20171122

Flipping Heaven

It has been a while since we looked at invertible scales and it might be worth a brief recap, starting with the Dorian mode of the diatonic pitch class set. This being the one with which we in the West are most familiar in its guises as the major and minor scales – or Ionian and Aeolian modes. From hereon in, we’ll abbreviate pitch class with PC.

By inversion, we mean reflection – or more accurately subtraction (see our earlier Pitch Axis Considered Harmful). By an invertible PC set, we specifically mean that the act of inversion – i.e. subtracting (modulo 12) each of its PCs from a certain fixed number - does not take you out of the set.

We start with the Dorian mode as it’s the most obviously invertible scale mode by anyone familiar with a piano, starting on a middle D and playing up and down the white keys either side of it. The piano keyboard is at its most obviously symmetric when you gaze at any D or A♭ key. If the D is PC 0 - usually represented on a twelve-hour clock-face as 12 o’clock (i.e. n00n!) - then the scale (which is to say the Dorian mode) seen as a seven note ‘motif’ is 0 2 3 5 7 9 10 (DEFGABC). Moving left (or anticlockwise) we also start at 0 but subtract and end up with the inverted sequence of negative numbers 0 –2 –3 –5 –7 –9 –10 (DCBAGFE) which, after adding 12 (an octave) to each (which is equivalent to adding 0, itself equivalent to doing nothing whatever to alter the essential nature of the inversion), gets us 0 10 9 7 5 3 2. And all of those numbers (and notes) remain a Dorian PC set’s PC numbers (and white keys).

dorian - the polar symmetric diatonic
interval skip pattern 2122212 D

That the fixed number one subtracts from may be different for different modes does not alter the essential invertibility of the set. For instance if one subtracts the major PC sequence (aka the Ionian mode 0 2 4 5 7 9 11) from 0, one would recover 0 -2 -4 -5 -7 -9 -11, which is 0 10 8 7 5 3 1 - decidedly not the same mode since, reordered, this is 0 1 3 5 7 8 10 – i.e. the Phrygian. The 2 and the 9 have been flattened to 1 and 8. To recover the same mode, one would have to subtract the Ionian PC numbers from 4, to yield 4 2 0 –1 –3 –5 -7 which – after adding a 12 (‘doing nothing’) where necessary to keep us in positive PC numbers – results in 4 2 0 11 9 7 5, the exact same Ionian PC numbers we started with – we just don’t start on the tonic note. There’s a table towards the end of Phrygian = subtonic – Dorian if you wish to see what happens to the modes if you subtract them from different values.

The invertible PC set however, regardless of mode, is the same particular ‘shape’ if you care to look at it as a polygon and – as such – does not alter at all. The shape just rotates around its centre. As such, its axis of symmetry is simply carried around with that rotation. There may of course be more than one axis of symmetry in other, non-diatonic, PC sets.

There are also scales which are not invertible such as the (pentatonic) Hirajoshi, the (hexatonic) Blues, the (heptatonic) Hungarian Major (which we’ve discussed), the (octatonic) Bebop Dominant Flat Nine and many, many more.

interval skip patterns
21414 321132 3121212 13122111
hirajoshi blues hungarian
major
bebop
dominant
flatnine

A Digressional Touch of Polemic

The 'interval skip pattern' notation, employed in the preceding diagrams, is simply a ‘key-signature’ independent representation of the semitone steps between each note of the scale, starting at its root or tonic note (twelve o'clock in the diagrams!), with the final step ‘returning’ you to the tonic (an octave above) regardless of whether or not the scale or mode is F Dorian, C# Hirajoshi or (if you’re one of those) a 432Hz-based-A Bebop Dominant Flatnine. This (completely scalable) interval skip pattern scheme works for systems other than twelve tone – indeed it works with any fixed octave-based microtonality and it doesn’t even need to be well-tempered (although the scales will sound different of course). Skip numbers simply have to total up to the number of microtones in your octave, and the number of skips (perforce not more than the number of microtones) must match the number of notes in your scale.

