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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.

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.