7-limit scales

First, a few points about this page: (0) This is a hodgepodge, and a work in progress. It is intended not as any sort of definitive treatise, but as some notes on some of the ideas I’ve been exploring. I wrote them up here in case my way of thinking about these things is useful to others, but don’t confuse this page with a textbook. (1) Very little if any of the ideas here are new, though a lot of it was new to me when I thought it up. (2) Herein, I generally consider two scales the “same” if one can be obtained from the other by starting on a different scale degree.

Introduction

One problem in thinking about microtonal music is the problem of scales — is a scale still a useful concept, and if so, how do you choose a scale?

For instance, let’s suppose you’re interested in 7-limit intervals — frequency ratios like 7/4, 7/5, 7/6, 8/7, 10/7, 12/7. Especially if you want to use these in a melodic context, you probably want a scale to work with; but conventionally constructed and tuned scales don’t include these intervals. So you’d need new, or at least retuned, scales.

That was my interest when I started this page, and maybe one should ask, why was it? Not necessarily for good reasons, as it turns out. Part of my interest in microtonal music stems from a desire to explore somewhat outside the bounds of the usual, while still keeping at least part of what I like in conventional music. I’ve explored listening to, and to some extent writing, some rather bizarre, experimental sorts of music, but I’ve found most such stuff ultimately dissatisfying when compared to good old tonalism based on diatonic scales. The quarter tone music of Ives and such never appealed much to me, but Blackwood’s “Twelve Microtonal Etudes” came as a pleasant shock: music in a positively antique mode, but based on anything but twelve notes per octave.

Observing that western music is based on 5-limit intervals, it seemed natural that a cautious explorer should concern himself first with 7-limit. Surely that would have the desired exotic quality without being too utterly foreign to grasp.

Well, maybe not. I’ve since learned that barbershop quartet singing is (supposedly) grounded in 7-limit just intonation, and that harmonicas until recently were usually tuned to 7-limit intervals (with a 4:5:6:7 chord sounding on the draw). Hardly exotic, most barbershop or harmonica music. And once I stopped playing mathematical games long enough to put together the hardware and software to play microtonal music, I discovered that indeed, 7-limit tetrads just sound, to my ear, too much like 5-limit dominant seventh chords to work, at least in the idioms I wanted to write in at that point, as stable consonances. Still, scales for 7-limit music strike me as at least intellectually interesting, whether or not they turn out to be useful.

Several approaches

How to approach the problem of constructing a scale?

One approach is to pick a set of just intervals, then take that set as defining a scale. But how to pick?

One option: take all frequency ratios n/m where n and m are odd numbers less than or equal to 11, multiplied as needed by powers of 2 to bring the ratio inside the 1/1:2/1 range. The result is an 11-limit Partch diamond, and it may be misleading to call it a “scale” but I will anyway. [As I understand it, this is better understood as a metascale from which a melodic scale is (rather arbitrarily) to be chosen.] Note that the step from the zeroth degree to the first is about 150 cents while that from the first to the second is less than 15 cents. The melodic useability of such a “clumpy” scale seems questionable. And indeed, it does seem to me a lot of what diamond-based music I’ve heard seems to stress harmony and brief motives rather than longer melody.

You can instead select for uniformity. For instance, see the “harmonic scale” in Carlos [1987]. Good uniformity, but arrived at rather arbitrarily; in addition, it has no inverse intervals above the tonic (no perfect fourth, for example).

An approach to the particular problem of a scale for 7-limit music that might seem promising but falls flat is as follows: We observe that a conventional (in the Ptolemy “Intense Diatonic Syntonon” tuning) diatonic scale consists of the union of all notes in three just major triads on roots a just fifth apart. Recall that a just major triad is three notes in the frequency ratio 4:5:6. Starting with a note at 0.0 cents (above whatever base pitch you want) you add the major third at 386.3 cents and perfect fifth at 702.0 cents; then you add the major third and perfect fifth a fifth higher (1088.3 cents and 203.9 cents), and the major third and perfect another fifth higher (590.2 cents and 905.9 cents). That’s your diatonic scale. Now we attempt to make a scale with 7-limit capabilities by adding a 7/4 interval to each triad, making three tetrads (where each tetrad is four notes in the frequncy ratio 4:5:6:7). So we add a note at 968.8 cents, another a fifth higher (470.8 cents), and another a fifth higher (1172.8 cents).

<td”>0 <td”>f<td”>9/8<td”>g

1/1 0.0 f
9/8 203.9 203.9 g
5/4 386.3 182.4 a
21/16 470.8 84.5
45/32 590.2 119.4 b
3/2 702.0 111.7 c
27/16 905.9 203.9 d
7/4 968.8 63.0
15/8 1088.3 119.4 e
63/32 1172.7 84.5
2/1 1200 27.3 f

Now, if the 1172.8 cent note weren’t there, this might not be bad. The intervals are a large whole tone (203.9 cents), a small whole tone (182.4 cents), a small semitone (84.5 cents), a large semitone (119.4 cents), a medium semitone (111.7 cents), another large whole tone, a very small semitone (63.0 cents), another large semitone, and a medium semitone. Okay, the very small semitone is really a quarter tone, and there’s only one of it, so that’s kind of ugly.

But with the 1172.7 note in place, the second medium semitone gets split into a small semitone and a 27.3 cent comma! So now we have note separated by as much as 203.9 cents and as little as 27.3 cents, with that and the second smallest step (63.0) occurring only once. This is much less nice than the two sizes of whole tone and one semitone we had before trying to put the harmonic sevenths in.