Other Heptatonic Flippers

The 2212221 skip pattern (of, specifically in that case, the Ionian mode of the ordinary ‘diatonic PC set’) simply rotates as 2122212 (Dorian), 1222122 (Phrygian), 2221221 (Lydian), 2212212 (Mixolydian), 2122122 (Aeolian) and 1221222 (Locrian) and carries its axis of symmetry around the clock as it does so. They are all regarded as the same PC Set (classified as Forte Number 7-35 in Allen Forte’s naming system). In that sense, modes are regarded as equivalences of the skip pattern, which is similarly regarded as a single pattern of interval skips, without being particularly bothered about which skip comes first.

For instance, with a seven note scale embedded within a twelve tone chromatic space, there are only two ways to have scales comprising only whole tone or semitone skips without two consecutive semitone skips. One of them is with three 2s and two 2s separated by 1s (they must, after all, add up to 12 in total). The only other possibility is four 2s and a single 2 separated by 1s. (Five 2s would leave only space for the forbidden two consecutive 1s). This alternative pattern (2222121, say) is also symmetric and also has seven modes. But it looks as if only six of them are in (reasonably, and variably so) common use.

various modes of interval skip pattern 2212122 (hindi)
javaneselocrian
natural
melodic
minor
hindiovertonelocrian
super

The axis of symmetry is marked with a yellow line. The Hindi scale, like the Dorian, is tonic-invertible in that subtraction from 0 will keep your PCs firmly inside the Hindi ‘mode’ of the PC set 7-34, the Forte Prime Form of which is known as the half-diminished scale (aka the Locrian Super in the above).

If you forego the restriction preventing your scales comprising only semitone or whole tone skips, you may permit yourself a sesquitonic (one and a half tone, or three semitone) step (or two). Such a pattern might be 1131213 – again, all of the numbers must add up to 12. This is PC set 7-22, turning up as the Double Harmonic scale. This PC set is invertible and here are three further modes which, due to historical and geographical accident, are also known as scales in their own right (despite their all being simply modes, or rotations, of each other).

three further modes of the symmetric sesquitonic scale PC set 7-22
hungarian minorgypsyoriental
interval skip
pattern
2131131
interval skip
pattern
1312131
interval skip
pattern
1312131

The distinctly non-PC (in the other sense) term ‘Gypsy scale’ is something that, alas, we can’t do very much about any time soon. Scale names are much too firmly embedded in musical language and history. Despite the obvious advantages of describing/classifying scales (and even modes) completely and unambiguously by numerical interval skip patterns, there’s pretty much no possibility of the more rational nomenclature prevailing any time soon. We just have to put up with the fact that the naming of musical scales is both haphazard and arbitrary. In short, the current space of scale names is clearly not scalable.

There are many other seven note PC sets. We (but we were by no means the first) counted 66 distinct (within modal, or rotational, equivalence) patterns back in that first post and we also noted that only 10 of those 66 were symmetric. In fact we observed that – regardless of the number of notes in a scale of any length (from 1 to 12 within the twelve-tone series) – the majority of PC sets are unsymmetric. Which is not to say that, worldwide, some musician has not come up with a scale which employs them (as, for example, the aforementioned Hungarian Major, demonstrates).

Octatonical Flips

How about eight note scales? We know there are 43 PC sets containing eight PCs, and that only 15 of them are symmetric. A few, like the heptatonic sets, can be constructed with the restriction that they contain only whole tone or semitone skips. But this time, because the scale is so crowded with notes, consecutive semitone skips are unavoidable. If three consecutive skips are whole tone then you must fit the remaining five notes – best case – within a span of six semitones (or fewer, if all three whole tone skips are not consecutive). If five skips are whole tone then that’s ten of the available PCs already accounted for, leaving you no room at all to fit in the other three PCs. Four 2s and four 1s seems to be a good compromise and the 21212121 pattern – which is doubly symmetric on two axes – is found in the usual pair of Jazz scales commonly called ‘octatonic’ – one of them starting with a whole tone, the other with a semitone. But there are other – single-axiswise symmetric – scales:

four symmetric octatonic scales - mostly bebop
modes with two consecutive semitones
bebop major flamenco
interval skip pattern
22121121
interval skip pattern
12112122
modes with three consecutive semitones
bebop minorbebop dominant
interval skip pattern
21221112
interval skip pattern
22122111

The two PC sets above are, respectively, 8-26 and 8-23, Forte-wise.

