Talk:Symmetric scale

I thought I'd comment on this, while the thoughts occur to me. I think this is a topic where there is considerable potential for confusion. Most of what I will say cannot be referenced, so it cannot go into the main article, and might even be "original research" - but it may help others get a bit of clarity on the topic as a whole.

This illustrates a problem I feel exists with the term "symmetrical" as applied to scales. I read Vincent Persichetti's book probably some time in the 1970s, and was quite impressed by the theoretical aspects of modern harmony it discussed, and it prompted me to do much thinking and theorizing about various scales. And at some point I came to realize that there were two quite different ways a scale could be regarded as symmetrical, which shouldn't be confused with each other, and that therefore this term was ambiguous and would better not be used. Instead, I started for myself the practice of calling the two types of scale periodic and self-reflective, thus avoiding the ambiguity. I only wish this usage were standard, but unfortunately it is not, and only I use it, to my knowledge. (I doggedly do so though, explaining the terms to others if necessary, since I value clarity and unambiguity more than I do conformity to established standards.)

Periodic scales are ones which divide the octave into two or more equal portions, and fill each portion with the same sequence of intervals exactly. These are the type described at the beginning of the article. Regarding Messiaen's 7 "Modes" of Limited Transposition (scales, really - modes are what you get when you start a scale on one or other of its particular notes) - these scales are all of of this type, although there are several others he didn't list for some reason. (All of the others can be found as *subsets*, though, of one or other of the 7 scales he did list.) I would therefore classify them all as periodic scales.

Reflective scales are ones which can be inverted (turned upside down), and, if the right point is chosen around which to invert the scale, the same set of notes will result, in the same key. For instance, the diatonic scale of C major is reflective: if you choose either a D or a G#/Ab and regard that as a mirror through which to reflect the scale, the same scale in the same key will result. (In detail, C and E reflect to each other; E; D reflects to itself (or another D an octave up or down); F and B reflect to each other; and G and A reflect to each other. If you choose the *wrong* point around which to reflect the scale, the reflection will still yield the same scale, but in a different key. (Reflecting a C major scale around the note C will give the Phrygian mode of the A-flat major scale, for example.) Depending on the exact scale in question, a proper reflecting point can sometimes be the quarter-tone point *between* two adjacent semitones. (Reflecting the C major scale around the quarter-tone between C and C# will give the Phrygian mode of the A major scale.) These are the type described towards the end of the article as having "inversional symmetry".

That this self-reflective quality is a different concept to the periodic scale as I described it above can be seen instantly by realizing that the diatonic scale is *not* periodic - and it is impossible for *any* 7-note scale to be periodic, because 7, as a prime number, cannot be divided into two or more equally-sized portions (only 7 - but you can't have an octave divided into 7 equal intervals in a 12-note chromatic scale).

Periodic scales are quite rare - there are only 17 of them (18 if you count a theoretical zero-note scale which would mathematically count as periodic), out of the 351 different possible scales of all sorts (not counting different *modes* of the same scale, nor simple transpositions to a different key). Messiaen's scales are 7 out of these 17.

Self-reflective scales are a minority of the total 351, but quite a large minority. I forget the exact number: I worked it out exactly some years ago, but don't recall - I think it was in the region of several dozen, maybe even close to a hundred. Most periodic scales are self-reflective, but not all. (Just two are not: the "Tritone major" (C Db E F# G Bb C) and the "Tritone minor" (C Db Eb F# G A C). (Those two names are probably just my own: I am not aware of any standard terms for these two scales.)

I find the term "asymmetric(al) scale" quite unhelpful - downright confusing, in fact - since it is not clear to me which type of symmetry is negated. Given that the two types of symmetry (periodic and self-reflective) are independent of each other, and can be present with or without the other, the two types of asymmetry can be independent of each other also. That is, a scale can be asymmetric in just one of the two senses, or in both of them.

The scale cited by ArdClose (C Db Eb F G# A Bb C) is not symmetrical in either of these senses: neither periodic nor self-reflective. I suspect he or she believed it to be symmetrical because of the balanced arrangement of HWW on the left side of the 3 placed in the centre, with HHW on the other side. But if W represents a whole-tone and H a half-tone (which I assume is the intended meaning), then, because those are different intervals, they do not indicate symmetry in either of the senses I just described. Different intervals cannot be viewed as mirroring to each other over a reflecting point, even if there are equal numbers of them on either side. I considered this scale for both periodicity and self-reflectivity, and it meets neither standard. I felt that even considering it within the realm of symmetrical scales as defined in this article would just muddy things up even further. And this thought was what prompted me to say any of this in the first place.