[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.53,0:00:04.22,Default,,0000,0000,0000,,the term temperament refers to the\Ntuning of musical intervals
Dialogue: 0,0:00:04.22,0:00:08.12,Default,,0000,0000,0000,,away from just intonation \Nwhich may be done for various reasons
Dialogue: 0,0:00:08.66,0:00:11.68,Default,,0000,0000,0000,,throughout history there have been many\Nsystems of temperament
Dialogue: 0,0:00:13.14,0:00:15.31,Default,,0000,0000,0000,,these include among others \Nequal temperament
Dialogue: 0,0:00:15.31,0:00:19.18,Default,,0000,0000,0000,,where one interval, usually the octave,\Nis divided into equally spaced pieces
Dialogue: 0,0:00:19.90,0:00:22.34,Default,,0000,0000,0000,,well temperament where \Nthe chromatic scale was made
Dialogue: 0,0:00:22.34,0:00:25.47,Default,,0000,0000,0000,,unevenly spaced so as to favor \Nthe purity of certain keys
Dialogue: 0,0:00:25.47,0:00:29.95,Default,,0000,0000,0000,,without completely sacrificing others as\Nin Bach's "Well-tempered clavier"
Dialogue: 0,0:00:29.95,0:00:33.72,Default,,0000,0000,0000,,and meantone temperament which was similar\Nto Pythagorean tuning
Dialogue: 0,0:00:33.72,0:00:37.21,Default,,0000,0000,0000,,in that it used stacked fifths to generate\Nits intervals
Dialogue: 0,0:00:37.21,0:00:39.41,Default,,0000,0000,0000,,but instead the fifths were flatted\N
Dialogue: 0,0:00:39.41,0:00:42.09,Default,,0000,0000,0000,,so that they stacked up \Nto produce pure thirds
Dialogue: 0,0:00:43.28,0:00:47.59,Default,,0000,0000,0000,,in this video however I will be\Nintroducing you to what is arguably
Dialogue: 0,0:00:47.59,0:00:51.95,Default,,0000,0000,0000,,today's most relevant and thoroughly to\Nfind system of tempering
Dialogue: 0,0:00:51.95,0:00:54.25,Default,,0000,0000,0000,,the regular temperament paradigm\N
Dialogue: 0,0:00:54.80,0:00:57.04,Default,,0000,0000,0000,,the fundamental concepts \Nof regular temperament
Dialogue: 0,0:00:57.04,0:00:59.48,Default,,0000,0000,0000,,are the generators and the mappings
Dialogue: 0,0:00:59.98,0:01:03.32,Default,,0000,0000,0000,,generators are intervals used \Nin combination with each other
Dialogue: 0,0:01:03.32,0:01:07.12,Default,,0000,0000,0000,,to create all other intervals \Nin a regularly tempered tuning
Dialogue: 0,0:01:07.65,0:01:10.59,Default,,0000,0000,0000,,the mapping is what ratios of just intonation
Dialogue: 0,0:01:10.59,0:01:12.79,Default,,0000,0000,0000,,we say those intervals are approximating
Dialogue: 0,0:01:13.56,0:01:17.46,Default,,0000,0000,0000,,You may notice that this method is\Nreminiscent of meantone temperament
Dialogue: 0,0:01:17.46,0:01:20.24,Default,,0000,0000,0000,,in fact meantone temperament\Nfits perfectly
Dialogue: 0,0:01:20.24,0:01:22.21,Default,,0000,0000,0000,,under the regular temperament paradigm
Dialogue: 0,0:01:22.21,0:01:25.92,Default,,0000,0000,0000,,where the generating intervals are\Nthe fifth and the octave
Dialogue: 0,0:01:25.92,0:01:29.56,Default,,0000,0000,0000,,and the are mapped to 2/3 and 2/1\Nrespectively
Dialogue: 0,0:01:30.35,0:01:34.04,Default,,0000,0000,0000,,then combinations of the two are\Nused to arrive at other intervals
Dialogue: 0,0:01:34.04,0:01:37.13,Default,,0000,0000,0000,,like the major third \Nwhich is mapped to 5/4
Dialogue: 0,0:01:37.13,0:01:41.12,Default,,0000,0000,0000,,or the minor third which is mapped to 6/5