Counting - The maths bit

We already counted, right back at the beginning, how many three note, four note, five note etc scales were inversions of themselves - we reproduce the distribution here:

image

The bluish bars represent the minority of invertible sets and the reddish the uninvertible ones within each set size. The sequence for k-note scales is 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1 for k=0 to 12. We didn't bother to chart the extremes (of 0 and 12) because, trivially there's only one way to have an empty musical scale (bound to be reflectionally symmetric), and also only one full 12 note chromatic scale - obviously very symmetric indeed.

If you add up all of these (bluish) numbers they total 96. Which is to say that, of the 352 (= 1 + 1 + 6 + 19 + 43 + 66 + 80 + 66 + 43 + 19 + 6 + 1 + 1) possible (differently 'shaped' rotationally equivalent) k-note scales (again, for k=0 through to k=12) representable inside a 12 hour clock, most of them (i.e. 256) are not reflectionally symmetric.

If you make the further equivalence between those shapes which are mirror images of each other (i.e. although you cannot rotate one to completely match the other, you can rotate its mirror image to match it) then those 352 shapes reduce to 224. As the 96 symmetric polygons already were 'equivalent in the mirror' they're not quite as rare amongst this reduced set.

If you remember your early geometry classes and remember your triangles, you may recall the difference between congruent triangles and similar triangles. In this regard, polygons are people too, as it were (notions of similarity also include not having to be the same size, but that's not relevant here). Additionally the general case of the haphazardly shaped polygon might remind you of those undistinguished scalene triangles. Isosceles triangles are analagous to our symmetric polygons.

More than 12 Notes

What about other microtonalities? Our '96', '224' and '352' counts are but one case (the '12' case) of well known integer sequences used to count bracelet or necklace arrangements. You may encounter the terms 'necklace' to model those patterns with rotational equivalences (our larger 352 case - think 'congruent triangles') and 'bracelet' to model a flippable necklace - i.e. patterns with the additional reflectional equivalences (the 224 case subset - think 'similar triangles') and - still further - bracelets which are also symmetric (think 'isosceles triangles' and even 'equilateral triangle').

The first two are, respectively and more formally, examples of Cyclic and Dihedral groups in Group Theory. The following tables show three entries in the Sloane Catalogue of Integer Sequences (the OEIS is the online version). The various headings for each sequence come from that catalogue.

12345678910111213141516
Number of n-bead necklaces with 2 colors when turning over is not allowed
also number of output sequences from a simple n-stage cycling shift register
also number of binary irreducible polynomials whose degree divides n
In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system
2346814203660108188352632118221924116
Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets)
23468131830467812622438068712242250
Number of necklaces with n beads and two colors that are the same when turned over and hence have reflection symmetry
2346812162432486496128192256384

It should be evident that each number in the third sequence is the corresponding number in the first sequence subtracted from twice the number in the second sequence. Twice the similars is the congruents plus the symmetrics – as it were.

But we are modelling scales (or, more abstractly, PC sets) as k-sided, or k-pointed (it's the same thing - a PC set with k members) polygons picked out as sub-polygons of the 12-sided 'chromatic dodecagon'. It turns out that the last series can be rather nicely captured by the generating function:

Generating Function for Symmetric Polygon Totals in n-sized tuning systems

where the pn (for n = 1 and upwards) are the exact same values in the third sequence above. (For completeness, p0 = 1.) This generating function is given in the Sloane catalogue, but without that rather mysterious 1 we've slipped in, with a comma, before the z. What’s that all about?

