Pentatonic Transformations

Join me in Paris (virtually) for the upcoming Analytical Approaches to World Music Conference 2020-2021, on June 9th, 5:30pm (UTC+2).

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Updated link to Session 2
Updated passcode: frM5Ln

Pentatonic Signature Transformations in Chinese Music


The pentatonic scale (CDEGA) is fundamental to Chinese music theory, and so are the transformations from one pentatonic scale to another (Du 2019). Yet, the largely pragmatic labels for these techniques are as diverse as the musical contexts they appear in. Similar moves can be described using from a multitude of different perspectives, resulting in overlapping and sometimes confusing terminology.

The theory of pentatonic signature transformations uses the precise language of mathematical music theory to enrich, complement, and shed light on existing Chinese terminologies. The theory does so by making explicit the general principles that guide musicians and contextualizing pentatonic transformations in the entirety of possibilities.

Pentatonic signature transformations expand on the original (diatonic) signature transformations based on actual key signatures (Hook 2008). Despite the subsequent generalization from diatonic scales to other scales (Tymoczko 2013), specific cases differ widely in their application.

For the pentatonic scale, there are 60 different transformations (12 notes × 5 scale positions), four of which are basic:

chromatic Tn (CDEGA → GABDE),
pentatonic tn (CDEGA → GACDE),
bian bn (CDEGA → BDEGA), and
qing qn (CDEGA → CDFGA).

The bian and qing transformations are generalized sharps or flats adapted from classical terminology. Successive applications of them can generate all pentatonic forms under enharmonic equivalence. This paper showcases four applications of the theory. First, the theory of pentatonic transformations provides fresh ways of understanding and listening to well-known songs. Second, it disambiguates key relations in debates on key names (weidiao and zhidiao) and modulations (xuangong fandiao). Third, it clarifies qupai modulations, especially the complex thirty-five-key cycle (chaoyuan) in Northeastern Shawm-and-Percussion Music (Dongbei Guchuiyue) (Li 1985). Four, it sheds light on the highly chromatic music of Li Yinghai and Bright Sheng that use pentatonic transformations as an organizing principle.

Book Launch!

New book chapter out in The Routledge Companion to Aural Skills Pedagogy! Do you sometimes wonder what people argue about when they say their solfege is better than others? A closer look finds that supporters of do-/la-minor solfege have more in common than not, and at the heart of the debate is an unanswered theoretical question about keys, not a pedagogical one. I’m giving a virtual flash presentation 8:30am EDT Wednesday May 19th if you’re curious.

The dashing purple poster should be enough reason to attend.

Event link:

Publisher website:

Pandemic Permutations

Boston pianist Cholong Park’s virtual performance of my 77 Canonic Variations on Twinkle Twinkle Little Star. Her selection ends on an especially poignant movement that remembers those lost to the pandemic. (Which is also a augmentation mirror canon–same type Bach’s holding in his portrait by Haussmann). The US may be saved by technology this time, but the country needs to urgently address the underlying issues that led to the massive, continuing deaths: the lack of healthcare, livable wage, critical thinking, and ability to manage anger and fear.

Intervals are Useless

Musicians learn three to four fundamental concepts separately in their early training: letter names, scale degrees, solfege, and intervals. They look wildly different on the page, yet they are in fact structurally the same.

Let’s compare notes (ha).

Letter namesCC#DEbEF#GbG
Inflected solfegedodirememifiseso
Scale degree namestonicsharp
Scale degree

That means you are learning the same thing four times because we have such a convoluted, patchwork music theory. If I were to put this in a more positive light, I’d say they weave a complex tapestry of history, an the rich variety of language gives us perspectives on yadda yadda. That is actually true for a history of theory class, but not fundamentals pedagogy. This is not the first time people found music theory confusing; 17th-c. musicians in Europe had it worse, and there’s a good reason we don’t pair 7 letters with 6 solfege names anymore.

In fact, the different names reflect the different reference points of each system. Letter names anchor to frequency (A = 440Hz), and the interval P1 anchors to the lowest sounding note. For scale degrees and do-minor solfege, do/^1/tonic anchor to tonal stability; for la-minor solfege, ti anchors to the leftmost sharp in a key signature, and fa anchors to the leftmost flat.

For students, if you understand one system–most likely letter names, you understand all others with a conversion chart. For teachers, this is a opportunity to streamline your teaching to make space for (gasp) actual music. For music theorists, we need to get our shit together.

Let’s look at how the assortment of concepts might conceal a straightforward calculation.

