Of Moonlight and Bridges

This post began with this work by Alan Theisen, where he figures out all the ways to combine four triads (major, minor, diminished or augmented) to exactly cover all twelve notes.

It ended up somewhere very different, but let's start there.

Exclusive Quartets of Triads

There are only six possibilities -- that is, six properly different ways to get the total chromatic scale out of the four common triads. For any of of them, there are 3 ways to split them into a pair of hexachords (i.e. to partition the four triads into two pairs). In many cases the hexachords have the same Forte number. Here's the breakdown:

• C Maj + D Maj + Ab Min + Bb Min
  • (C Maj + D Maj) [f6-33] + (Ab Min + Bb Min) [f6-33]
  • (C Maj + Ab Min) [f6-20] + (D Maj + Bb Min) [f6-20]
  • (C Maj + Bb Min)[f6-z29] + (D Maj + Ab Min) [f6-z50]
• C Maj + F# Maj + D Min + Ab Min
  • (C Maj + F# Maj) [f6-30] + (D Min + Ab Min) [f6-30]
  • (C Maj + D Min) [f6-32] + (F# Maj + Ab Min) [f6-32]
  • (C Maj + Ab Min) [f6-20] + (F# Maj + D Min) [f6-20]
• D# Dim + G Dim + B Dim + C Aug
  • (D# Dim + G Dim) [f6-z49] + (B Dim + C Aug) [f6-z28]
  • (D# Dim + B Dim) [f6-z49] + (G Dim + C Aug) [f6-z28]
  • (G Dim + B Dim) [f6-z49] + (D# Dim + C Aug) [f6-z28]
• C Maj + Bb Min + D# Dim + G# Dim
  • (C Maj + Bb Min) [f6-z29] + (Eb Dim + Ab Dim) [f6-z50]
  • (C Maj + D# Dim) [f6-27] + (Bb Min + G# Dim) [f6-27]
  • (C Maj + G# Dim) [f6-31] + (Bb Min + D# Dim) [f6-31]
• C Maj + Ab Min + Db Aug + Bb Aug
  • (C Maj + Ab Min) [f6-20] + (Db Aug + D Aug) [f6-20]
  • (C Maj + Db Aug) [f6-31] + (Ab Min + Bb Aug) [f6-31]
  • (C Maj + Bb Aug) [f6-34] + (Ab Min + Db Aug) [f6-34]
• C Maj + Eb Min + G# Dim + Db Aug
  • (C Maj + Eb Min) [f6-z49] + (G# Dim + Db Aug) [f6-z28]
  • (C Maj + G# Dim) [f6-31] + (Eb Min + Db Aug) [f6-31]
  • (C Maj + Db Aug) [f6-31] + (Eb Min + G# Dim) [f6-31]

There are 11 different hexachords across all these:

• [f6-20] Augmented Hexatonic (self-complement)
• [f6-27] Whole Half subset (self-complement)
• [f6-z29] Harmonic Minor subset
• [f6-z28] Harmonic Minor subset
• [f6-30] Whole Half subset (self-complement)
• [f6-31] Harmonic Minor subset (self-complement)
• [f6-32] Diatonic subset (self-complement)
• [f6-33] Diatonic subset (self-complement)
• [f6-34] Melodic Minor subset (self-complement)
• [f6-32] Diatonic subset
• [f6-z49] Whole-Half subset
• [f6-z50] Whole-Half subset

I've marked "self-complement" where a set of notes you choose is the same as the set you don't choose, like with the whole tone scale. As I said, I'm hoping to revisit some of the more interesting ones in detail in due course.

A practical question: How are we to learn these? We don't just want to "commit them to memory" but to internalise them at the level they can be available for improvisation, but the first step to the latter is the former, to get ourselves off the page and start really listening. One way to get started is to focus on the movement of the roots of the triads (almost like the "bassline" of the idea) and the way the thirds and fifths interlock above those roots.

Alan Theisen has tasting notes that are very enlightening and help in trying to think about the identity of each of these. I'm intrigued by the hexachords that come from each adjacent pair of triads, too.

Finding Moonlight

For now, though, something else strikes me about these. Look at these two lines:

• (C Maj + Bb Min)[f6-z29] + (D Maj + Ab Min) [f6-z50]
• (C Maj + Bb Min) [f6-z29] + (Eb Dim + Ab Dim) [f6-z50]

These show there are two different ways to partition f6-z50 into a pair of common triads. That seems like it's probably an unusual property for a hexachord, although I haven't checked. Here's those two partitions visualized -- in the top diagram, Eb Dim is red and Ab dim is green, in the bottom one D Maj is red and Ab Min is green:

In my notes I've labelled this hexachord "Moonlight" because I associate it with Debussy’s Preludes, Book 2 no 7, "Terrace of Moonlit Audiences" -- that's a bit spurious as the complement, f6-z29, appears more explicitly but I like the name so there you have it. Here's the piece:

At the very least, 6-z50 is a subset of the Half-Whole Diminished scale and Debussy's music is full of those.

