0:00:00.530,0:00:04.220
the term temperament refers to the[br]tuning of musical intervals

0:00:04.220,0:00:08.120
away from just intonation [br]which may be done for various reasons

0:00:08.660,0:00:11.679
throughout history there have been many[br]systems of temperament

0:00:13.139,0:00:15.309
these include among others [br]equal temperament

0:00:15.308,0:00:19.179
where one interval, usually the octave,[br]is divided into equally spaced pieces

0:00:19.899,0:00:22.339
well temperament where [br]the chromatic scale was made

0:00:22.339,0:00:25.469
unevenly spaced so as to favor [br]the purity of certain keys

0:00:25.469,0:00:29.948
without completely sacrificing others as[br]in Bach's "Well-tempered clavier"

0:00:29.949,0:00:33.719
and meantone temperament which was similar[br]to Pythagorean tuning

0:00:33.719,0:00:37.210
in that it used stacked fifths to generate[br]its intervals

0:00:37.210,0:00:39.409
but instead the fifths were flatted[br]

0:00:39.409,0:00:42.089
so that they stacked up [br]to produce pure thirds

0:00:43.279,0:00:47.589
in this video however I will be[br]introducing you to what is arguably

0:00:47.590,0:00:51.949
today's most relevant and thoroughly to[br]find system of tempering

0:00:51.948,0:00:54.249
the regular temperament paradigm[br]

0:00:54.799,0:00:57.038
the fundamental concepts [br]of regular temperament

0:00:57.039,0:00:59.479
are the generators and the mappings

0:00:59.979,0:01:03.320
generators are intervals used [br]in combination with each other

0:01:03.320,0:01:07.118
to create all other intervals [br]in a regularly tempered tuning

0:01:07.648,0:01:10.590
the mapping is what ratios of just intonation

0:01:10.590,0:01:12.790
we say those intervals are approximating

0:01:13.560,0:01:17.460
You may notice that this method is[br]reminiscent of meantone temperament

0:01:17.459,0:01:20.240
in fact meantone temperament[br]fits perfectly

0:01:20.240,0:01:22.210
under the regular temperament paradigm

0:01:22.209,0:01:25.919
where the generating intervals are[br]the fifth and the octave

0:01:25.920,0:01:29.560
and the are mapped to 2/3 and 2/1[br]respectively

0:01:30.350,0:01:34.040
then combinations of the two are[br]used to arrive at other intervals

0:01:34.040,0:01:37.130
like the major third [br]which is mapped to 5/4

0:01:37.129,0:01:41.119
or the minor third which is mapped to 6/5

0:01:42.289,0:01:45.629
a more mathematically robust definition[br]of the mapping

0:01:45.629,0:01:49.158
specifies how many of each generator [br]are required to reach

0:01:49.159,0:01:52.768
each prime number of just intonation[br]taken into account

0:01:52.768,0:01:54.419
thereby defining how to arrive

0:01:54.420,0:01:58.090
at any just intonation ratio[br]in terms of those generators

0:01:59.070,0:02:02.250
a useful convention is to refer to[br]the larger generator

0:02:02.250,0:02:05.530
in a temperament with only two[br]generators as the period

0:02:05.530,0:02:10.949
so then in the meantone 2/1 [br]is mapped to one period - the octave

0:02:11.319,0:02:16.189
3/1 is mapped to one period[br]plus one generator - a perfect twelve

0:02:16.189,0:02:21.399
and 5/1 is mapped to four generators[br]a major third of two octaves

0:02:23.450,0:02:26.030
an interesting repercussion [br]of mappings like this one

0:02:26.030,0:02:29.120
is that commas, small ratios close to 1/1,

0:02:29.120,0:02:33.749
cease to exist in the system and are[br]equated with 1/1 or the unison.

