to create all other intervals 
in a regularly tempered tuning

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the mapping is what ratios of just intonation

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we say those intervals are approximating

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You may notice that this method is
reminiscent of meantone temperament

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in fact meantone temperament
fits perfectly

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under the regular temperament paradigm

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where the generating intervals are
the fifth and the octave

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

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

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like the major third 
which is mapped to 5/4

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or the minor third which is mapped to 6/5

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a more mathematically robust definition
of the mapping

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specifies how many of each generator 
are required to reach

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each prime number of just intonation
taken into account

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thereby defining how to arrive

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at any just intonation ratio
in terms of those generators

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a useful convention is to refer to
the larger generator

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in a temperament with only two
generators as the period

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

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

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

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an interesting repercussion 
of mappings like this one

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

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

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this is called "tempering
out that comma"

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in most mathematical contexts it would be
gibberish to equate

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

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it is one of the most revolutionary
concepts in recent history

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in meantone temperament
the comma 81/80

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also known as the Syntonic comma,
is tempered out

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

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

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

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

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

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

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the meaning behind this seemingly
meaningless statement

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is that in a tuning that tempers out 81/80

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all ratios that differ by that comma

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will be represented by the same interval

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this has two important implications

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

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now return to the same note you started on


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eliminating chromatic drift
for that comma

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

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one dimension of just intonation
the dimension of the prime 5

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in terms of two other primes - 
2 and 3

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in doing so we are approximating
a three-dimensional structure

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with only two dimensions
therefore lowering the complexity

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it is only an approximation however

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

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as our 80/16 or 5/1

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

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

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and using 81/64 as our 80/64

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but not to worry 
as history already dictates

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


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that is to say 
regularly temper them, 


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

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here is that same chord progression in
several different meantone tunings

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that form a compromise between the
ratios of 3

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and ratios of 5

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

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to reach a compromise and purity between

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all three primes we're representing

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

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for such a simple two-dimensional
structure

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

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tuned to an optimized meantone tuning

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as we'll find out in the next video

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the use of generators 
to define a tuning's intervals

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has additional benefits
not related to harmonic purity

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or chromatic drift