Note, by the way, that “explaining” the diatonic scale as three triads inverts what happened historically — triadic harmony was developed based on the existing diatonic scale, not vice versa. That is, the diatonic scale was not developed explicitly to support 5-limit harmony. But this is not to say we cannot be permitted to develop scales to support 7-limit, or other, harmony.

A more sophisticated selection process is called for. One better approach to generalizing the “diatonic = 3 triads on a chain of fifths” idea is the periodicity block concept (Erlich [1999] ). That’s discussed somewhat elsewhere on this site.

Here we consider yet another approach: “linear” (or, perhaps more appropriately, “rank-2”), “2-step” (or, as they are more commonly known, Moment of Symmetry) scales.

Octave-based rank-2 2-step (MOS) scales for the 7-limit

A Pythagorean approach

We start with a way of generating scales that, it seems, goes back to Pythagoras. We take a fundamental note (frequency = 1.0) and its octave (2.0), and add the fifth above (1.5), the fifth above that (1.52), and so on. Any time adding a fifth takes us outside the octave above the starting note, we drop that note down an octave.

With the first fifth we get a “scale” of two notes, C and G, with a step of 702 cents from C to G and a step of 498 cents from G to C’:

Degree Name Pitch Step size
1 C 0
702
2 G 702
498
1′ C’ 1200

Adding another fifth (and lowering the result by an octave) splits the 702 into two steps of 204 and 498 cents:

Degree Name Pitch Step size
1 C 0
204
2 D 204
498
3 G 702
498
1′ C’ 1200

Adding another splits one of the 498s into 204 and 294; adding another splits the other 498 likewise. The next fifth splits one of the 294s into 90 plus 204.

And so on. Generally each scale contains three step sizes between consecutive notes — “small”, “medium”, and “large”, where large = small + medium, and adding a fifth splits one of the large steps into a small step and a medium step. When we run out of large steps then the next fifth splits a middle step into a small plus a small-prime = medium – small (which may be larger or smaller than the small step), and then the pattern continues with the new small, medium, and large being the old small-prime, small, and medium (or small, small-prime, medium). As follows:

2 notes = 498.045 1 times, 701.955 1 times
3 notes = 203.910 1 times, 498.045 2 times
4 notes = 203.910 2 times, 294.135 1 times, 498.045 1 times
5 notes = 203.910 3 times, 294.135 2 times
6 notes = 90.225 1 times, 203.910 4 times, 294.135 1 times
7 notes = 90.225 2 times, 203.910 5 times
8 notes = 90.225 3 times, 113.685 1 times, 203.910 4 times
9 notes = 90.225 4 times, 113.685 2 times, 203.910 3 times
10 notes = 90.225 5 times, 113.685 3 times, 203.910 2 times
11 notes = 90.225 6 times, 113.685 4 times, 203.910 1 times
12 notes = 90.225 7 times, 113.685 5 times
13 notes = 23.460 1 times, 90.225 8 times, 113.685 4 times
14 notes = 23.460 2 times, 90.225 9 times, 113.685 3 times
15 notes = 23.460 3 times, 90.225 10 times, 113.685 2 times
16 notes = 23.460 4 times, 90.225 11 times, 113.685 1 times
17 notes = 23.460 5 times, 90.225 12 times
18 notes = 23.460 6 times, 66.765 1 times, 90.225 11 times

Note where we go from three step sizes to two: at five, seven, and twelve notes. These are familiar scales — pentatonic, diatonic, and chromatic, with step size ratios of approximately 3/2, 2/1, and 1/1, respectively. Stopping at five, seven, or twelve fifths is a natural thing to do because at these points the octave is “filled” relatively uniformly, without as much clustering as you get in between when some of the larger steps have not yet been split down to smaller steps.

Once we’ve seen the pattern we can compute how it continues without actually doing the powers-of-3/2 calculations. Suppose i0 notes give us a 2-step scale, with nl(i0) larger and ns(i0) smaller steps. Then the next 2-step scale will be at i1 = i0 + n1(i0), with either nl(i1) = nl(i0) + ns(i0) and ns(i1) = nl(i0) or vice versa. One step in the latter scale will be the smaller step in the former scale, and the other step in the latter scale will be the difference between the larger and smaller steps in the former scale. So the first several of these “Pythagorean 2-step scales” are:

# Notes Small step Times Large step Times
3 203.910 1 498.045 2
5 203.910 3 294.135 2
7 90.225 2 203.910 5
12 90.225 7 113.685 5
17 23.460 5 90.225 12
29 23.460 17 66.765 12
41 23.460 29 43.305 12
53 19.845 12 23.460 41
94 3.615 41 19.845 53
147 3.615 94 16.230 53
200 3.615 147 12.615 53
253 3.615 200 9.000 53
306 3.615 253 5.385 53
359 1.770 53 3.615 306
665 1.770 359 1.845 306
971 0.076 306 1.770 665

If we take this procedure seriously then we can consider 17, 29, 41, 53… as “natural” sizes of microtonal scales generalizing the pentatonic, diatonic, and chromatic scales. For these the large/small step ratios are roughly 4/1, 3/1, 2/1, and 1/1. (If we temper the intervals to make these ratios exact, and then divide the large steps into four, three, two, or one small ones, then in every case we get a 53-equal scale.) The table continues this up to a 971-note scale, though of course by 94 notes when the small step goes to 3.6 cents, the scales become decidedly nothing more than an arithmetic exercise!

