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