0:00:00.000,0:00:04.559
in addition to providing us with an[br]efficient system for approximating JI

0:00:04.559,0:00:09.119
two-dimensional regular[br]temperaments also generate scales with

0:00:09.119,0:00:14.250
very specific and useful melodic properties[br]these scales are called

0:00:14.250,0:00:21.420
"moment of symmetry scales"[br]a moment of symmetry scale or MoS for short

0:00:21.420,0:00:25.679
is defined by having exactly two different[br]intervals that span one step in the scale

0:00:25.679,0:00:31.259
one large, one small and by being[br]derivative of a continuous chain of generators

0:00:31.260,0:00:36.630
that is the same way the[br]diatonic scale contains only a major and

0:00:36.630,0:00:41.399
minor second and can be derived from[br]seven continuous fifths repeated within

0:00:41.399,0:00:46.530
the octave, all MoS scales must have only[br]two types of second and they must be

0:00:46.530,0:00:51.239
derivative of a continuous chain of some[br]generator repeated within some .

0:00:52.880,0:00:58.760
since it has those defining properties[br]the diatonic scale is an MoS. as it turns

0:00:58.759,0:01:02.780
out our other standard Western scales[br]are also moment of symmetry scales

0:01:02.780,0:01:07.310
including the pentatonic scale whose two[br]step sizes are the major second in the

0:01:07.310,0:01:12.230
minor 3rd and the chromatic scale whose[br]to step sizes are the minor second and

0:01:12.230,0:01:16.850
the Augmented unison. both derived from a[br]continuous chains of fifths from the

0:01:16.849,0:01:21.679
circle of fifths. so we don't know for[br]sure why historically we arrived so

0:01:21.680,0:01:27.050
naturally at MoS scales we can use that[br]fact to theorize that other scales that

0:01:27.049,0:01:32.810
share similar properties would be[br]similarly fruitful. MoS scales are

0:01:32.810,0:01:38.450
categorized by their number of large and[br]small seconds. for example the diatonic scale

0:01:38.450,0:01:42.978
scale would be prefer to as the 5[br]large and 2 small MoS, because it has five

0:01:42.978,0:01:48.379
major seconds and two minor seconds[br]each MoS has what is referred to as a valid

0:01:48.379,0:01:54.409
tuning range. the range of generator[br]sizes over which that MoS is generated

0:01:54.409,0:01:58.099
as you vary the size of the generator[br]across that range the large and small

0:01:58.099,0:02:03.169
steps vary in size compared to each other[br]at one extreme a small steps

0:02:03.170,0:02:06.890
shrink towards being non-existent, while[br]the large steps take over the whole

0:02:06.890,0:02:10.699
scale at which point you reach an equal[br]temperament equal to the number of large steps

0:02:10.699,0:02:15.500
at the other extreme the small steps[br]get larger and a large steps

0:02:15.500,0:02:19.580
shrink until both are the same size at[br]which point you reach an equal

0:02:19.580,0:02:22.850
temperament equal to the number of total[br]steps in the scale

0:02:23.460,0:02:28.590
outside of those parameters the scales[br]generated with no longer be of the same

0:02:28.590,0:02:37.890
MoS structure. for example the diatonic[br]scale a.k.a. the 5 large 2 small MoS is

0:02:37.889,0:02:40.859
valid with the generator ranging from[br]approximately 685 cents

0:02:40.860,0:02:46.980
to 720 cents. in the middle of[br]these two around 700 cents you arrive at

0:02:46.979,0:02:51.539
scales or the scale structure is[br]obviously LLsLLLs

0:02:51.539,0:02:57.719
as you approach 720 cents[br]however the small steps shrink towards

0:02:57.719,0:03:01.740
nothing until the minor second[br]disappears completely and you get 5-tet

0:03:01.740,0:03:05.879
here is what that progression in generator size

0:03:05.879,0:03:08.650
sounds like

0:03:59.909,0:04:05.988
conversely if your generator approaches[br]685 cents your major and minor seconds

0:04:05.989,0:04:09.829
approach each other in size until[br]they're equal at which point you have 7-tet