First, understand that we can add intervals.

P1 + M3 + m3 = P5

The addition above traverses a major triad. The numbers look wonky, but the math checks out (story for another time).

Second, to show that they are only different symbolically, we can replace the intervals above with letter names or scale degrees.

C + E + Eb = G
1 + 3 + b3 = 5

Weird. But OK.

Now try this question.

A flute player is trying to match the trumpet player’s pitch (and, OK, for whatever reason they can’t hear each other).
– The flute player is playing an alto flute in G, and in fact, doing so quite in tune.
– The trumpet player is playing a Bb trumpet that is a minor 2nd too sharp.
– The trumpet player is playing an arrangement of “Maria” from West Side Story, and in their score, they just reached scale-degree #4 in G major.
What note should the flute player play?

One could solve this step by step:
– scale-degree #4 in G major is C#;
– written C# on a Bb instrument is actually B (natural);
– B tuned up a minor 2nd is C;
– concert C on a G instrument is G.

Or, because they are structurally the same, we could
– add all of them up: G + Bb + m2 + #4 + G = G
– or write them as intervals: P5 + m7 + m2 + A4 + P5 = P5 = G
– or write the intervals as successive notes from C: G, F, Gb, C, G

The flute player plays a G.

Realistically, the question might look like this: “what’s the Bb clarinet’s G in concert pitch if I were to play it on my Eb saxophone,” which I’ve done countless times in a school band setting. But this goes to show that intervals are useless …not! Intervals in fact underlie all fundamental pitch concepts. But if you know letter names already, then you can already work out a lot more than you think.

Time & Pitch Multiplication

Animated version of two pairs of augmentation canons from my 77 Canonic Variations on Twinkle Twinkle Little Star. In staff notation, the y axis represents pitch, and the x axis represents time. Time multiplication makes a tune go slower/ faster, pitch multiplication contracts/expands the range. These miniatures demonstrate multiplication in both axes in a relatively simple way.

Rebarring Für Elise

Rick Cohn’s rhythmic analysis of Beethoven’s “Für Elise” (recording the bottom) blew my mind in 2014 when I was a budding theorist, and it still does today. In a very concrete way, it showed the wealth of topics on rhythm and meter beyond the mere categorization of time signatures in music theory curriculums.

A tricky transition in “Für Elise”

Despite “Für Elise” being a relatively easy piece that many pianists have played, Cohn found recordings where the transition back to the A’ section stumps the best of pianists.

Looking at the score (above), it’s easy to see how one could get lost among the many E’s and D#’s. My rebarring (below) based on Cohn’s analysis puts the tricky transition into context. Of course, time signatures were never used in this way, especially during Beethoven’s time, but I find it a great visual aid nonetheless.

My alternative barring based on Cohn’s analysis.

Starting from the notated 3/8, the triple meter gradually expands in duration, ultimately arriving at 3/2. In this imaginary 3/2 bar, each half-note beat has a different function.

  • The first half-note beat consists of rising E’s.
  • The highest E starts the second half-note beat, and it is followed by the dreaded trill.
  • The third beat get back to the original “upbeat” on the third half note.

In retrospect, the prolonged upbeat should have been a clue that things are not always as they seem, and I’m still astonished I knew this piece for so long without noticing its nuanced rhythmic trajectory.

Diatonic Slide Rule

I refer to the diatonic slide rule in my dissertation and a few upcoming essays. The slide rule was previously hosted under this now-broken link:

The grey box shows all seven members of a diatonic mode along the line of 5ths. The bottom horizontal strip show scale degrees (do-minor movable-do solfege); scale degree 1 should always be in the grey box. The middle strip show letter names (fixed-do solfege), and the top strip show what I call diatonic positions (la-minor movable-do solfege). You can slide each strip by clicking “flatwise” and “sharpwise.” To find, for example, F aeolian/natural minor, align la (the minor tonic), F, and scale degree 1.

As far as slide rules go, this one is super rudimentary, but it does clarify two aspects of diatonicism that musicians often find confusing.

First, the three strips signify independent and essential properties of diatonic notes. Some music theorists have argued for the use of one solfege over another in music curriculums (usually shouting over and/or willingly ignoring each other), and la-minor is pretty much ignored in the field of music theory. I argue that all three mainstream solfege are equally important in defining a rudimentary key label like F aeolian. (I call this the essentiality argument in a forthcoming book chapter in the Routledge Companion to Aural Skills Pedagogy.) Without scale degrees you loose the tonic (F aeolian could well be Ab major). Without letter names you loose fixed pitch (it’ll be just aeolian). Without the grey box and diatonic positions, you loose the diatonic collection (it’ll be “in F”, but the mode won’t be specified). In other words, a key label like “F aeolian” implies all all three strips/solfege.