Dialogue: 0,0:01:42.29,0:01:45.63,Default,,0000,0000,0000,,a more mathematically robust definition\Nof the mapping
Dialogue: 0,0:01:45.63,0:01:49.16,Default,,0000,0000,0000,,specifies how many of each generator \Nare required to reach
Dialogue: 0,0:01:49.16,0:01:52.77,Default,,0000,0000,0000,,each prime number of just intonation\Ntaken into account
Dialogue: 0,0:01:52.77,0:01:54.42,Default,,0000,0000,0000,,thereby defining how to arrive
Dialogue: 0,0:01:54.42,0:01:58.09,Default,,0000,0000,0000,,at any just intonation ratio\Nin terms of those generators
Dialogue: 0,0:01:59.07,0:02:02.25,Default,,0000,0000,0000,,a useful convention is to refer to\Nthe larger generator
Dialogue: 0,0:02:02.25,0:02:05.53,Default,,0000,0000,0000,,in a temperament with only two\Ngenerators as the period
Dialogue: 0,0:02:05.53,0:02:10.95,Default,,0000,0000,0000,,so then in the meantone 2/1 \Nis mapped to one period - the octave
Dialogue: 0,0:02:11.32,0:02:16.19,Default,,0000,0000,0000,,3/1 is mapped to one period\Nplus one generator - a perfect twelve
Dialogue: 0,0:02:16.19,0:02:21.40,Default,,0000,0000,0000,,and 5/1 is mapped to four generators\Na major third of two octaves
Dialogue: 0,0:02:23.45,0:02:26.03,Default,,0000,0000,0000,,an interesting repercussion \Nof mappings like this one
Dialogue: 0,0:02:26.03,0:02:29.12,Default,,0000,0000,0000,,is that commas, small ratios close to 1/1,
Dialogue: 0,0:02:29.12,0:02:33.75,Default,,0000,0000,0000,,cease to exist in the system and are\Nequated with 1/1 or the unison.
Dialogue: 0,0:02:34.39,0:02:37.31,Default,,0000,0000,0000,,this is called "tempering\Nout that comma"
Dialogue: 0,0:02:37.31,0:02:41.32,Default,,0000,0000,0000,,in most mathematical contexts it would be\Ngibberish to equate
Dialogue: 0,0:02:41.32,0:02:45.98,Default,,0000,0000,0000,,any non 1/1 ratio with 1/1,\Nbut in tuning theory
Dialogue: 0,0:02:45.98,0:02:49.62,Default,,0000,0000,0000,,it is one of the most revolutionary\Nconcepts in recent history
Dialogue: 0,0:02:51.46,0:02:54.75,Default,,0000,0000,0000,,in meantone temperament\Nthe comma 81/80
Dialogue: 0,0:02:54.75,0:02:57.52,Default,,0000,0000,0000,,also known as the Syntonic comma,\Nis tempered out
Dialogue: 0,0:02:57.91,0:03:02.79,Default,,0000,0000,0000,,we can show this starting with how 5/1\Nis mapped to 4 generators
Dialogue: 0,0:03:03.08,0:03:07.65,Default,,0000,0000,0000,,that means\N3/2 to the fourth equals 5/1
Dialogue: 0,0:03:08.22,0:03:12.92,Default,,0000,0000,0000,,expanding 3/2 to the fourth\Nwe get 81/16 equals 5/1
Dialogue: 0,0:03:13.61,0:03:18.57,Default,,0000,0000,0000,,finding a common denominator\Ngives us 81/16 equals 80/16
Dialogue: 0,0:03:18.73,0:03:23.09,Default,,0000,0000,0000,,then we can multiply both sides by 16\Nand get 81 equals 80
Dialogue: 0,0:03:23.09,0:03:28.97,Default,,0000,0000,0000,,then finally dividing both sides by 80 \Ngives us 81/80 equals 1/1
Dialogue: 0,0:03:31.65,0:03:34.73,Default,,0000,0000,0000,,the meaning behind this seemingly\Nmeaningless statement
Dialogue: 0,0:03:34.73,0:03:38.54,Default,,0000,0000,0000,,is that in a tuning that tempers out 81/80
Dialogue: 0,0:03:38.54,0:03:41.62,Default,,0000,0000,0000,,all ratios that differ by that comma
Dialogue: 0,0:03:41.62,0:03:45.06,Default,,0000,0000,0000,,will be represented by the same interval
Dialogue: 0,0:03:45.06,0:03:47.77,Default,,0000,0000,0000,,this has two important implications