In fact it is an application of a more general generating function of two variables, u and z, evaluated at the value of u = 1. This function is:

Generating Function for Symmetric Polygons of k-sized scales in n-sized tuning systems

The apparently unnecessary double summation tacked on to the end of that function is there for a reason - it allows you to pick out the individual counts for k-sized scales inside n-sized tuning systems for any n and for any k (perforce less than or equal to n). All you have to do is expand the rational polynomial as a power series in u and z. Which is pretty straightforward.

k-sized scale counts in 0 to 13 note tuning systems

In particular the penultimate line of the expression - representing the familiar 12-tone chromaticism - shows the coefficients of the individual powers of u, i.e. the a12 kuk, as the 1, 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1 we presented earlier (including the u0 and u12 cases). For example we see that there are 10 symmetric heptatonic (and of course pentatonic) scales within 12 tone chromaticism.

In general, for musical systems with an even number of divisions (like our familiar 12 tone), we have that:

P2m(u) = (1 + u + u2)(1 + u2)m-1

And that for musical systems with an odd number of divisions (e.g. 17 or 19 tone systems):

P2m+1(u) = (1 + u)(1 + u2)m

You may also wish to verify that - if you want to count all invertible scales, regardless of k, for each of those values of n, just set u = 1 (pn = Pn(1)) in the above.

The above 'triangular series' which appears as the ever lengthening lines of coefficients of u for increasing powers of z is also documented in the Sloane Catalogue as series A119963. Dr Petros Hadjicostas has attached a generating function Sum_{n,k >= 1} RE(n,k)*x^n*y^k = (1+x*y-x^2)*x*y/((1-x)*(1-x^2-x^2*y^2)) which, in our letterings (u for y and z for x), would translate to our G(u,z)-1/(1-z) which, taking into account our start at n = 0, is the same thing.

As a fun application, consider all of the triads available to a musician playing with the aforementioned 17 note system, which can be represented with a 17-gon. We pick out

P17(u) = (1 + u)(1 + u2)8

and expand powers of u to obtain

k-sized scale counts in a 17 note tuning systems

and we can immediately see (from the coefficient of u3) that 8 (out of the possible 40 from the well known dihedral symmetry enumerations) triads are self-inversional. Below is a diagram showing all 40, with the 8 self-inversional triads as isosceles triangles in blue and the remaining 32 non-inversional, as 16 mirror pairs, in salmon.

20170917

The Third Degree

As the previous article mentions, the following ‘motif’ (scare quotes because of what follows) begins with a long run of thirds:

This, the Berg Violin Concerto's 'defining tone row' is the one catalogued by the ‘34433443222’ interval class index in Wikipedia’s list of tone rows (hereinafter referred to as wLOTR), which cites this concerto as its principle reference. Fripertinger's more technically exciting Database on tone rows and tropes also indexes the works discussed here with the exact same tone row or interval sequence keys (it's not clear if the Wiki page is based on this). So it might be interesting to investigate how this particular sequence came to be the one elected to characterise the entire opus.

As a more general question, of all of the instantiations of a tone-row within a work, how does one decide which of them is ‘prime’? For it is a well-known problem, as Peter Castine reminds us, at the beginning of §2.9 in his Set Theory Objects (Europaïscher Verlag der Wissenschaften; Musicology, Peter Lang 1994):

“The main difficulty in set theoretic musical analysis is not so much that of recognizing relations between pc sets, it is deciding which notes in the score should be analyzed as pc sets. This is called segmentation. Once segmentation has been established, much of the remaining work is mechanical; [etc] …”

We should also consider Babbitt’s contributions to the concepts of serialism and what might be called ‘tone-rowism’. He writes (of Schoenberg - whom he knew - and of the nuances of translating German into English)

“The word Reihe bothered him because it became row. And to him row suggested left to right – something in a row – and that’s what it does connote. And that connotation, he thought, was part of all these misunderstandings about the twelve-tone notion having to do with some sort of thematic, motivic thing that went from left to right. It upset him, so he asked various friends about it.

[omitted paragraph on how series was suggested, and rejected, as translation for Reihe]

And I regret to tell you, I am guilty. I suggested the word set, which had absolutely no meaning in music as yet. It came out of mathematics (not that that pleased me particularly) and it seemed to be a neutral term. Of course, a set does not mean anything ordered, but if you append twelve-tone or twelve-pitch-class to the word set, then that implies an ordered set and that’s a very familiar structure, too, in abstract relation theory. So there we were.”