Now what? Well, consider what use history has made of these scales. The pentatonic scale is used somewhat in western music; also there are non-western musics that use pentatonic scales, though not necessarily this pentatonic scale, and in any case I know very little about non-western music. But the Scala scale archive includes a “Chinese pentatonic from Zhou period” equivalent to this scale.

The diatonic scale, or its tempered equivalent, has of course been the basis of most western music in the past several centuries. Its advantages over the pentatonic scale are, I think (1) the presence of a leading tone (at least in the major mode) and (2) the presence of more major and minor thirds (there are only one major and two minor thirds in the pentatonic scale versus three major and four minor thirds in the diatonic). The former assists in establishing a tonal center. The latter permits development of rich triad-based harmony.

The chromatic scale has relatively little direct use in western music. You find chromatic scale passages, and occasional truly chromatic musical works, in pre-20th century music, but for the most part these are decidedly secondary to diatonicism. The reason simply is that the chromatic scale has essentially no structure: all the scale steps are more or less the same, so there’s nothing to support a tonal center. (Not to belittle Schönberg and his followers, but I think the verdict of history will be that atonality is generally not a desireable feature.) What the chromatic scale is good for is as a meta-scale, out of which various diatonic scales can be drawn. Using just fifths, this becomes problematic, but with tempered fifths many or all scale degrees will have acceptable diatonic scales available, allowing free modulation. Of course the 12-equal temperament of this scale is the basis of present-day western tuning.

Next let’s consider the 17-note scale. Its two steps are a limma (90.2 cents) and a (Pythagorean) comma (23.5 cents); the steps occur in this order (L = limma, C = comma):

      Degree:     0123456789abcdefg
      Step:       LCLLCLLCLLLCLLCLL

When I first started investigating this I assumed this scale probably had been analyzed before, perhaps even used for composition, by some theorist or another of the past century. I was wrong by only about 700 years. If you start on degree 2 of the above sequence you get the 17-note division of the octave established by Safi al-Din in his Kitab al-adwar written probably in 1252 (Wright [1978]) and used for some centuries thereafter as the theoretical framework for Arab and Persian musical theory. This apparently was, however, a meta-scale, from which selected notes could be chosen to produce the various melodic modes.

So it’s clear the Pythagorean procedure gives us at least four scales that have been historically found useful. Can we use any of these 2-step scales for 7-limit music? Initially they do not show much promise. The 7-limit intervals are not particularly well approximated in the 17-note scale (or, of course, the smaller ones). The 29-note scale does better, but it’s hard to believe 29 notes is not too many for a “diatonic-like” scale, that is, a scale one would use directly rather than as a “chromatic-like” metascale. Another drawback to these scales is the large step size ratios of 4:1 and 3:1, making them melodically “jagged”.

Tempering

Another problem is that these scales don’t even approximate 5-limit intervals very well: many of the major thirds are the lousy 21.5 cent sharp ones characteristic of the Pythagorean diatonic scale. Historically one cure for that problem is to temper the fifths, using an interval slightly different from 701.96 cents to generate a “linear” (or “regular”) tuning. We can try that, and maybe by tempering the fifths we can find better 7-limit intervals too. And given that the small step in the above microtonal scales is a Pythagorean comma — the difference between twelve fifths and seven octaves — it follows that a small change in the size of the fifth will have a big effect on this step size, potentially addressing the step size ratio problem.

It might seem a good way to address the problems with the just intonation scale I started with: three tetrads a fifth apart. Recall the biggest problem was the comma between F and the harmonic seventh above G. Maybe that can be tempered away?

Now I’ll just interrupt myself here to say two things. First, several months after first writing this stuff up, I learned Erv Wilson calls 2-step scales generated in this way “moments of symmetry”, or MOS. I happen to think this is a perfectly dreadful choice of nomenclature, but it seems to be the standard term. Second, it’s possible to temper the octaves too. In fact, there’s nothing in any of this that requires the “fifth” to be anywhere near 700 cents or the “octave” to be anywhere near 1200 — instead of “fifth” and “octave” call them “generator” and “period”, and you can generate scales using generators and periods of any size. These are called “rank-2” scales; in the special case where the period is equal to, or nearly equal to, an octave, these are called “linear” scales. (Some writers use “linear” to mean the same thing as “rank-2”, but I’ll reserve “linear” to refer to octave-period rank-2 scales). On this page I consider only the linear (octave-period) case, and where I say “scale” on this page from here on I probably mean “linear 2-step (MOS) scale” unless the context indicates otherwise.

Suppose we squeeze the fifth all the way down to about 685.7 cents. Now seven fifths make an exact (just) octave. Additional fifths will then just generate notes we already have, so this method yields no scales larger than 7. Of course this is a 7-equal? scale, of dubious use harmonically, but undeniably equivalent scales can be played on all scale degrees. Therefore (1) you can’t get a 12-note scale this way and (2) you don’t need to — or at least, the reasons you want a 12-note scale with more conventional fifths no longer apply here.

Likewise, if you stretch the fifth out to 720 cents, then five fifths make an octave (a 5-equal? scale); our procedure gives us no scales larger than 5. Again, with this temperament, no 12-note scales.

