Tuning Theory 3: Moment of Symmetry ("Microtonal" Theory)

Up until now we've talked about tuning intervals so that they approximate Just Intonation to some degree. But as I hope this video will show, a scale's melodic resources are AT LEAST as important as its harmonic resources.

Here are the previous videos in this series:

Tuning Theory 0: A Primer
https://www.youtube.com/watch?v=DB2aHGW45fY

Tuning Theory 1: Just Intonation
https://www.youtube.com/watch?v=S7YRYm-trAs

Tuning Theory 2: Temperament
https://www.youtube.com/watch?v=ZoAuVgndmbU

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

two-dimensional regular
temperaments also generate scales with

very specific and useful melodic properties
these scales are called

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

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

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

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

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

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

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

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

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

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

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

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

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

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

share similar properties would be
similarly fruitful. MoS scales are

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

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

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

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

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

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

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

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

at the other extreme the small steps
get larger and a large steps

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

temperament equal to the number of total
steps in the scale

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

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

valid with the generator ranging from
approximately 685 cents

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

scales or the scale structure is
obviously LLsLLLs

as you approach 720 cents
however the small steps shrink towards

nothing until the minor second
disappears completely and you get 5-tet

here is what that progression in generator size

sounds like

conversely if your generator approaches
685 cents your major and minor seconds

approach each other in size until
they're equal at which point you have 7-tet

here's what that progression in generator size

sounds like

the sizes and structures of the
different MoS scales you reach given a

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

those two intervals. though the
intricacies of that relationship are

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

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

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

previous cardinalities. also the number
of large and small steps in your new

scale will match those two previous
cardinalities. for example when using a

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

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

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

a perfect fifth and a perfect fourth

our next MoS has reached at three
notes a.k.a. 2+1 giving us

2L1s MoS scale including two
perfect fourth and a major second

after that we had an MoS of five notes a.k.a.
2+3 giving us a 2L3s MoS scale

which is our pentatonic scale
the next MoS is reached seven notes just

5+2 giving us a 5L 2s MoS scale
which is our diatonic scale

and then 12 notes - 7+5 giving us a 5L 7s MoS scale
and uneven

chromatic scale. after that the bag
pattern continues onto 17 notes,

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

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

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

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

the closer to generators are in size the
more scale structures they'll have in

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

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

same MoS scales as when generating with
the pure 3/2 until we get to the 12

notes scale which will instead have
seven large steps and five small steps

making it an uneven chromatic scale with
a reverse structure

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

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

scale structure sound like

those are just several examples among an
infinite number of MoS scales which are

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

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

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

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

represent these musical structures
physically for musical performance