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.)