0:04:09.829,0:04:13.790
here's what that progression in generator size

0:04:13.789,0:04:16.789
sounds like

0:04:58.050,0:05:02.490
the sizes and structures of the[br]different MoS scales you reach given a

0:05:02.490,0:05:06.810
period and a generator combination are[br]determined by the ratio in size between

0:05:06.810,0:05:10.889
those two intervals. though the[br]intricacies of that relationship are

0:05:10.889,0:05:16.379
beyond the scope of this video. to put it[br]as simply as possible the number of

0:05:16.379,0:05:22.439
notes in each successive MoS scale a.k.a.[br]their cardinalities follow a Fibonacci

0:05:22.439,0:05:27.089
ESCA pattern or each new scales[br]cardinality is an addition of two

0:05:27.089,0:05:31.979
previous cardinalities. also the number[br]of large and small steps in your new

0:05:31.980,0:05:38.670
scale will match those two previous[br]cardinalities. for example when using a

0:05:38.670,0:05:43.620
pure 3/2 and a pure 2/1[br]approximation 702 cents and 1200 cents

0:05:43.620,0:05:49.019
as our generator and period we have MoS[br]scales after one note, which in this case

0:05:49.019,0:05:54.269
is just octaves, then at two notes a[br]scale of one large one small

0:05:54.269,0:05:55.979
a perfect fifth and a perfect fourth

0:05:55.980,0:06:00.750
our next MoS has reached at three[br]notes a.k.a. 2+1 giving us

0:06:00.750,0:06:05.970
2L1s MoS scale including two[br]perfect fourth and a major second

0:06:05.970,0:06:11.640
after that we had an MoS of five notes a.k.a.[br]2+3 giving us a 2L3s MoS scale

0:06:11.639,0:06:17.339
which is our pentatonic scale[br]the next MoS is reached seven notes just

0:06:17.339,0:06:22.979
5+2 giving us a 5L 2s MoS scale[br]which is our diatonic scale

0:06:22.980,0:06:28.590
and then 12 notes - 7+5 giving us a 5L 7s MoS scale[br]and uneven

0:06:28.589,0:06:32.849
chromatic scale. after that the bag[br]pattern continues onto 17 notes,

0:06:32.850,0:06:40.080
29 notes etc. if you generate with the[br]pure 6/5 and a 2/1 however you

0:06:40.079,0:06:43.740
arrive at a different combination of MoS[br]scales that they follow a similar

0:06:43.740,0:06:50.460
pattern. they would occur at one note,[br]two notes, three notes (just 2+1), 4 notes

0:06:50.459,0:07:00.839
(3+1), 7 notes (3+4), 11 notes (7+4)[br]15 notes (11+4) and so on

0:07:00.839,0:07:04.409
the closer to generators are in size the[br]more scale structures they'll have in

0:07:04.410,0:07:09.000
common before they diverge. for example[br]when generating scales using a flatted 3/2

0:07:09.000,0:07:14.189
or 696 cents, as often happens in[br]mintone temperament, we arrived at the

0:07:14.189,0:07:18.839
same MoS scales as when generating with[br]the pure 3/2 until we get to the 12

0:07:18.839,0:07:22.889
notes scale which will instead have[br]seven large steps and five small steps

0:07:22.889,0:07:26.339
making it an uneven chromatic scale with[br]a reverse structure

0:07:26.949,0:07:34.509
that scale is followed by MoS scales of[br]19 notes (a.k.a. 12+7), 26 notes (19+7)

0:07:34.509,0:07:40.240
and then the pattern diverges from then on[br]here is what several alternative MoS

0:07:40.240,0:07:43.240
scale structure sound like

0:08:25.288,0:08:31.188
those are just several examples among an[br]infinite number of MoS scales which are

0:08:31.189,0:08:35.958
theoretically ripe for exploration.[br]but how on earth should one go about playing

0:08:35.957,0:08:39.467
these new musical structures if they can[br]differ on such a basic level from our

0:08:39.469,0:08:43.490
familiar scales? certainly many of them[br]would not be playable on traditional

0:08:43.490,0:08:48.110
musical instruments. next video I'll be[br]covering the most ergonomic way to

0:08:48.110,0:08:51.560
represent these musical structures[br]physically for musical performance