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What is a Moment of Symmetry Scale (MOS)? - Draft
Answer:
The most consonant interval is the octave.
Ex.. a to a', 440 Hz to 880 Hz = double the frequency. As a ratio:
2/1.
Next most consonant interval is 3/2, e.g 440 Hz to 660 Hz.
All cultures recognise the octave as a consonance, and the 3/2 is also
universally recognised, in fact, often when a group of people sing a
song, some of those who are musically untrained may sing at an interval
of 3/2 instead of 2/1, mistaking it for an octave, giving motion in
parallel 3/2s.
Now, to build a MOS scale, start with two notes at a ratio of 3/2,
such as our 440 Hz and 660 Hz.
Keep adding new notes at an interval of 3/2 above the previous note,
and reduce by an octave (halve the frequency) if it goes above 880 Hz.
If you carry through this construction, you will find that usually
there are three step sizes in the scale, but there is a" moment of
symmetry" at five notes, when the scale has only two step sizes.
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There are more moments of symmetry at 41, then 53 notes, ...
Scales built in this way from 3/2s are known as" Pythagorean".
If you go up two notes in any of the scales, again one has two
interval sizes. E.g. in the diatonic scale, the interval will either be
a major or a minor third (A to C = minor third, C to E = major third in
C major).
In fact, you have two sizes of intervals for all the numbers of steps.
A scale with this property is known as a Myhill scale. All MOS scales
are Myhill, e.g. in the pentatonic scale the two step interval is
either a major third (C to E say) or a fourth (D to G say), and so on.
In the same way, the Pythagorean twelve tone has two sizes of
semitone, two sizes of tone, two sizes of minor third, two sizes of
major third, and so on.
The Pythagorean twelve tone major thirds are 81/64 and 8192/6561.
Actually the sweetest major third you can play is a 5/4. So, in early
times (especially 17th century and earlier), it was common to temper
the scales by reducing the size of the 3/2 to get pure 5/4s.
Tempering to give a pure 5/4 gives the scale known as the quarter
comma meantone. Tempering to give a pure minor third at 6/5 gives the
third comma meantone scale.
Comma here = syntonic comma = the ratio between the pythagorean 81/64
major third and the sweet" just intontation" 5/4. It is 81/80.
In the construction, four 3/2s get you from 1/1 to 81/64, so to get a
pure 5/4, you need to temper each one a little flat, by a quarter of
the syntonic comma.
Quarter here means a quarter of the value of the comma in cents, not a
quarter of its value as a ratio.
Three 3/2s get you to 27/16, which is a pythagorean minor third below
the octave at 2/1. To get pure minor thirds, one wants it to be at 5/3,
i.e. a pure minor third below the octave.
You will find that you ned to divide by 81/80 again to do this. Since
there are three 3/2s this time, you need to temper each one by a third
of the syntonic comma.
This makes sense since 1/1 5/4 3/2 gives a pure triad with the minor
third between the 5/4 and the 3/2. So if you need to flatten the
pythagorean major third by a comma to get the sweet j.i. triad, then
you need to sharpen the minor third by the same amount.
These scales all have sharp fifths (flat by 5.38 cents for the quarter
comma meantone 3/2). A musician will be able to hear this, but they are
still reasonably consonant and perfectly usable.
However, at a moment of symmetry (MOS), all the intervals come in two
sizes, so corresponding to the pure 3/2 is a wolf fifth, which for the
Pythagorean scales is 678.495 cents instead of the pure 3/2 at 701.955
cents - this is very noticeably flat. For the quarter comma meantone,
the wolf fifth is very sharp rather than flat, 737.647 cents.
Luckily there is only one wolf fifth. However, one in three of the
major thirds is in the alternative interval size, and one in four of
the minor thirds is. Unfortunately, in quarter comma meantone, the wolf
fifth also makes a triad with one of the impure major thirds. This
impure triad has wild beating, and is completely unusable for a chord
intended to sound as a moment of consonance and rest. So the effect is
that in quarter comma meantone, one of the twelve scales can't be used
in the normal way at all. At the time, this was thought a small price
to pay, as music didn't tend to modulate to distant keys anyway.
Quarter comma meantone gradually fell out of fashion from about the
time of Bach onwards. Musicians came to relish the ability to modulate
to distant keys more and more. Quarter comma meantone continued to be
used for church organs for a long time.
In its place, various scales were used that temper the 3/2 in the
opposite direction, flat rather than sharp. This time the aim is to
facilitate modulation, and to keep the 3/2s as pure as one can in most
of the scales. Such scales are known as well tempered scales.
The sweet 5/4 was no longer targetted - by this time musicians had got
used to having sharp major thirds, and came to accept them as a
consonance, and even to like them.
The modern tempering in which all semitones are equal in size is
actually, surprisingly, a very early development, and was in use for
lutes. Equal temperament major thirds sounds much sweeter on a lute
than on a harpsichord, because of the harpsichord's prominent fifth
partial. Also the lute is easier to tune to equal temperament by
geometrical positioning of the frets. Tuning a keyboard to equal
temperament is far harder, and in fact at the time, the best method to
do so would perhaps be to tune a lute, then tune the keyboard to it.
When one tempers the 3/2, then the positions of the moments of
symmetry change. So, for some temperings, one will get a MOS at 19
notes and 31 notes instead of 17 and 21.
If you temper the 3/2 to 696.774 cents then you get 31 tone equal
temperament as a MOS - with all thirty one notes equal in size. This is
a particularly popular scale amongst microtonalists, because it has
relatively sweet major and minor thirds.
19 tone equal temperament is also popular, and has especially nice
approximations to the septimal (or bluesy) minor third of 7/6.
In 19 tone scales of this type, one makes a distinction between sharps
and flats, with Eb> D#. There is a sharp key for E and B too, with E# =
Fb and B# = Cb.
You also get 17 tone scales - these have a distinction between sharps
and flats, but E# = F and B# = C.
In 31 note scales, one uses half sharps and flats, giving six notes
between C and D : C, C half sharp, C sharp, D flat, D half flat and D.
The pythagorean 17 note scale can be notated using sharps and flats
too, - it can be presented as a twelve note scale with each of the
accidentals doubled. This time, the C, again one has a distinction
between sharps and flats, but this time, the Db is flatter than the C#.
It is tuned so that the interval Db to F is a Pythagorean major third
at 81/64, and the interval A to C# is also an 81/64, which places the
C# above the Db.
So, some musicians think of C# as sharper than Db, while others think
of Db as sharper than C#. Perhaps it is more common to use the 19 or 17
tone type convention where the order of the notes is C C# Db D. But, if
playing in Pythagorean intonation, one would make the order of the
notes C Db C# D.
Pythagorean intonation is one of the natural tunings that can be used
on strings - if the open strings of a quartet are tuned to perfect
3/2s, then the high E of the violin will be a Pythagorean major third
(plus some octaves) above the low C of the 'cello.
The fifth harmonic of the 'cello C (which one can easily sound by
touching the string lightly one fifth of the way across while playing
the open string) will then be flatter than the E string of the violin,
by 21.5063 cents, quite a large amount really, and one then hears the
81/80 syntonic comma in action!
So, it would be quite natural for a string quartet to fall into
Pythagorean intonation, and one would then find that the order of the
notes was C Db C# D.