WEBVTT

00:00:00.000 --> 00:00:04.559
in addition to providing us with an
efficient system for approximating JI

00:00:04.559 --> 00:00:09.119
two-dimensional regular
temperaments also generate scales with

00:00:09.119 --> 00:00:14.250
very specific and useful melodic properties
these scales are called

00:00:14.250 --> 00:00:21.420
"moment of symmetry scales"
a moment of symmetry scale or MoS for short

00:00:21.420 --> 00:00:25.679
is defined by having exactly two different
intervals that span one step in the scale

00:00:25.679 --> 00:00:31.259
one large, one small and by being
derivative of a continuous chain of generators

00:00:31.260 --> 00:00:36.630
that is the same way the
diatonic scale contains only a major and

00:00:36.630 --> 00:00:41.399
minor second and can be derived from
seven continuous fifths repeated within

00:00:41.399 --> 00:00:46.530
the octave, all MoS scales must have only
two types of second and they must be

00:00:46.530 --> 00:00:51.239
derivative of a continuous chain of some
generator repeated within some .

00:00:52.880 --> 00:00:58.760
since it has those defining properties
the diatonic scale is an MoS. as it turns

00:00:58.759 --> 00:01:02.780
out our other standard Western scales
are also moment of symmetry scales

00:01:02.780 --> 00:01:07.310
including the pentatonic scale whose two
step sizes are the major second in the

00:01:07.310 --> 00:01:12.230
minor 3rd and the chromatic scale whose
to step sizes are the minor second and

00:01:12.230 --> 00:01:16.850
the Augmented unison. both derived from a
continuous chains of fifths from the

00:01:16.849 --> 00:01:21.679
circle of fifths. so we don't know for
sure why historically we arrived so

00:01:21.680 --> 00:01:27.050
naturally at MoS scales we can use that
fact to theorize that other scales that

00:01:27.049 --> 00:01:32.810
share similar properties would be
similarly fruitful. MoS scales are

00:01:32.810 --> 00:01:38.450
categorized by their number of large and
small seconds. for example the diatonic scale

00:01:38.450 --> 00:01:42.978
scale would be prefer to as the 5
large and 2 small MoS, because it has five

00:01:42.978 --> 00:01:48.379
major seconds and two minor seconds
each MoS has what is referred to as a valid

00:01:48.379 --> 00:01:54.409
tuning range. the range of generator
sizes over which that MoS is generated

00:01:54.409 --> 00:01:58.099
as you vary the size of the generator
across that range the large and small

00:01:58.099 --> 00:02:03.169
steps vary in size compared to each other
at one extreme a small steps

00:02:03.170 --> 00:02:06.890
shrink towards being non-existent, while
the large steps take over the whole

00:02:06.890 --> 00:02:10.699
scale at which point you reach an equal
temperament equal to the number of large steps

00:02:10.699 --> 00:02:15.500
at the other extreme the small steps
get larger and a large steps

00:02:15.500 --> 00:02:19.580
shrink until both are the same size at
which point you reach an equal

00:02:19.580 --> 00:02:22.850
temperament equal to the number of total
steps in the scale

00:02:23.460 --> 00:02:28.590
outside of those parameters the scales
generated with no longer be of the same

00:02:28.590 --> 00:02:37.890
MoS structure. for example the diatonic
scale a.k.a. the 5 large 2 small MoS is

00:02:37.889 --> 00:02:40.859
valid with the generator ranging from
approximately 685 cents

00:02:40.860 --> 00:02:46.980
to 720 cents. in the middle of
these two around 700 cents you arrive at

00:02:46.979 --> 00:02:51.539
scales or the scale structure is
obviously LLsLLLs

00:02:51.539 --> 00:02:57.719
as you approach 720 cents
however the small steps shrink towards

00:02:57.719 --> 00:03:01.740
nothing until the minor second
disappears completely and you get 5-tet

00:03:01.740 --> 00:03:05.879
here is what that progression in generator size

00:03:05.879 --> 00:03:08.650
sounds like

00:03:59.909 --> 00:04:05.988
conversely if your generator approaches
685 cents your major and minor seconds

00:04:05.989 --> 00:04:09.829
approach each other in size until
they're equal at which point you have 7-tet