A more linear way to find Moonlight is to start with the minor pentatonic scale (1-b3-4-5-b7), raise the 4th to a #4 (1-b3-#4-5-b7) and add a b9 (1-b2-b3-#4-5-b7). This is a somewhat similar construction to the one that started our investigation of Suvarnangi Minor 6. Thinking about this in relation to the Dmaj-Abmin partition, we start with Ab minor pentatonic and the Dmaj is the major triad built on the b5. If we think of the Dmaj as the root, the minor pentatonic is built at the b5 instead. This sounds more complicated than it looks on paper.

This led me to see Moonlight in another way, too: as Abm7 + Bm7. This is an instance of the minor up a minor third trick.

The Moonlight Hypermodes

Six-note structures are nice, but it's always interesting to know how they embed in seven-note scales since those provide a bit more scope of harmonization and tend to have fewer large gaps (so make a certain kind of melodic writing easier). In the following we take Moonlight to be rooted at the root of the minor triad, so in the example we've been using, Ab would be the root and Moonlight would be spelled 1-b2-b3-#4-5-b7. We look at what adding each possible note in turn produces, and the results are three Forte set classes, each of which comes in a pair that are inversions of each other. I've included all the modes in the group that have "proper" names in the book, although some of them are made up by me:

• 7-z18 (contains three consecutive semitones)
  • 7-z18A (Add natural 7): Divyamani
  • 7-z18B (Add natural 2): Dhavalambhari
• 7-29 (contains two consecutive semitones)
  • 7-29A (Add flat 6): Bhavapriya / Vagadhisvani / Naganandini
  • 7-29B (Add 4): Superaugmented / Ratnangi / Marva
• 7-31 (no consecutive semitones: Half-Whole subsets)
  • 7-31A (Add natural 6): Nasika Bhushani / Augmented Minor / Dorian Marva
  • 7-31B (Add natural 3): Marva Dominant

The Bridge Chord

If 6-z50 is Moonlight it makes some sort of sense to call its complement, 6-z29, "Sunlight", but it has a more traditional name, the Bridge Chord. It's named after composer Frank Bridge -- here's his Piano Sonata to set the mood:

It can be visualized as a C major triad combined with a Bb minor triad. It has a very natural C7 sound, made somewhat jazzy by the b9 (the third of Bbm) but somewhat un-jazzy by the natural 11 (the fifth of Bbm). Here C major is in red, Bb minor is in green:

Again we can take a more linear approach by thinking of a pentatonic, this time Bb minor 6 pentatonic (1-b3-4-5-6). We again raise the 4, giving 1-b3-#4-5-6, and add a 2, but this time we add the natural 2 resulting in a Bridge chord, 1-2-b3-#4-5-6.

The Moonlight Chord and the Bridge Chord are "z-partners", meaning they're not modes or inversions of each other but they share all the same intervals. In 12EDO z-partners are relatively unusual and always come in pairs (in other tuning systems, other phenomena are possible).

The Bridge Hypermodes

We'll do the same exercise as before, finding all the heptatonics we can make by adding a note to the Bridge chord. Again we'll take the pentatonic-derived spelling that I think guitar players will find most natural, 1-2-b3-#4-5-6.

• 7-z19 (contains three consecutive semitones)
  • 7-z19A (Add flat 2): Gavambodhi / Hatakambari
  • 7-z19B (Add b6): Ganamurti / Visvambhari / Syamalangi
• 7-25 (contains two consecutive semitones)
  • 7-25A (Add major 3): Super Locrian bb3 bb5 bb6
  • 7-25B (Add 4): Midlocrian
• 7-32 (contains no consecutive semitones)
  • 7-32A (Add b7): Harmonic Minor
  • 7-32B (Add 7): Sarasangi / Lydian Minor

These observations along with the similar ones for Moonlight give us ways to think about adding notes to the basic hexachords, which I think helps expand our understanding of them a bit. While each of these heptatonics is a subset of seven different hexatonics, this exact combination of sounds -- which I've called in the past the "7-spectrum" -- is unique to each hexatonic chord.

Moonlight and Bridge

Returning to the original idea, we can combine a Moonlight and a Bridge to cover the whole chromatic scale. Another way to say this is that when we made heptatonics by adding various notes to the Bridge chord, those notes were precisely the notes of a Moonlight chord and vice versa.

To be specific, the complementary pair is a Bridge with a Moonlight a whole tone below it. To remember which way round they go, imagine being surprised to look down from a bridge and see the moon reflected in the water. Moonlight below a Bridge is far more memorable than above it! But realistically, we'll learn these sounds by ear and eye more than by calculation, which is just there to get us off the page as quickly as we can.

The twelve hypermodal groups I've listed out here point towards deeper waters than we can reach right away. It would be a good idea to first internalise the sounds of Bridge and Moonlight -- especially, but not only, in their interlocked form that covers the whole chromatic scale.