We’ve defined a range of values for tempered fifths which generate 12-note MOS scales, at the end points of which there are no 12-note scales generated. (Likewise the intervals in this range, but not at its endpoints and beyond, generate recognizable diatonic tunings — see Blackwood [1985] .) Near the bottom end of the range (685.7 cents), the 12-note scales contain seven large scale steps and five much smaller steps, which go to zero size and leave a 7-equal scale at the end of the range. Likewise, near the other end of the range (720 cents), you get five large steps and seven much smaller steps, degenerating to a 5-equal scale at the end of the range. In a small fraction of this range near the middle, the large and small steps are roughly equal in size and the scale is approximately 12-equal. It’s only by “luck” that just fifths generate a near-equal scale with this few notes.

Much the same sort of thing happens when you consider the range of values for tempered fifths which generate 17-note scales, with strong repercussions for the step size ratio problem. As indicated earlier, if we temper the fifth a little bit, the 17-note small step is affected twelve times as much. If we temper the fifth down to 700 cents, the small step vanishes — five notes then disappear from the scale and we’re left with only twelve notes at equal intervals: the 12-equal scale. Adding another tempered fifth just sends you around the circle again and you never get more than twelve notes. On the other hand, if we temper the fifth to a larger value, the small step gets bigger (and the big step gets smaller), reducing the jaggedness of the scale. At about 705.9 cents the small and large steps become equal in size and we have a 17-equal scale. For larger temperaments the once-smaller five steps become larger than the once-larger twelve steps; at 720 cents the latter vanish and we’re left with a 5-equal scale.

So by tuning the temperament within the 700-720 cents range we can get any step size ratio we like from 1:infinity through 1:1 to infinity:1. What does not vary across the range is the number of steps in the scale, and the interleaving pattern of the two different step sizes. At the boundaries of this range we no longer get 17-note scales, getting “stopped” at 12-equal or 5-equal scale respectively, and if we go beyond the ends of the range we find we get scales of different sizes. For example, with temperaments slightly below 700 cents the MOS that follows the 12-note is not 17 notes but 19.

Likewise the other scales (including the five, seven, and twelve note scales) have “validity ranges” outside of which those scale lengths are no longer generated by stacking fifths. These ranges are bounded by the nearest bracketing pair of temperaments which generate smaller equally tempered scales.

Here’s a graphic showing validity ranges for all scale lengths from 5 to 55 and generators from the 7-equal fifth (685.714 cents) to the 5-equal fifth (720 cents). (Any lengths that aren’t shown, e.g. 24, do not arise using generators in this range.) The numbers at the left are generators in cents which lead to equally tempered scales of the size shown. Different intervals can generate the same equal scale; for instance, 689.36 and 714.89 both generate 47-equal. Then the validity ranges are shown in the main body of the chart as vertical lines terminated by horizontal lines; scale sizes are shown at the equal temperament points. For example, the range for 31-note scales is from 694.74 cents (the 19-equal temperament point) to 700.00 cents (12-equal temperament), with 31-equal temperament at 696.77 cents. By reading off the vertical lines that intersect 701.96 cents you get the scale lengths generated by just fifths: 5, 7, 12, 17, 29, 41, 53… Likewise you can read off the scale lengths generated by any generator. (Larger version covering 600 to 1200 cents — the 0 to 600 range is a mirror image.)equalgens_smallIn summary, what we now have is a recipe for generating a continuum of microtonal scales, non-clumpy in the sense of having only two step sizes, varying from equally tempered to near-equal to “diatonic-like” with moderate step size ratios to “jagged” scales with large step size ratios; what’s constrained is the number of notes per octave available (depending on the generator) and the interleaving of the two step sizes. These constraints form a part of the desired barrier against artificiality in scale construction; we’re using the mathematical properties of our generator to obtain these constraints. Our freedom is to choose a generator (and, as noted above, a period, but the following disregards that possibility), and to choose from its corresponding set of scale lengths, in order to obtain a scale with desireable properties. Question is, is that freedom enough to get the kind of scale we need?

Searching for 7-limit MOS scales

Well, what do we need?

It seems reasonable that a melodic scale should have a step size ratio that’s big enough but not too big. That is, we want to stay away from the almost-equal temperament scales, whose tonic center cues are weak, and from the “jagged” scales where some steps are much larger than others. For example, suppose we were to demand a step size ratio of (precisely) 2:1. If we choose a generator whose sequence of MOS scales ends with an equal temperament scale, then the previous scale will have a 2:1 step size ratio. The scales previous to that will have other step size ratios, and may or may not be useful as well. E.g., using 709.09 cents, we get a 22-equal scale — which Erlich likes as the smallest equal scale with good 7-limit intervals — and before that, a 17-note scale (“17-of-22”) with 2:1 steps, a 12-note scale (“12-of-22”) with 3:1 steps, a 7-note scale (“7-of-22”) with 4:1 steps, and so on.

It seems to me that 17 notes is too many for a 7-limit analog of the 5-limit, 7-note diatonic scale — too many notes for the listener to grasp what’s going on. Twelve notes is better. But in the 12-of-22 scale there is a good 4:5:6:7 tetrad on only one of the twelve notes, and a good 1/4:1/5:1/6:1/7 tetrad on only one note. This isn’t very useful.