00:04:09.829 --> 00:04:13.790
here's what that progression in generator size

00:04:13.789 --> 00:04:16.789
sounds like

00:04:58.050 --> 00:05:02.490
the sizes and structures of the
different MoS scales you reach given a

00:05:02.490 --> 00:05:06.810
period and a generator combination are
determined by the ratio in size between

00:05:06.810 --> 00:05:10.889
those two intervals. though the
intricacies of that relationship are

00:05:10.889 --> 00:05:16.379
beyond the scope of this video. to put it
as simply as possible the number of

00:05:16.379 --> 00:05:22.439
notes in each successive MoS scale a.k.a.
their cardinalities follow a Fibonacci

00:05:22.439 --> 00:05:27.089
ESCA pattern or each new scales
cardinality is an addition of two

00:05:27.089 --> 00:05:31.979
previous cardinalities. also the number
of large and small steps in your new

00:05:31.980 --> 00:05:38.670
scale will match those two previous
cardinalities. for example when using a

00:05:38.670 --> 00:05:43.620
pure 3/2 and a pure 2/1
approximation 702 cents and 1200 cents

00:05:43.620 --> 00:05:49.019
as our generator and period we have MoS
scales after one note, which in this case

00:05:49.019 --> 00:05:54.269
is just octaves, then at two notes a
scale of one large one small

00:05:54.269 --> 00:05:55.979
a perfect fifth and a perfect fourth

00:05:55.980 --> 00:06:00.750
our next MoS has reached at three
notes a.k.a. 2+1 giving us

00:06:00.750 --> 00:06:05.970
2L1s MoS scale including two
perfect fourth and a major second

00:06:05.970 --> 00:06:11.640
after that we had an MoS of five notes a.k.a.
2+3 giving us a 2L3s MoS scale

00:06:11.639 --> 00:06:17.339
which is our pentatonic scale
the next MoS is reached seven notes just

00:06:17.339 --> 00:06:22.979
5+2 giving us a 5L 2s MoS scale
which is our diatonic scale

00:06:22.980 --> 00:06:28.590
and then 12 notes - 7+5 giving us a 5L 7s MoS scale
and uneven

00:06:28.589 --> 00:06:32.849
chromatic scale. after that the bag
pattern continues onto 17 notes,

00:06:32.850 --> 00:06:40.080
29 notes etc. if you generate with the
pure 6/5 and a 2/1 however you

00:06:40.079 --> 00:06:43.740
arrive at a different combination of MoS
scales that they follow a similar

00:06:43.740 --> 00:06:50.460
pattern. they would occur at one note,
two notes, three notes (just 2+1), 4 notes

00:06:50.459 --> 00:07:00.839
(3+1), 7 notes (3+4), 11 notes (7+4)
15 notes (11+4) and so on

00:07:00.839 --> 00:07:04.409
the closer to generators are in size the
more scale structures they'll have in

00:07:04.410 --> 00:07:09.000
common before they diverge. for example
when generating scales using a flatted 3/2

00:07:09.000 --> 00:07:14.189
or 696 cents, as often happens in
mintone temperament, we arrived at the

00:07:14.189 --> 00:07:18.839
same MoS scales as when generating with
the pure 3/2 until we get to the 12

00:07:18.839 --> 00:07:22.889
notes scale which will instead have
seven large steps and five small steps

00:07:22.889 --> 00:07:26.339
making it an uneven chromatic scale with
a reverse structure

00:07:26.949 --> 00:07:34.509
that scale is followed by MoS scales of
19 notes (a.k.a. 12+7), 26 notes (19+7)

00:07:34.509 --> 00:07:40.240
and then the pattern diverges from then on
here is what several alternative MoS

00:07:40.240 --> 00:07:43.240
scale structure sound like

00:08:25.288 --> 00:08:31.188
those are just several examples among an
infinite number of MoS scales which are

00:08:31.189 --> 00:08:35.958
theoretically ripe for exploration.
but how on earth should one go about playing

00:08:35.957 --> 00:08:39.467
these new musical structures if they can
differ on such a basic level from our

00:08:39.469 --> 00:08:43.490
familiar scales? certainly many of them
would not be playable on traditional

00:08:43.490 --> 00:08:48.110
musical instruments. next video I'll be
covering the most ergonomic way to

00:08:48.110 --> 00:08:51.560
represent these musical structures
physically for musical performance