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.

Fifths Are Everywhere

Staff lines, letter names, and intervals emphasize pitch proximity: diatonic steps and semitone alternations; but fifths occupy an equally important position in the fundamental organization of pitch.

Letter names & intervals

Fifths untangle the messy order of intervals and scales.

Where does the ordering PMMPPMMP come from?

Perfect fifths (or their inverse, perfect fourths) generate all letter names; other intervals can’t do that.

Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# F##

It is no surprise that intervals can be also ordered by perfect fifths. Doing so reveal the spectrum of qualities. (Since letter names and intervals can be generated using fifths, we can alternatively express intervals as combination of fifths and octave displacements.)

Intervals from middle C; top notes are generated by fifths, then folded into the same octave.

Scales & key signatures

Seven fifths make a diatonic scale. The tone/semitone pattern of TTSTTTS can be more efficiently described as seven fifths, which key signatures imply.

Seven fifths make a diatonic scale.

The circle of fifths show how each shift swaps one note for another, and crucially, why the tonic moves by a fifth. Because the ordering by fifths traces back to scales and letter names themselves.

Intervals & chords in diatonic scales

Perfect and diminished fifths also reveal the hidden structure behind interval and chord qualities in diatonic scales.

Where does the ordering MmmMMmdim come from?

In a diatonic scale, all fifths are perfect except for B to F, a diminished fifth. From the perspective of fifths, this little asymmetry creates the rich variety of intervals and chords in diatonic scales. When intervals in diatonic scales are arranged in generic fifths, the quality changes whenever B transposes to F (arrows below).

Intervals in diatonic scales arranged in fifths.

Chords are different intervals combined. Chords (or any set of notes) in a diatonic scale have as many qualities as there are notes, since each of those B’s have to loop back to F’s at different points. That’s why there’s three types of triads and four types of seventh chords in diatonic scales. Of course, adding accidentals like raised leading tones in minor can further alter these chords.

Chords in diatonic scales arranged in fifths.

TL;DR: Symbolically, abstract pitch concepts emphasizes pitch proximity (steps and semitones); but they are underpinned by a deeper organization based on fifths.

A Pythagorean Postlude

The fifth-based approach above is loosely Pythagorean. Pythagorean tuning uses pure octaves (2:1 frequency, 1:2 string length) and pure fifths (3:2 frequency, 2:3 string length) to generate all notes. I say ‘loosely’ because the pitches above can be mapped to any kind of tuning, and Pythagoreans did other things like worshipping numbers and abstaining from beans. Other tunings imply different kinds of pitch structure. For example, in equal temperament (12TET), F# = Gb, which means that music notation contains redundancies in 12TET. In Pythagorean tuning, the difference between F# and Gb is real (a Pythagorean comma). It doesn’t mean that staff notation and intervals must be tuned in a Pythagorean way, but it best correlates with the methods and structure of Pythagorean tuning.

Midi, Zoom, & Me

Here are some easy, lightweight, plug & play options for sharing music from a MIDI keyboard over Zoom.

MIDI keyboards don’t usually generate sound. You need to plug it into a synthesizer or computer. Here are 4 broad types of options.

For the best chances of success, plug the keyboard in before opening an app. For web apps, use Chrome if possible. In many cases, you also have to select your MIDI keyboard model as the MIDI Input. Audio Output should be the model of your speaker or headphone.

1. Online rooms

EASIEST. With a shared link, people on the other end can listen and even jam from the browser. No zoom is required at all.

My pick: https://musiclab.chromeexperiments.com/Shared-Piano/

Others: https://www.multiplayerpiano.com/

2. Online synths

These web apps generate sound from your MIDI keyboard, but you’d have share it through Zoom.
In Zoom, select share screen, select your browser, and tick shared computer sound.
For sound only, select share screen, go to the advanced tab, select share computer sound only.

My pick: https://dotpiano.com/

https://www.caseyrule.com/projects/piano https://virtualpiano.eu/
https://recursivearts.com/virtual-piano/ https://midi.city/
https://synth.playtronica.com/ https://www.webaudiomodules.org/wamsynths/ https://www.gsn-lib.org/apps/cardboardsynth/index.html

Online notation apps like Flat also have in-app keyboards: https://flat.io/help/en/music-notation-software/midistart.html. To play instead of edit, turn off write while playing.

Note: Web apps are not secure; some explicitly broadcast your music. So don’t encode your bank account number as musical notes.

3. Downloadable synths

Mac users rejoice!

GarageBand comes with Macs; it is a basic DAW (Digital Audio Workstation) that can handle MIDI and much more. See https://support.apple.com/kb/PH25066 and https://support.apple.com/kb/PH25039.

There’s still all sorts of piano apps for Windows users, like https://sourceforge.net/projects/freepiano/.


If you don’t like default sound(s) of the apps, there’s a whole wide world of VST instruments. Many are free, some cost you an arm and a leg. You can load VST’s into your DAW, or use a VST host like https://www.tone2.com/nanohost.html or https://www.cantabilesoftware.com/. Here’s some free piano VSTs: https://blog.landr.com/best-free-piano-vsts/. Some, like https://www.native-instruments.com/en/catalog/free/ also provide a standalone app. This is a rabbit hole I’m not diving any further into…

Audio Routing

Last but not least, audio applications often override or bypass normal sound settings, meaning Zoom sometimes doesn’t pick it up the audio at all. (This is why I’m kinda enamored by the web synths.) Test your setup over Zoom with a friend. If it doesn’t work, try the following.

For Mac users, Zoom has a ZoomAudioDevice that your DAW can output to.

If nothing else works, you may have to route your computer sounds (and microphone output) into a virtual input, and then get Zoom to recognize the virtual output as its input. These are super finagly. Here are two popular audio routing apps: https://rogueamoeba.com/loopback/ (Mac), https://www.vb-audio.com/Cable/ (Win).

4. Go Acoustic

Swap your MIDI keyboard for a portable electronic piano (which usually includes MIDI out) or a melodica. Make sure you enable and turn on original sound. https://support.zoom.us/hc/en-us/articles/115003279466-Enabling-option-to-preserve-original-sound

The melodica is young Jacob Collier’s instrument of choice.