Can we come up with better scales? We need to develop criteria — which are necessarily somewhat arbitrary — and search the parameter space. We can choose any generator from 600 to 1200 cents (0 to 600 gives the same results) and choose any MOS generated by each interval. (Again, I’m here considering only octave periods.) I’ve looked at scales with all generators from 600 to 1200 cents incrementing by 1 cent, selecting those with the following attributes:

  • Step size ratio <= 3:1
  • Each step size accounts for at least 1/5 of all steps
  • Number of degrees <= 24

For each degree of each scale I find the best approximation to a 4:5:6:7 tetrad, and I calculate a “score” for the tetrad equal to the magnitude of the largest deviation from a just interval. (For example, in a 12-equal scale the best tetrad on any degree has a score of 31.2, because the approximate 7/4 interval is 31.2 steps away from just. Note that Carlos [1987] does something similar but uses the sum of the squares of the deviations as the score. I’ve tried this but feel it’s more useful to have a score that indicates how bad the worst interval is.) In general there will be several scale degrees with equally good tetrads. I keep track of the score for the best tetrad in each scale, and the number of such tetrads. For the aforementioned 12-of-22 scale the best tetrad score is 13 cents, and there is, as I said earlier, only one such tetrad in the scale. There are nine in the 17-of-22 scale.

I’ll give you the punch line now: There aren’t many good 2-step linear scales (with octave period) meeting these criteria. In fact, arguably there aren’t any.

The lowest-scoring 9-note scale I find has a generator of 919 cents, with three “good” tetrads. They aren’t very good at all, though: score is 29 cents. (The thirds are that flat.)

Likewise the best 10-note scale has thirds 24 cents flat in the “good” tetrads. With eleven notes about the best we can do is to have the fifths 12 cents sharp, the thirds 9 cents sharp, and the 7/4 intervals 12 cents flat, in a scale with a generator of 881 cents. All three intervals that far out of tune doesn’t seem promising. For 12-note scales the best generator is near the just fifth, and the 7/4 intervals are poor. And so on. With fewer than 17 degrees I find no scores less than about 15 cents, except for the 11-note scale mentioned above and a related 15-note scale. And as stated before, scales of 17 or more notes are almost certainly too long to be useful.

Linear scales on the circle of fifths

So linear MOS scales don’t seem to get us what we want. Why not, and can we learn something from why they don’t?

Consider the circle (or spiral) of fifths in Pythagorean tuning; part of it can be represented like this:

        ...  Eb Bb F  C  G  D  A  E  B  F# C# G# D# A# E# B# Fx  ...

A Pythagorean scale of n notes is then n consecutive notes of this circle of fifths; conventionally I’ll start with F.

Beginning with the completely obvious, in the circle of fifths, consecutive notes are a P5 apart. (In the following I’ll use conventional interval names like P5, M3, m3, and so on; I’ll also use H7, “harmonic seventh”, and SM6, “septimal major sixth”, to refer to intervals with frequency ratios near 7/4 and 12/7, respectively.) So of course in a 1-note Pythagorean scale:

        ...  -- -- F  -- -- -- -- -- -- -- -- -- -- -- -- -- --  ...

there are no P5s, but there is one in a 2-note scale (in this diagram and in the following ones, @—-* labels an interval, with @ marking the root and * marking the second note. Later I’ll generalize this symbolism to chords, e.g. *—-@–*—-* .) :

        ...  -- -- F  C  -- -- -- -- -- -- -- -- -- -- -- -- --  ...
                P5=@--*

and a second one in a 3-note scale:

        ...  -- -- F  C  G  -- -- -- -- -- -- -- -- -- -- -- --  ...
                   P5=@--*

and a third one in a 4-note scale:

        ...  -- -- F  C  G  D  -- -- -- -- -- -- -- -- -- -- --  ...
                      P5=@--*

and so on. Generally, in an n-note scale, there are n-1 perfect fifths, one on each of the first n-1 notes. (By “first” I mean “first to be added”, that is, “first note of the scale in the circle of fifths”.)

In this tuning, four perfect fifths make a (not especially good) approximation to a major third. So there is no M3 in a 4-note scale, because the first and last notes are only three P5s apart, but in a 5-note scale there is one:

        ...  -- -- F  C  G  D  A  -- -- -- -- -- -- -- -- -- --  ...
                M3=@-----------*

In a 6-note scale we add a second one:

        ...  -- -- F  C  G  D  A  E  -- -- -- -- -- -- -- -- --  ...
                   M3=@-----------*

and in a 7-note scale we add a third:

        ...  -- -- F  C  G  D  A  E  B  -- -- -- -- -- -- -- --  ...
                      M3=@-----------*

and so on. Generally, in an n-note scale, there are n-4 M3s, one on each of the first n-4 notes. Of course there are P5s on all of these notes (and the next three, too), so there are n-4 major triads in an n-note Pythagorean scale.

Or there would be if going up four fifths were our best option for a M3, but if we can go down eight fifths that’s a better M3. So in a scale of eight or fewer notes, there are no M3s of this kind. With nine notes we get an one on the last note added, it being the only note for which we have a note eight fifths below it. Of course that note has no P5 above it, so there are no major tetrads in this scale using this type of M3. In a scale of nine or more notes there are M3s of this better sort on the last n-8 notes (and major triads on all but the last of these). These are in addition to the first type of M3 found on the first n-4 notes. Note that a Pythagorean 12-note scale therefore has M3s of one sort or another on all twelve notes, and is the shortest such Pythagorean scale.