This is from chapter 1, “The Twelve Tone Tradition” of Milton Babbitt’s “Words About Music”, Dembski and Straus (eds), from the University of Wisconsin. You may read much of this book in google’s academicals, but not – unfortunately - this particular bleeding chunk since pages 11 and 12 are ‘not shown in this preview’. You’ll either have to take my word for it or find a real copy, but as Babbitt’s generally a fun read, it’s worthwhile.

So Babbitt claims that Schoenberg never intended that all twelve tones must be played out (exhausted if you will) in order, before you were allowed to proceed to any of its transformations (i.e. its repeats or retrogrades or inversions). Although this does not seem to square (advance pun warning) well with later attentions paid to the construction of P-R-I-RI grids, aka Babbitt Squares, it just means that these are possible playthings, not necessary ones - and a comparatively recent invention unused in the composition of the pair of works discussed here.

In any case. musicians may write what they please, even the (Babbittally argued) most intellectual of the Berg-Schoenberg-Webern group, Berg himself, to whose concerto’s tone-row ‘signature’ we now return.

Berg in Threes

In the score’s introduction (Universal Edition Philharmonia Partituren #426), “F.S.” (Friedrich Saathen) describes the above tone row as being ‘the one from which the Concerto is made’ (presumably what we might call its prime ‘P0’ form). Attention is drawn to the G D A E, the four consecutive perfect fifths of the violin’s open strings, embedded within its head and also to the terminating tritone resulting from the last three whole tone steps. And of course the concerto begins with those four open strings. Clarinets and harp provide the interstitial B, F# and C in the first two bars. They also throw in some Fs, but as these could conceivably have come from some unheard previous row’s instantiation (n.b. F is at the end of the model above), we concede its legitimacy (no, we’re not serious).

We bring up these matters only because at what might seem to be one of the focal points of the concerto, in the second half, we have this at bar 195:

Berg Violin Concerto - Adagio Bars 195ff

where the soloist comes in (the p at the middle of bar 196) with an entire tone row (interval classes at the top, zeroed-out pitch classes at the bottom) :

- presumably intended to convey angelic little Manon Gropius heavenward. It’s damn’ poignant, and it seems notable that, at such a critical moment, the ‘principal tone row’ isn’t quite there. It’s certainly a close relative, but the 34433443222 jump pattern - even if rotated to recover 43222344334 - cannot get you to 32222344334 with its four 2s and concomitantly constrained tone rows. Any claim that sliding the ‘serial selection window’ left or right will reveal the true constructor is easily dismissed by inspection (the intervals will not fit).

The ‘Es ist Genug’ violin solo manifestation at the start of the Adagio (bar 136) isn’t even twelve-tone but is straightforwardly tonal. The ‘prime form’ of which Saathen speaks is almost, but again not quite, found in the bassoons, who are concurrently executing an ascending 11 note sequence G# B D# F G# B D# G B♭ C D.

As the concerto has never been claimed as a strict serial composition then we’ll just grind to a halt right there … we just thought it interesting that the concerto is indexed in wLOTR with an expression that isn’t obviously evident in the actual piece. It may very well be somewhere in the concerto, but not easily available to a listener or score-reader.

Webern in Threes

Another piece of serialism, also from the B-S-W trinity, is indexed in both wLOTR and Fripertinger as 34343443431. As you can see, it’s much thirdier; almost, but not quite, as thirdy as you can get.

In wLOTR (loc cit) it’s the immediately preceding entry (if sorted by interval class, at the time of writing – there’s lots of possible room for expansion, so who knows how long this will stand). And the tone row entry that it falls under is indexed as 0 3 e 2 t 1 9 5 8 4 7 6 (or 03B2A1958476 in the alternative popular tone-row labelling scheme), which looks like this (if you regard the leading F# as the 0).

The interval class jumps (+ or – omitted as obvious) annotate the bottom of the illustration. Remember that, interval classwise, descending a major or minor sixth is the same as ascending (respectively) a minor or major third (and vice-versa). It’s why interval classes (which is one of the ways the wLOTR Wiki is indexed) are in the range 0 to 6 and not 0 to 11. I.e. the indexing is actually a string of ±3±4±4±3… where the ± is not shown but is to be taken as ‘understood’.