Dialogue: 0,0:03:47.77,0:03:52.46,Default,,0000,0000,0000,,one is that chord changes that\Nwould drift by 81/80 in just intonation
Dialogue: 0,0:03:52.46,0:03:55.37,Default,,0000,0000,0000,,now return to the same note you started on\N
Dialogue: 0,0:03:55.37,0:03:58.01,Default,,0000,0000,0000,,eliminating chromatic drift\Nfor that comma
Dialogue: 0,0:03:59.28,0:04:03.42,Default,,0000,0000,0000,,the second important implication\Nis that we are giving away to represent
Dialogue: 0,0:04:03.42,0:04:07.27,Default,,0000,0000,0000,,one dimension of just intonation\Nthe dimension of the prime 5
Dialogue: 0,0:04:07.27,0:04:10.62,Default,,0000,0000,0000,,in terms of two other primes - \N2 and 3
Dialogue: 0,0:04:11.39,0:04:14.89,Default,,0000,0000,0000,,in doing so we are approximating\Na three-dimensional structure
Dialogue: 0,0:04:14.89,0:04:18.91,Default,,0000,0000,0000,,with only two dimensions\Ntherefore lowering the complexity
Dialogue: 0,0:04:20.82,0:04:23.37,Default,,0000,0000,0000,,it is only an approximation however
Dialogue: 0,0:04:23.37,0:04:27.20,Default,,0000,0000,0000,,as you may notice\Nif we treat a pure 81/16
Dialogue: 0,0:04:27.20,0:04:30.01,Default,,0000,0000,0000,,as our 80/16 or 5/1
Dialogue: 0,0:04:30.01,0:04:34.61,Default,,0000,0000,0000,,by definition it'll be out of tune\Nwith respect to 5/1
Dialogue: 0,0:04:35.22,0:04:39.78,Default,,0000,0000,0000,,here is the 1-6-2-5 progression\Nusing pure fifths
Dialogue: 0,0:04:39.78,0:04:43.80,Default,,0000,0000,0000,,and using 81/64 as our 80/64
Dialogue: 0,0:04:58.95,0:05:01.89,Default,,0000,0000,0000,,but not to worry \Nas history already dictates
Dialogue: 0,0:05:01.89,0:05:03.96,Default,,0000,0000,0000,,if we temper all 3/2's in the system, \N
Dialogue: 0,0:05:03.96,0:05:06.49,Default,,0000,0000,0000,,that is to say \Nregularly temper them, \N
Dialogue: 0,0:05:06.49,0:05:10.71,Default,,0000,0000,0000,,then stacking four generators \Narrives at a much pure 5/1
Dialogue: 0,0:05:11.49,0:05:14.93,Default,,0000,0000,0000,,here is that same chord progression in\Nseveral different meantone tunings
Dialogue: 0,0:05:14.93,0:05:18.40,Default,,0000,0000,0000,,that form a compromise between the\Nratios of 3
Dialogue: 0,0:05:18.40,0:05:20.71,Default,,0000,0000,0000,,and ratios of 5
Dialogue: 0,0:05:56.09,0:05:59.76,Default,,0000,0000,0000,,in addition\Nwe could slightly temper 2/1 - the octave
Dialogue: 0,0:05:59.76,0:06:01.83,Default,,0000,0000,0000,,to reach a compromise and purity between
Dialogue: 0,0:06:01.83,0:06:03.73,Default,,0000,0000,0000,,all three primes we're representing
Dialogue: 0,0:06:03.73,0:06:08.31,Default,,0000,0000,0000,,doing so can result in a remarkably good \Napproximation of just intonation
Dialogue: 0,0:06:08.31,0:06:11.82,Default,,0000,0000,0000,,for such a simple two-dimensional\Nstructure
Dialogue: 0,0:06:11.82,0:06:14.53,Default,,0000,0000,0000,,here's the 1-6-2-5 chord progression
Dialogue: 0,0:06:14.53,0:06:17.35,Default,,0000,0000,0000,,tuned to an optimized meantone tuning
Dialogue: 0,0:06:31.40,0:06:33.92,Default,,0000,0000,0000,,as we'll find out in the next video
Dialogue: 0,0:06:33.92,0:06:37.07,Default,,0000,0000,0000,,the use of generators \Nto define a tuning's intervals
Dialogue: 0,0:06:37.07,0:06:40.17,Default,,0000,0000,0000,,has additional benefits\Nnot related to harmonic purity
Dialogue: 0,0:06:40.17,0:06:44.00,Default,,0000,0000,0000,,or chromatic drift