Second, the slide rule implies a modal key space (below), which includes the usual circle of 5ths‘s major and minor keys. The slide rule, however, shows all seven modes in an interactive way that displays all seven members of a key.

7#F# lydC# majG# mixD# dorA# aeoE# phrB# loc
6#F# majC# mixG# dorD# aeoA# phrE# locB lyd
5#F# mixC# dorG# aeoD# phrA# locE lydB maj
4#F# dorC# aeoG# phrD# locA lydE majB mix
3#F# aeoC# phrG# locD lydA majE mixB dor
2#F# phrC# locG lydD majA mixE dorB aeo
1#F# locC lydG majD mixA dorE aeoB phr
0F lydC majG mixD dorA aeoE phrB loc
1bF majC mixG dorD aeoA phrE locBb lyd
2bF mixC dorG aeoD phrA locEb lydBb maj
3bF dorC aeoG phrD locAb lydEb majBb mix
4bF aeoC phrG locDb lydAb majEb mixBb dor
5bF phrC locGb lydDb majAb mixEb dorBb aeo
6bF locCb lydGb majDb mixAb dorEb aeoBb phr
7bFb lydCb majGb mixDb dorAb aeoEb phrBb loc
the modal key space

The flatwise and sharpwise clicks are actually key relations. (Note that moving one strip is the same as moving the other two strips in the opposite direction, so there’s really only two axes of movement.) Each of these key relations imply one changing solfege while the relationship between the two other solfege remain put.

  • Moving letter names = going around the usual circle of 5ths (C major, G major etc.).
  • Moving scale degrees = moving between relative keys (C major, A aeolian, D dorian etc.).
  • Moving diatonic positions = moving between parallel keys (C major, C aeolian, C dorian etc.).

These moves correspond to those in Julian Hook’s signature transformations and spelled heptachords.

Try out these key relations with the earworm that defined 2013.

  • Intro & Verse (The snow glows white…)
    – Alternation between F aeolian (4b key signature) and F dorian (3b)
    (Move diatonic positions sharpwise.)
  • Pre-chorus (Don’t let them in…): Eb mixolydian (4b)
    (Move diatonic positions flatwise back to F aeolian, move scale degrees sharpwise twice.)
  • Chorus (Let it go…): Ab major (4b)
    (Move scale degrees sharpwise.)

Sheet Music in Print!

I’m very happy I found a good medium to share my sheet music, available for purchase at Subito Music. The first batch I’m publishing are canons and similar kinds of musical puzzles from the past three years. (You can hear the tone darken from 2017 to 2020 as local and global crises unfold.)

The music is lovingly set in Bravura font. The font mimics 19th- and 20th-c. engraving and it’s very comfortable to read at a distance. All chamber and solo scores are printed on high quality 9 x 12 inch paper.

A closeup of Subtactus Canons for clarinet and bari sax.

Canon at Any Interval

This canon from 77 Canonic Variations on Twinkle Twinkle Little Star, mvt. 74 squeezes sixty copies of Twinkle Twinkle Little Star into two minutes. By adding two extra rests, the tune can accompany itself pleasingly at any pitch interval (transposition) and time interval (delay). Think of this as the ultimate 1st-species counterpoint exercise.

Above: The illustrious Jihye Chang playing the opening part of the canon. Below: My remix. You can buy the piano score here. Pianists–let me know if you’re interested in performing it.

To cycle through the different combinations, the top part shifts in time, and the bottom part shifts in pitch.

  • The top part shifts earlier by one note at each repetition (phase). Accenting the original length of the tune reveals a long tune (the green bloom in the video), making this a tempo canon.
  • The bottom part shifts down by a step at each repetition (sequence). It actually descends by more than 4 octaves. In this remix, I’ve used Shepard tones (octave cross fade) to keep it in the same octave.

The tune is built mainly out of stepwise descent, and as Scott Murphy mentions in his blog post on “Annie” time or pitch interval canons, a scale is the simplest tune that can accompany itself at all time and pitch intervals. So it didn’t take a lot of tweaking to get this melody to work as a canon; rather, I spent most of my time trying to make it interesting. The tempo canon was a happy accident that grew out of the phasing, and it provided a large-scale focus for the 26 mini canons.