The work cited by that index (or indexed by that citation) is the second of Webern’s Drie Lieder (Op 18) of 1927(ish), "Erlösung" which begins thus:

Bars 1 to 6 of Webern's Erlösung

We’ve coloured up the first seven (red, green, blue, red, green, blue, red) of the tone-row (actually set) block instantiations. Apart from the slight ambiguity of which of two pitches in a guitar’s tremolo comes first, the sequence of pitch first-appearances across the instrumentation is – in all seven cases [except the last ‘interesting’ one] F# C F G# E G E♭ B D B♭ C# A.

One might notice the overlaps. For instance the first ‘green’ set begins before the earlier ‘red’ one has quite finished, and these overlaps continue as the piece progresses. But there’s no change in the particular sequence of pitch classes.

So, out of bars 1 to 3, one may pull out the first red (bars 1 and 2) and green (bars 2 and 3) blocks to confirm the (tremolos notwithstanding) sequence. You may painstakingly, if you wish, verify that the following blocks follow the same tone row (i.e. no inversions or retrogrades):

Erlösung's Tone Row

As the annotations show (interval jumps underneath the stemless abstractions), the intervals aren’t quite the same as the one ‘attached’ to this work in wLOTR. And, of course, 0 3 e 2 t 1 9 5 8 4 7 6 (also from wLOTR) is not (from the above, assuming F# → 0) 0 6 e 2 t 1 9 5 8 4 7 3.

You can certainly see the similarity – it’s only at the edges (in bold underlining) that there is a difference

0 3 e 2 t 1 9 5 8 4 7 6
0 6 e 2 t 1 9 5 8 4 7 3

(although the 0 hardly needs to be underlined as its presence is demanded by the nature of the presentation). So what’s with this 3/6 head/tail swap? The worst possible place to happen for a lookup, wrecking both the tone-row and the interval-sequence index? The reference authority cited for this tone row is, in fact, a footnote in a journal article by the one and only David Lewin:

(from A Theory of Segmental Association in Twelve-Tone Music, D Lewin, Perspectives of New Music V#1 N#1, Autumn 1962) where one may clearly see the A after the initial F# and the terminal C in the upper (Φ0) line in contrast to the actualité of Webern’s C and F#, easily seen (multiple times, as if to nail it firmly into your head) in the above score-snippet. Lewin’s exposition is – by the way - quite legitimate (his article concerns hexachords and segmentation). As he says, earlier on (in regard to Schoenberg’s Op 36 Violin Concerto):

“These examples are of considerable value in cautioning us against the naïve but plausible assumption that all effective associative relations in such music as this must be presented explicitly. The reader is urged to keep this moral in mind throughout the sequel. Of course, the extent to which we will recognize any such relations, whether explicit or not, is heavily dependent on the extent to which the compositional presentation of the notes involved supports or obscures the abstract relation, and/or the extent to which the sonorities involved have been explicitly established as referential.” [our emph]

What appear to be the 'obvious' tone rows (or, equivalently, interval-class sequences) - by which we mean those sets or sequences readily available to a listener or a score-reader (bearing in mind Castine’s caution), in contrast to those other, rather abstruse, representations available to a reader of somewhat rarefied musical journals - may not be terribly useful when interrogating a database to discover them. Fortunately, other rather more instantly available - even to a non-musician - indexes such as the composer's name or the work's name are also present.

20170905

The Third Way

This post was triggered by a brief exchange with Jan-Willem van Ree (on musescore) about the use of certain intervals in structured or otherwise constrained music. Having believed I had seen, long ago, examples of tone rows built exclusively from major and minor thirds (or their interval class equivalent, minor and major sixths), I’d formed the impression that either Webern or Berg had actively sought such sequences.

Dr van Ree kindly reminded me of the Berg Violin Concerto which includes the following run of eight successive thirds (abstracted and annotated below, with both note names and semitone interval classes between notes), and which finishes off with a triple whole tone tritone run. In the score’s introduction (Universal Edition Philharmonia Partituren #426), “F.S.” describes it as being the one from which the Concerto is made (presumably its prime ‘P0’ form - although it's not a strict 12 tone work).

