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in addition to providing us with an
efficient system for approximating JI

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two-dimensional regular
temperaments also generate scales with

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very specific and useful melodic properties
these scales are called

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"moment of symmetry scales"
a moment of symmetry scale or MoS for short

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is defined by having exactly two different
intervals that span one step in the scale

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one large, one small and by being
derivative of a continuous chain of generators

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that is the same way the
diatonic scale contains only a major and

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minor second and can be derived from
seven continuous fifths repeated within

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the octave, all MoS scales must have only
two types of second and they must be

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derivative of a continuous chain of some
generator repeated within some .

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since it has those defining properties
the diatonic scale is an MoS. as it turns

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out our other standard Western scales
are also moment of symmetry scales

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including the pentatonic scale whose two
step sizes are the major second in the

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minor 3rd and the chromatic scale whose
to step sizes are the minor second and

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the Augmented unison. both derived from a
continuous chains of fifths from the

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circle of fifths. so we don't know for
sure why historically we arrived so

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naturally at MoS scales we can use that
fact to theorize that other scales that

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share similar properties would be
similarly fruitful. MoS scales are

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categorized by their number of large and
small seconds. for example the diatonic scale

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scale would be prefer to as the 5
large and 2 small MoS, because it has five

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major seconds and two minor seconds
each MoS has what is referred to as a valid

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tuning range. the range of generator
sizes over which that MoS is generated

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as you vary the size of the generator
across that range the large and small

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steps vary in size compared to each other
at one extreme a small steps

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shrink towards being non-existent, while
the large steps take over the whole

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scale at which point you reach an equal
temperament equal to the number of large steps

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00:02:10,699 --> 00:02:15,500
at the other extreme the small steps
get larger and a large steps

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shrink until both are the same size at
which point you reach an equal

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00:02:19,580 --> 00:02:22,850
temperament equal to the number of total
steps in the scale

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outside of those parameters the scales
generated with no longer be of the same

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MoS structure. for example the diatonic
scale a.k.a. the 5 large 2 small MoS is

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valid with the generator ranging from
approximately 685 cents

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to 720 cents. in the middle of
these two around 700 cents you arrive at

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00:02:46,979 --> 00:02:51,539
scales or the scale structure is
obviously LLsLLLs

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00:02:51,539 --> 00:02:57,719
as you approach 720 cents
however the small steps shrink towards

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00:02:57,719 --> 00:03:01,740
nothing until the minor second
disappears completely and you get 5-tet

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00:03:01,740 --> 00:03:05,879
here is what that progression in generator size

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00:03:05,879 --> 00:03:08,650
sounds like

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00:03:59,909 --> 00:04:05,988
conversely if your generator approaches
685 cents your major and minor seconds

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00:04:05,989 --> 00:04:09,829
approach each other in size until
they're equal at which point you have 7-tet

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00:04:09,829 --> 00:04:13,790
here's what that progression in generator size

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00:04:13,789 --> 00:04:16,789
sounds like

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00:04:58,050 --> 00:05:02,490
the sizes and structures of the
different MoS scales you reach given a

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period and a generator combination are
determined by the ratio in size between

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those two intervals. though the
intricacies of that relationship are

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beyond the scope of this video. to put it
as simply as possible the number of

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notes in each successive MoS scale a.k.a.
their cardinalities follow a Fibonacci

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ESCA pattern or each new scales
cardinality is an addition of two

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00:05:27,089 --> 00:05:31,979
previous cardinalities. also the number
of large and small steps in your new

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00:05:31,980 --> 00:05:38,670
scale will match those two previous
cardinalities. for example when using a

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pure 3/2 and a pure 2/1
approximation 702 cents and 1200 cents

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as our generator and period we have MoS
scales after one note, which in this case

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is just octaves, then at two notes a
scale of one large one small

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a perfect fifth and a perfect fourth

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00:05:55,980 --> 00:06:00,750
our next MoS has reached at three
notes a.k.a. 2+1 giving us

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00:06:00,750 --> 00:06:05,970
2L1s MoS scale including two
perfect fourth and a major second

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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

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00:06:11,639 --> 00:06:17,339
which is our pentatonic scale
the next MoS is reached seven notes just

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00:06:17,339 --> 00:06:22,979
5+2 giving us a 5L 2s MoS scale
which is our diatonic scale

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00:06:22,980 --> 00:06:28,590
and then 12 notes - 7+5 giving us a 5L 7s MoS scale
and uneven

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00:06:28,589 --> 00:06:32,849
chromatic scale. after that the bag
pattern continues onto 17 notes,

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29 notes etc. if you generate with the
pure 6/5 and a 2/1 however you

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arrive at a different combination of MoS
scales that they follow a similar

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pattern. they would occur at one note,
two notes, three notes (just 2+1), 4 notes

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(3+1), 7 notes (3+4), 11 notes (7+4)
15 notes (11+4) and so on

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00:07:00,839 --> 00:07:04,409
the closer to generators are in size the
more scale structures they'll have in

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common before they diverge. for example
when generating scales using a flatted 3/2

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or 696 cents, as often happens in
mintone temperament, we arrived at the

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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

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00:07:18,839 --> 00:07:22,889
notes scale which will instead have
seven large steps and five small steps

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making it an uneven chromatic scale with
a reverse structure

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that scale is followed by MoS scales of
19 notes (a.k.a. 12+7), 26 notes (19+7)

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and then the pattern diverges from then on
here is what several alternative MoS

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scale structure sound like

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00:08:25,288 --> 00:08:31,188
those are just several examples among an
infinite number of MoS scales which are

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theoretically ripe for exploration.
but how on earth should one go about playing

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these new musical structures if they can
differ on such a basic level from our

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familiar scales? certainly many of them
would not be playable on traditional

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musical instruments. next video I'll be
covering the most ergonomic way to

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represent these musical structures
physically for musical performance