0:02:34.389,0:02:37.310
this is called "tempering[br]out that comma"

0:02:37.310,0:02:41.319
in most mathematical contexts it would be[br]gibberish to equate

0:02:41.319,0:02:45.979
any non 1/1 ratio with 1/1,[br]but in tuning theory

0:02:45.979,0:02:49.619
it is one of the most revolutionary[br]concepts in recent history

0:02:51.460,0:02:54.750
in meantone temperament[br]the comma 81/80

0:02:54.750,0:02:57.520
also known as the Syntonic comma,[br]is tempered out

0:02:57.910,0:03:02.790
we can show this starting with how 5/1[br]is mapped to 4 generators

0:03:03.079,0:03:07.649
that means[br]3/2 to the fourth equals 5/1

0:03:08.219,0:03:12.919
expanding 3/2 to the fourth[br]we get 81/16 equals 5/1

0:03:13.609,0:03:18.569
finding a common denominator[br]gives us 81/16 equals 80/16

0:03:18.729,0:03:23.089
then we can multiply both sides by 16[br]and get 81 equals 80

0:03:23.089,0:03:28.970
then finally dividing both sides by 80 [br]gives us 81/80 equals 1/1

0:03:31.650,0:03:34.730
the meaning behind this seemingly[br]meaningless statement

0:03:34.729,0:03:38.539
is that in a tuning that tempers out 81/80

0:03:38.539,0:03:41.620
all ratios that differ by that comma

0:03:41.620,0:03:45.060
will be represented by the same interval

0:03:45.060,0:03:47.769
this has two important implications

0:03:47.769,0:03:52.459
one is that chord changes that[br]would drift by 81/80 in just intonation

0:03:52.459,0:03:55.369
now return to the same note you started on[br]

0:03:55.369,0:03:58.010
eliminating chromatic drift[br]for that comma

0:03:59.280,0:04:03.419
the second important implication[br]is that we are giving away to represent

0:04:03.419,0:04:07.269
one dimension of just intonation[br]the dimension of the prime 5

0:04:07.270,0:04:10.620
in terms of two other primes - [br]2 and 3

0:04:11.390,0:04:14.890
in doing so we are approximating[br]a three-dimensional structure

0:04:14.889,0:04:18.909
with only two dimensions[br]therefore lowering the complexity

0:04:20.819,0:04:23.370
it is only an approximation however

0:04:23.370,0:04:27.199
as you may notice[br]if we treat a pure 81/16

0:04:27.199,0:04:30.010
as our 80/16 or 5/1

0:04:30.010,0:04:34.610
by definition it'll be out of tune[br]with respect to 5/1

0:04:35.220,0:04:39.780
here is the 1-6-2-5 progression[br]using pure fifths

0:04:39.779,0:04:43.799
and using 81/64 as our 80/64

0:04:58.950,0:05:01.889
but not to worry [br]as history already dictates

0:05:01.889,0:05:03.959
if we temper all 3/2's in the system, [br]

0:05:03.959,0:05:06.490
that is to say [br]regularly temper them, [br]

0:05:06.490,0:05:10.708
then stacking four generators [br]arrives at a much pure 5/1

0:05:11.488,0:05:14.928
here is that same chord progression in[br]several different meantone tunings

0:05:14.928,0:05:18.399
that form a compromise between the[br]ratios of 3

0:05:18.399,0:05:20.709
and ratios of 5

0:05:56.089,0:05:59.759
in addition[br]we could slightly temper 2/1 - the octave

0:05:59.759,0:06:01.830
to reach a compromise and purity between

0:06:01.830,0:06:03.730
all three primes we're representing

0:06:03.730,0:06:08.310
doing so can result in a remarkably good [br]approximation of just intonation

0:06:08.310,0:06:11.819
for such a simple two-dimensional[br]structure

0:06:11.819,0:06:14.529
here's the 1-6-2-5 chord progression

0:06:14.529,0:06:17.349
tuned to an optimized meantone tuning

0:06:31.399,0:06:33.920
as we'll find out in the next video

0:06:33.920,0:06:37.070
the use of generators [br]to define a tuning's intervals

0:06:37.070,0:06:40.170
has additional benefits[br]not related to harmonic purity

0:06:40.170,0:06:44.000
or chromatic drift