Berg Violin Concerto 'Es ist Genug' Tone Row

Being the sort of thing I was looking for, the start of the sequence might be represented by the curved (red) path in the following transition-by-thirds (dark/light blue arrows for minor/major thirds respectively) diagram:

But with this we may seek circuits, i.e. closed paths. For instance one might start at E, move to G, then B, then to D, to F, to A♭, to C, to E♭, to F#, to B♭, to C# and back to the starting E. That’s a circuit of eleven pitches, visiting each note only once. The one missed is the A. Is there any way to visit all 12, and end up where we started?

Unhappily not. There's no path which can complete such a circuit. You cannot find a path from any note back to itself visiting every other exactly once on the way.

You may, however, visit all notes exactly once – you just don’t get to return home. Here is one such path, again the curved one, starting at C and finishing at B:

The length of the route is 3+3+3+4+3+3+3+4+3+3+3 = 35. If the diagram would allow you to step 1 semitone then you could make it home from B to C (a nicely cadential leading tone) at 36.

There are four such paths (or 48, 12 each, if you care about your start position, but in pitch class world we really don’t). The jump patterns (and path lengths thereof) are:

  1. 35 = 3+3+3+4+3+3+3+4+3+3+3(+1)
  2. 37 = 4+3+3+4+3+3+3+4+3+3+4(-1)
  3. 39 = 4+3+3+4+4+3+4+4+3+3+4(-3)
  4. 41 = 4+4+3+4+4+3+4+4+3+4+4(-5)

The numbers in parentheses at the end are the extra steps you'd need to return to your start position. If you insist on upward jumps only then you must change the last three 'homing' jumps to +11, +9 and +7.

Completenesswise, journey number 4 also has a resolving flavour with its dominant/tonic termination. Journey 3 doesn’t, but the final jump home is at least another third, it’s just in the ‘wrong’ (with this diagram) direction.

Another representation of such circuits is by drawing them on the 12 hour pitch class clock:

0369147A258B 047A258B3691 047A269158B3 048B37A26915

where the modulo 12 labelling describes the pitch class visit-order – all of them starting from 0 (at 12 o’clock) and tracing the black lines strictly clockwise in their 3 or 4 ‘hourly’ jumps, except for that final jump, back to 0 (A and B being 10 and 11 respectively). You may click on the coloured areas to watch the traversals.

It will not have escaped your attention that, plotted this way, bilateral polygonal symmetry is apparent. Musically, this simply admits that their retrogrades and inversions are identical (descending thirds instead of ascending ones). The inverse of a minor third is a major sixth (3 + 9 = 12), and vice versa (4 + 8 = 12). The differences are all interval class 3 or 4. The third clock diagram is distinguished green only because, unlike the others, the closing leap happens to be in that very interval class (+9 ≡ –3, modulo 12).

But once we’ve admitted the equivalent interval classes of sixths and thirds, we may dispense with the arrowheads, since we may now move from node to node in either direction (major 3rd being equivalent to minor sixth, minor third to major sixth, either rising or falling). We may also take the opportunity to dispense with note names and abstract to pitch classes. The following transition diagram results:

Where the A and the B represent pitch classes 10 and 11 (not notes A and B!).

Now, since one may move in either direction, the possibilities for complete circuits (visiting all twelve pitch classes exactly once in a closed loop) would appear to dramatically increase. As in fact they have, to 252 in sheer numbers. But as far as the actual patterns of such loops are concerned, where (musical) transposition of a circuit is the same as starting at a different point, retrograde is circuiting in the opposite direction, and inversion as reflection, etc, it turns out that there are only 11 distinct patterns. In ‘alphabetical’ order, these are