Now, to get a decent approximation to a H7 with Pythagorean tuning, we have to go down 14 fifths! (It’s about 3.8 cents too sharp; the next best approximation using fewer fifths is down two fifths, which is 27.3 cents sharp — I’ll regard this possibility as too inaccurate.) Of course you have the better version of M3 on all notes that have such an H7, but the last note has no P5. So the smallest Pythagorean scale with a major tetrad is 16 notes, and it has one tetrad, on the 15th note:

        ...  F  C  G  D  A  E  B  F# C# G# D# A# E# B# Fx Cx  ...
                                                    P5=@--*
             *-----------------------------------------@=H7
                               *-----------------------@=M3

Note: Notationally, Fx to B is not a major third, it’s a diminished fourth. But in Pythagorean tuning this is closer to a 5/4 frequency ratio than the major third, Fx to Ax. I could instead say, not that eight fifths down makes a major third, but that eight 3/2 down makes a 5/4. Approximately! Of course it really makes 256/6561, or 8192/6561 after octave shifting, so this terminology is problematic too. So I’ll just use terms like “major third” to mean “close approximation to frequency ratio of 5/4”, regardless of whether it’s notationally a third or not.

Adding another note gives you another major tetrad, on the 16th note, and so on; generally a Pythagorean scale of n notes has n-15 major tetrads of this form.

Obviously Pythagorean scales have to get rather long before they contain enough tetrads to be of much use.

Now, what if we temper our fifths? In particular, suppose we use a quarter comma tempered fifth. Two things happen. Let’s consider one at a time.

First, while we still use four fifths to make an M3 (and this time it’s the best choice, indeed this M3 is essentially just), now the smallest number of fifths for a H7 within 10 cents of just is ten — going up this time, not down. That means we get our first H7 in a scale with only eleven notes, on the first note, which already has a P5 and an M3, so we get a major tetrad:

        ...  F  C  G  D  A  E  B  F# C# G# D#  ...
          P5=@--*
          M3=@-----------*
          H7=@-----------------------------*

In a 12-note scale we’d get two H7s and two major tetrads, in 13 notes we’d get three, and so on.

But not ad infinitum! That’s the second consequence of tempering, in particular, using a fifth that generates an equal division of the octave. Consider what happens when you add the 31st note, which if you start with F would be Gxx (G quadruple sharp). This not only is a P5 above the 30th note (Cxx) but is also a P5 below the first note (F), because we’ve “wrapped around” the 31-equal closed circle of fifths. So while adding one note to a scale usually gives you one more P5, adding the 31st note gives you two more.

Likewise, when we go from 27 to 28 notes, the 28th note (Bx#) not only is a M3 above the 24th (Gx#), it also is a M3 below the first note. So while adding one note to a scale usually gives you one more M3, adding the 28th note gives you two more. So does adding the 29th (Fxx, giving a M3 from Dx# to Fxx and a M3 from Fxx to C), the 30th, and the 31st. Of course when you have all 31 notes in a 31-equal division of the octave, you have 31 M3s.

And likewise, adding the 22nd note gives you a H7 from note 12 to note 22 and another from note 22 to note 1. Again, from 22 notes on adding one new note gives you two new H7s.

So in summary, for fewer than eleven notes you get no major tetrads; for 11 to 21 notes you get one to eleven tetrads, and for 22, 23, 24… 31 notes you get 13, 15, 17… 31 tetrads.

How about minor tetrads? For those you need a P5, a m3, and a septimal major sixth, SM6. For m3 go down three fifths and for SM6 go down nine. Then in an 11-note scale there are two SM6s, on the 10th and 11th notes, but the latter has no P5, so there is only one minor tetrad, on the 10th note. And as with the major tetrads, adding one note to the scale adds one minor tetrad until you get close enough to 31 notes to “wrap around”.

If we arbitrarily say we want half the notes in our scale to have either a major or a minor tetrad (or both!), how many notes do we need? With 14 notes there would be major tetrads on notes 1-4 and minor tetrads on notes 10-13, total of eight tetrads. That’s the shortest linear scale in quarter comma temperament meeting that criterion. It is, of course, a three-step scale.

        ...  F  C  G  D  A  E  B  F# C# G# D# A# E# B#  ...
             @--*--------*-----------------*            Major
                @--*--------*-----------------*          tetrads
                   @--*--------*-----------------*        ...
                      @--*--------*-----------------*      ...
             *-----------------*--------@--*            Minor
                *-----------------*--------@--*          tetrads
                   *-----------------*--------@--*        ...
                      *-----------------*--------@--*      ...

And the small step here is a single 31-equal step. So here again, the scales with H7s in them have commas.

Looking at it another way: Recall the problem with the comma between F and the harmonic 7th above G. Equivalently, there’s a comma between two perfect fourths and one harmonic 7th. To get a linear tuning that contains good diatonic scales (in which the major third is four perfect fifths) requires the fifths be negatively tempered — made narrower, that is. But tempering out the P4/H7 comma requires the fifths be positively tempered. You can’t do both! So you have to give up on linear tunings that have good diatonic scales and in tune harmonic sevenths and no commas.

What you can think about is linear tunings that have good P5s, M3s, and H7s, but not good diatonic scales: Ones where the P4/H7 comma is tempered out by using a positively tempered fifth, and where the M3 is not four P5s but some other number of P4s or P5s. A fifth of about 709 cents gives you a 22-equal scale, with good M3s given by nine, not four, P5s. Then you can make a 9-note scale out of three major tetrads a fifth apart, with no commas. Success! But… there are no minor tetrads in such a scale.

