We can also notice that (3/2)2 = 9/4, and due to octave equivalence (3/2)2 = 9/8. Secondarily, we notice that (5/4) / (9/8) = 10/9. 9/8 and 10/9 are fairly similar. We could similarly do a thing whereby we represent the two of them by the same interval, and this interval is some form of weighted average between the two of them. Thus, we have an 'average' tone between the two tones that add up to (something along the lines of) 5/4!
We can go on and notice that (5/4)3 is (125/64), which is not particularly far off from 128/64 = 2. So, we can represent our 5/4 by something that is really (5+s)/4, where s = (128/125)1/3. Since each third is a stack of four fifths, we then will need four fifths per such third, and end up with 12 fifths.
We could, however, do other things as well. We may notice, for instance, that 125/64 is a bit flat of an octave - about 0.4 semitones, and that (5/4)2 = 25/16, 25/16 > 3/2 by a slight bit (about 0.7 semitones). 7.7 + 11.6 = 19.3. If we were to lower 5/4 a bit more - which we obtain by lowering 3/2 just a slight bit, we could potentially get this snake to bite its own tail.
25/16 is obviously two thirds, four fifths each. 125/64 is three thirds, four fifths each. And we've planned that combining these we'd obtain a fifth - thus 2*4 + 3*4 = 1 → a stack of 20 fifths gives us a fifth. Thus the nineteenth fifth has to be wrap back on an octave! It turns out that 19-tone equal temperament has a rather low major third (1200/19 ≃ 63, 6*63 = 378), although it is closer to the just intonation ideal than 12-tone equal temperament achieves. It has way worse fifths, however.
The result we see above is a rather fascinating thing - it means if you were to do something along the lines of C → E → G# → C → E → G#, and you stuck to major thirds, you'd end up rather doing C → E → G# → Cb → Eb → G instead - in both cases stacking what sounds like and functions as the very same interval, viz. the major third, 5/4.
We can now introduce a concept that has been sort of kicking around in the background for a while - Moments of Symmetry. Although not all scales that occur in human cultures are moments of symmetry, they are surprisingly common.
Remember how I have used strings of L and s to describe scale structures? A Moment of Symmetry (from now on MOS) can be described using only two letters - if you need mid-sized or supersized or subsized intervals, it's not a MOS. However, there is a second requirement: if we were to look at the distances from any tone to the tone after the next tone after it, we would also only find two sizes for that interval. Similarly, we would always find that any interval - except the period (which for us mostly will be the octave), will come in two sizes.
This does happen for 19tet with regards to our regular major scale:
C c# db D d# eb E e#/fb F f# gb G g# ab A a# bb B b#/cb CFinding that the bold letters in fact form a MOS is left to the reader [small appendix at the end, though, explaining in detail]. In 19-edo, there is a reason why c| < dd (etc), but this rule does not apply to all temperaments. (Hint: it's basically because meantone is the naming-scheme that we use: as you ascend the circle of fifths, you add #s, as you descend it, you add bs. There are circumstances where c# > db, and there may be circumstances where it's not even consistent.) Tasks for diligent students:
- What tones are in D major in 19-tet?
- Does C D E G A - the major pentatonic - have the MOS property in 12tet? Does it have it in 19tet?
- Can you find a six-tone scale in 12-tet that has the MOS property?
- Instead of using the C-G interval to build your scale, consider using the CE or CE𝄳 intervals. (So, e.g. CEG𝄲B...) Can you find scales that have the MOS property? Can you find such for any other interval? (Notice how I've already laid out the start for one such series above, viz.
C → E → G# → Cb → Eb → G. Is that a MOS? If you need to, how many notes do you need to add to turn C Eb E G G# cbc into a MOS?)
There is one obvious set of MOSes in any temperament: those generated by any length of single steps are always MOSes: C C# D D# E [gap ] c, C C# D D# E F [gap] c.
However, in 19edo, we also get rather many examples of LLL...s and sss...L, i.e. one single type of step that goes almost all the way to the octave, and then a single step of another kind wrapping it up at the end. I mark L by an underline, and s by absence of space at all in the following:
ssssL = CDEF#A#_C,
LLLLLs = C_D_E_F#_A#_B#C
ssssssssL = CDbD#EFGbG#ABb_ C
LLLLLLs = C_Db_D#_E_F_Gb_G#_A_Bb_CbCThe Pythagorean chromatic scale also has this property - which has the interesting property of making some of the major and minor scales available in it lack this property. However, using a MOS that divides the octave into a large number of steps (something like ten or greater) and pretending it is an equal temperament - especially when the difference is subtle - and embedding a MOS in it -- seems to enable some pretty neat effects.
Turns out some Japanese scales actually is the pentatonic MOS (small-small-large-small-large) embedded in a seven-tone MOS (large-large-small-large-large-large-small). Thus wherever in the seven-tone MOS you start, you include the next tone if the five-MOS interval is small, and you skip it if it's large. So you get a large number of scales derived from what essentially is very similar to our major and minor scales by pretending it's an equal temperament and picking subsets out of it.
APPENDIX: Is the major scale a 19-TET MOS? (Note: non-member notes are written out to show how the member notes, in bold large fonts, relate to the temperament)
C c# db D d# eb E e#/fb F f# gb G g# ab A a# bb B b#/cb CWe notice there are two sizes of second:
C c# db DD d# eb EF f# gb GG g# ab AA a# bb Bvs.A size of a second clearly is either 'two intervening elements (e.g. c# db)', or 'one intervening element' (e.g. b#/cb). Since sevenths are inverses of seconds, the same will apply.
B b#/cb CE e#/fb F
As for fifths, due to the way this thing was generated F → C → G → D → A → E → B, they all come in one size except the one counting from B, which is slightly smaller. The same applies, inversely, for fourths. (There, F will be the end of the line, and the distance from it to B is sharpened.)
We have the thirds and sixths left, then. We could count them individually - and find that some come with something along the lines of [c# db D d# eb] intervening, some come with something slightly smaller - [d# eb E e#], i.e. 5 vs 4.
We could do another trick here: since sixths are fifths + seconds, and all but one fifth are of uniform size, we can tell that all the regular fifths produce sixths of two sizes - due to there only being seconds of two sizes. In that case, we only need to look at the exceptional fifth - B→F, and we find that there are [b# C c# db D d# eb E e#] intervening. This is one less than the regular fifth. To get the sixth, we add [F f# gb], which gives us the same length as e.g. A→F (basically, all we have to do at that point is notice that regular fifth + small second = small fifth + large second, and that the small fifth is surrounded by large seconds, making it impossible to generate an exceptional sixth using it.
APPENDIX: Another MOS, now in 11-edo
Consider [0,2,3,5,6,8,9,(11)]
K k# L M m# N O o# P Q q# K1) Identify the generator
2) Show that this is a MOS.
Yet another MOS, that should be trivial to show is a MOS:
K k# L l# M m# N O o# P p# K1) Identify the generator
2) Show that this is a MOS
Notice that in both these scales, note names have now been given so that each member has a name that lacks accidentals, and the complement - the 11 tet note names missing in the MOS itself are given accidentals. With regards to the second MOS, it's been rotated a little bit - this is not a scale you will obtain by just stacking the generator - you have to move some intervals from one end of the line to the other to obtain it.
Is this a MOS?
C C# D D# E f F# G G# A bb B CHow about this:
C C# D D# E F F# G G# A bb B C
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