  1. 0362591A7B84 → +3+3-4+3+4+4-3-3+4-3-4(-4) ≡ +3+3+8+3+4+4+9+9+4+9+8(+8)
  2. 0362A147B859 → +3+3-4-4+3+3+3+4-3-3+4(+3) ≡ +3+3+8+8+3+3+3+4+9+9+4(+3)
  3. 0362A1958B74 → +3+3-4-4+3-4-4+3+3-4-3(-4) ≡ +3+3+8+8+3+8+8+3+3+8+9(+8)
  4. 0362A7B84159 → +3+3-4-4-3+4-3-4-3+4+4(+3) ≡ +3+3+8+8+9+4+9+8+9+4+4(+3)
  5. 0362B7A14859 → +3+3-4-3-4+3+3+3+4-3+4(+3) ≡ +3+3+8+9+8+3+3+3+4+9+4(+3)
  6. 0362B847A159 → +3+3-4-3-3-4+3+3+3+4+4(+3) ≡ +3+3+8+9+9+8+3+3+3+4+4(+3)
  7. 0369152A7B84 → +3+3+3+4+4-3-4-3+4-3-4(-4) ≡ +3+3+3+4+4+9+8+9+4+9+8(+8)
  8. 036A1952B748 → +3+3+4+3-4-4-3-3-4-3+4(+4) ≡ +3+3+4+3+8+8+9+9+8+9+4(+4)
  9. 036A259147B8 → +3+3+4+4+3+4+4+3+3+4-3(+4) ≡ +3+3+4+4+3+4+4+3+3+4+9(+4)
  10. 037B26A19584 → +3+4+4+3+4+4+3-4-4+3-4(-4) ≡ +3+4+4+3+4+4+3+8+8+3+8(+8)
  11. 037B2A691584 → +3+4+4+3-4-4+3+4+4+3-4(-4) ≡ +3+4+4+3+8+8+3+4+4+3+8(+8)

0362591A7B84 0362A147B859 0362A1958B74 0362A7B84159 0362B7A14859 0362B847A159 0369152A7B84 036A1952B748 036A259147B8 037B26A19584 037B2A691584

The above 'pitch class clock diagrams' show all 11 species of ‘dodecacircuits’ where every jump size is either ±3 or ±4 semitones. Musically, these steps are minor or major thirds (up or down) in pitch class (or, respectively, major or minor sixths, down or up). Of the 11 patterns, 10 turn up 24 times (rotations & reflections leaving their essential paths unaltered) and 1 turns up only 12 times (because it has two axes of symmetry) - the last shown, in dark blue. Thus accounting for the 252.

Two are coloured yellow; unlike the others, they are unsymmetric. The pair are, however, related – one being a reflection of the other (musical inversion) most easily perceived by flipping (say) the second one (0369152A7B84) in its horizontal (3 o’clock to 9 o’clock) axis (i.e. it cannot rotate into the other, only flip).

All others are symmetric – the brighter blue pair are so distinguished only because their single axis of symmetry happens to cut through an axis on whole hours rather than the majority whose axes of symmetry lie over half-hours. For instance the symmetry axis on 0362A147B859 is at 2 o’clock to 8 o’clock, but lies at 2:30 to 8:30 on 036A1952B748. That last one (number 8 in the above list, with the interval path +3+3+4+3-4-4-3-3-4-3+4 is depicted below:

Pitch class Sequence 036A1952B748 as a 12-Tone tone row

And at the end it will take the step of a major third to start the row afresh.

Notice that the ninth dodecagon, labelled 036A259147B8 (the first one in the third row here) is our old friend from the quartet of ‘third-up-only circuiters’ from before, labelled 047A269158B3. It turns up here because its final jump back (of –3) is in the interval class allowed by our new bidirectional interval jumping rule. To see that this is indeed the case, move the first character of the earlier label to the front, thus 047A269158B3 → 47A269158B30 (equivalent to rotating the polygonal pattern by one ‘hour’). Whereupon you now subtract 4 (or, equivalently modulo 12, add 8) to each character of this rotated label to restart it at zero. Thus 47A269158B30 → 036A259147B8, precisely the label of the ninth one above.

The labels above are chosen because they are – alphabetically – the ‘lowest valued’ ones of the (generally 24) possible ‘pitch class path’ labellings for each shape, i.e. with the longest runs of minor third runs at the start, then the major thirds, minor sixths and major sixths. Consequently the polygons drawn are the ones determined by that label.