Of scales and tetrachords

Yesterday I discussed the Byzantine scales and their construction from two equal tetrachords. This actually inspired another post.

What happens if we

  • do construct tetrachords that span a perfect fourth, but
  • only allow whole tones and semitones, and
  • we allow two different tetrachords in the scale?

Well, if we only allow whole tones and semitones, and the beginning and ending notes are fixed at a perfect fourth, leaving only the two middle notes movable, then you basically only have the following choices. I will notate their interval sequences in semitones (half-steps):

  • major: 2-2-1 (i.e., whole-whole-half — major second, major third, perfect fourth)
  • minor: 2-1-2 (whole-half-whole — major second, minor third, perfect fourth)
  • upper minor: 1-2-2 (half-whole-whole — minor second, minor third, perfect fourth)
  • harmonic: 1-3-1 (half-third-half—minor second, major third, perfect fourth)
  • [an odd-duck fifth member is the augmented tetrachord 2-2-2, which ends on an augmented rather than a perfect fourth]

From mixing and matching pairs of those four tetrachords (plus one odd duck), we can assemble the following:

  • major+major: Ionian mode or classical major scale. In medieval church music, this was actually called the “lascivious mode” for its association with sprightly dances.
  • major+minor: mixolydian mode (major with a flattened seventh). Very common in Anglo folk tunes and in rock and pop music derived from it. Example: “Get Back” by The Beatles. (BTW, it also contains all five notes of the major pentatonic.) [On the piano, playing a scale on the white keys but starting from G rather than C gives you G mixolydian  — G major would have had F# rather than F natural.]
  • minor+minor: dorian mode (minor with a raised sixth). Very common in Anglo folk tunes and in rock and pop music derived from it. Has a more ‘minor’ feel than the mixolydian. Examples: “What shall we do with the drunken sailor”,  “Scarborough fair”,… (BTW, it also contains all five notes of the minor pentatonic, which is the backbone of the blues “scale” — which is really more like an Indian raga rather than a scale.) [On the piano, playing a scale on the white keys, but starting from D rather than C, gives you D dorian — D minor would have had Bb rather than B natural.]
  • minor+upper minor: aeolian mode, or natural minor scale.
  • minor+augmented: harmonic minor scale. In Western classical music that follows common-practice harmony, the need for major chords on the dominant (fifth) step automatically requires accidentals to temporarily raise the seventh. Hence a classical piece “in X minor” actually will pop in and out of harmonic (and melodic) minor rather than stay in natural minor. Using minor chords on the dominant instead automatically will give the piece a “modal” (“churchy” or “folky”) feel.
  • minor+major: melodic minor scale (ascending). Unlike the harmonic minor scale, the “un-flattened” sixth eliminates the minor third. Generally, classical melodies in a minor scale follow melodic minor when ascending, and natural minor when descending, although the locally prevailing harmony may dictate variances from this.
  • upper minor+upper minor: Phrygian mode. This “more minor than minor” or “martial” scale has occasionally been used in Western classical music (e.g., “Mars” from Holst’s The Planets), and is fairly common in darker heavy metal tunes. (E.g., “Harvester of Sorrow” by Metallica, the opening of “Sails of Charon” by The Scorpions, the fast middle section of “Seventh Son” by Iron Maiden) Among film composers, Phrygian for battle scenes is something of a cliché.
  • harmonic+upper minor: Phrygian major, a.k.a. Flamenco scale, a.k.a. “Jewish scale”. This Phrygian mode with a raised third is indeed a staple of Flamenco music, but can also be heard in the synagogue (the ahava rabba mode), in klezmer music, and indeed in some heavy metal tunes — for example: “Forty-six and two” by Tool, or the opening of “Killing in the name of…” by Rage Against The Machine.
  • harmonic+harmonic: double-harmonic or “Byzantine” or “Arabic” scale. It can be derived from the previous mode by flattening the sixth. Used for an exotic feel by classical composers (Debussy was rather fond of it), and sometimes in hard rock and metal. Dick Dale‘s instrumental “surf music” classic “Misirlou” is a very nice illustration.

An odd duck in the above list is the Lydian mode, which is built from augmented+major tetrachords. In a sense “more major than major”, it’s rarely used for a whole piece (the theme from The Simpsons is one example of a popular tune in the Lydian mode), but episodically can be used to set a mood of hope or anticipation. An example is the verse of “Freewill” by Rush.

OK, I guess this is an excuse for posting two of my favorite tunes:


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