Non-octave-based rank-2 2-step (MOS) scales for the 7-limit

Non-octave-based scales

Now at this point it might occur to us that by staying within 14 consecutive notes of the 31-note circle of fifths, we don’t take advantage of the “wrap around” by which adding one note to the scale adds not one but two new H7 and two new major tetrads (and similarly for SM6 and minor tetrads). Unfortunately the “wrap around” doesn’t kick in until the 22nd note in the circle of fifths, and 22 notes is too many. Can we find a way around this?

One obvious possibility is to not use a contiguous set of notes from the circle of fifths, but only use the ones that contribute to a suitable set of tetrads. Given the importance of fifths in establishing tonality, we probably want a fair number of consecutive notes on the circle of fifths in our scale, but a few gaps might be useful. Note that in the scale above if we were interested only in major tetrads we could omit C# and G# from our scale and have a (non linear) scale of twelve notes containing four major tetrads. It would also contain one minor tetrad, on D#. Total of five tetrads for twelve notes, not quite half.

Another thing we can do is to try to “wrap around” by using tetrads that are not one but several fifths away from one another. Again, due to the importance of fifths, we probably would want at least some of our tetrads a single fifth apart, but we can try some skipping around too.

And a third thing we can do is to use a smaller division of the octave. The smallest equal division to give decent approximations to just intervals through the 7-limit is 22-equal.

In 22-equal the fifth is positively tempered with a value of 709.1 cents, and the M3 is best approximated not by four fifths but by nine. The H7 in this temperament is best approximated by two fifths going down. Thus you need a 12-note span on the circle of fifths to get a major tetrad, on the third note of that span:

        ...  F  C  G  D  A  E  B  F# C# G# D# A# ...
             *-----@--*-----------------------*

Here’s the full circle of fifths (and I’ll compact it by indicating accidentals above the note names), beyond which it wraps around, with a major tetrad shown on the third note:

                             #
               #######xxxxxxxx
        FCGDAEBFCGDAEBFCGDAEBF
        *-@*-------*

Suppose now we add a tetrad a fifth higher, and then two more, also a fifth apart from each other but eleven fifths above the first pair. Then discard the notes not so used:

                             .
               ....###xx......
        FCGDA......AEBFC......
        *-@*-------*
         *-@*-------*
        *          *-@*-------
        -*          *-@*------

The second two tetrads wrap around, giving us four major tetrads using only ten notes. We have here two pentatonic scales, one starting on F and one on A#, together constituting a 10-note non-octave-based scale. (“Non-octave-based” meaning it’s not constructed from a generator and period where the period is an octave. This is a rank-2 scale, constructed from a generator (the tempered fifth) and a non-octave period (a tritone).)

By contrast you would need eleven notes for four tetrads in an unbroken chain of P5s:

                             .
               ....###x.......
        FCGDAEB....AEBF.......
        *-@*-------*
         *-@*-------*
          *-@*-------*
           *-@*-------*

or twelve for an analogous pattern in 31-equal:

                             #.........
               ...###.xx....xx.........
        FCG.AE....DAE.FC....BF.........
        @*--*-----*
         @*--*-----*
    	          @*--*-----*
                   @*--*-----*

You might think having all four tetrads in a chain of fifths would be worth the extra note. But as mentioned before, a scale of three major tetrads in a chain of fifths contains no minor tetrads.  Neither does a scale of four major tetrads.

But the above ten note scale contains four minor tetrads, two of them wrapping around:

                             .
               ....###xx......
        FCGDA......AEBFC......
        *-@*-------*            Major
         *-@*-------*            tetrads
        *          *-@*-------    ...
        -*          *-@*------     ...
            *-------@*-*        Minor
           *-------@*-*          tetrads
        -@*-*          *------    ...
        @*-*          *-------     ...

This is Erlich’s symmetrical scale (for which he allows four modes). His pentachordal scale uses a different offset between the two pentatonic scales, containing fewer tetrads:

                             .
               ...####x.......
        FCGDA.....DAEBF.......
        *-@*-------*            Major
         *-@*-------*            tetrads
        *          *-@*-------    ...
          *-------@*-*          Minor
           *-------@*-*          tetrads
        @*-*          *-------    ...

but having other advantages: more “strong version” acceptable modes and less tonal ambiguity.

Certainly 31-equal has better approximations to just intervals than 22-equal does, and it would be nice to take advantage of that; but these diagrams show you’d need more notes in that tuning to get the same number of tetrads, since you’d need larger numbers of tetrads to take any advantage of wrap around.

Here’s another way to look at this. If you consider all the 7-limit intervals you notice one of them, 7/5, is 582.5 cents — close to half an octave, 600 cents. Now, in any equal temperament with an even number of notes to the octave there will be notes 600 cents apart, and these may be the best approximations to a 7/5 interval. In fact they will be if, and only if, the step size is more than twice (600-582.5) cents, or 35 cents. This corresponds to 34 or fewer notes to the octave. In any such equal temperament, a 7/5 above a 7/5 is exactly an octave: in Erlich’s notation, 2(S-T) = U. This identity implies two tetrads 600 cents apart will have two notes in common, so we’ll need only six notes to make two tetrads. (Well, actually it’s more complicated than that. 7/5 is the interval between two of the notes of the tetrad, the ones 7/4 and 5/4 above the root, but in some equal temperaments the best approximation to 7/4 minus the best approximation to 5/4 is not the best approximation to 7/5. Such scales are “inconsistent”. For example, in 14-equal the best approximations to 5/4, 7/4, and 7/5 are five, eleven, and seven steps respectively. So a tetrad consisting of a root and notes 5/4, 3/2, and 7/4 above it will not contain a 7-step interval where you’d expect it between the second and fourth notes of the tetrad.) As Erlich points out there are only two equal divisions with an even number of notes, fewer than 35 notes, with good 7-limit intervals: 22 and 26. Erlich discusses the 22 case — let’s examine 26.

In 26-equal the major tetrad is: one fifth up (P5), four fifths up (M3), nine fifths down (H7). Two tetrads a fifth apart, and two more 600 cents higher, would look like this:

                             .###.
               ..###.#x.......xxx.
        FC.......GDA.BF.......CGD.
        *--------@*--*              Major
         *--------@*--*              tetrads
        *            *--------@*--    ...
        -*            *--------@*-     ...
        @*--------*            *--  Minor
                  *--@*--------*     tetrads

This is an analog to Erlich’s symmetrical scale, and it again uses ten notes for four major tetrads. There are only two minor tetrads (as shown), though, and the major thirds are even worse than in 12-equal. On the other hand, the H7s are nearly just. Note, however, than whereas in 22-equal you end up with two linear pentatonic scales interleaved (that is, two sets of five contiguous notes on the circle of fifths), here we have no more than three contiguous notes anywhere. We could add two more tetrads 600 cents apart, another fifth higher, to get two interleaved diatonic scales as follows:

                             .####
               ..#####xx......xxxx
        FCG......GDAEBFC......CGDA
        *--------@*--*
         *--------@*--*
          *--------@*--*
        *            *--------@*--
        -*            *--------@*-
        --*            *--------@*

(only major tetrads shown). That makes for a simpler scale structure, but drives the number of notes in the scale up to 14. From that viewpoint, 22 notes has the advantage over 26.

(Note, by the way, what we have here: A set of three major tetrads in a chain of fifths, and another half an octave higher. It’s two interleaved tempered versions of the “horrible” three-tetrad scale discussed above! Each of the two fills in the gaps in the other to make a non-clumpy scale.)

One thing I initially found bothersome about Erlich’s scales is their close relationship with 22-equal tuning. The diatonic scale is not as closely bound to an equal temperament. Reasonably useful versions of diatonic scales are found in a fairly wide range of tunings including Pythagorean, 12-equal, 19-equal, and 31-equal. In all of these four fifths make a major third, so the pattern of a major triad on the circle of fifths is the same; and no use is made of wraparound, so the temperament does not influence the number of notes needed to make three major triads.

It’s otherwise with Erlich’s scales. Of course if you think of e.g. the symmetric scale as two pentatonic scales a tritone apart, then you can construct such a scale in any even-size equal division of the octave greater than 8. In particular you could do it in 26-equal:

                             .....
               ......#xxxx........
        FCGDA........BFCGD........

but this only works if the pattern of a major tetrad is analogous to what it is in 22-equal, with two fifths down making a H7 and eleven fifths up (instead of nine) making a M3:

                             .....
               ......#xxxx........
        FCGDA........BFCGD........
        *-@*---------*              Major tetrad?

and it’s not — G to B# is a bad M3, and G to F is a bad H7. 32-equal will work, because in 32-equal two fifths down is H7 and 14 fifths up is M3, and 2 + 14 = 16 = 32/2. But the accuracy of these intervals is worse than in 22-equal. These are two of the only three equal temperaments where two fifths down is H7 and n/2-2 fifths up is M3. The third is 12-equal, which of course has very bad H7s. Hence the closer ties between the scales and the temperament than in the diatonic case.

Blackwood’s book discusses mainly diatonic tunings, principally as applied to performance of existing repertoire, so his comments about 22-equal don’t entirely apply to these scales. In any case, he writes that “a 22-note major triad is considerably less consonant than a 19-note triad, but noticeably better than a 12-note triad” but “Subjectively, this [22-equal diatonic] scale sounds badly out of tune…” Note carefully he is saying 22-equal gives consonant harmony but out of tune scales — the important distinction between harmonic and melodic usefulness of a tuning. Three observations: first, the perceived intonation of the 22-equal diatonic scale may have little relevance for new scales. Second, I’m not sure to what degree the perceived tuning deficiencies of the 22-equal diatonic scale is learned response due to familiarity with 12-equal scales (as opposed to consonance of triads, which presumably relates to beats and is less subjective). Third, it’s probably too much to ask (though it might be nice) for a not-too-large scale constructed for 7-limit music to include an excellent 5-limit diatonic scale as a subset.

Bibliography

  1. Barbour, James Murray. 1951. Tuning and Temperament: A Historical Survey
  2. Blackwood, Easley. 1985. The structure of recognizable diatonic tunings
  3. Carlos, Wendy. 1987. “Tuning: At the Crossroads”. Computer Music Journal 11(1): 29-43.
  4. Erlich, Paul. 1999. “A gentle introduction to Fokker periodicity blocks“.
  5. Erlich, Paul. 2002. “Tuning, Tonality, and Twenty-Two Tone Temperament” (revised from an article in Xenharmonicon 17 Spring 1998).
  6. Wright, O. 1978. The Modal System of Arab and Persian Music, A.D. 1250-1300.