The Modes in “Circle of Fifths” Order

I mostly won’t be analyzing the modes in their traditional order, since I’m analyzing how lowering a regular pattern of notes by a half-step each walks us through every mode on every key. A few notes (pun intended):

• Most of this analysis focuses on twelve-tone equal temperament (12-TET for short), the tuning system the vast majority of Western music has used for hundreds of years. Where exceptions to what I write exist in other temperaments, I will attempt to note them, but I can’t promise to have thought of them all, especially because my understanding of other tuning systems is spotty at best.
• For the most part, I will attempt to avoid bogging this analysis down in heavy mathematics, but I must begin this analysis by noting that in 12-TET, every note on the chromatic scale vibrates at exactly 2^¹⁄₁₂ its immediate neighbor’s frequency – that is, the notes are spaced exactly evenly. This is an important precondition for the analysis I’ve conducted here: much of it wouldn’t apply otherwise.
• In 12-TET, A♯ and B♭ represent the same pitch. Likewise B and C♭, C♯ and D♭, and so on. However, in terms of music theory, A♯ and B♭ can be semantically quite different in some contexts – be careful.
• Key signatures with seven sharps or seven flats exist, but I’ve mostly omitted them from my analysis of Ionian’s modes. (There are even monstrosities with double-sharps [𝄪] and double-flats [𝄫] … which I mostly ignore. I can at least tolerate the double-flat symbol, but the double-sharp is a monstrosity that doesn’t resemble even slightly the thing it’s supposed to double.) I feel it was extravagant enough just to include both six-flat and six-sharp key signatures.
• If we start from Lydian mode, lowering the correct pitches one at a time by a half-step each gives us the next mode in the cycle. I’ll use C Lydian as an example to demonstrate the principle:

• In other words:
  • Lowering C Lydian’s F♯ to F gives us C Ionian (i.e., C major).
  • Lowering C Ionian’s B to B♭ gives us C Mixolydian.
  • And so on, until we reach C Locrian – whence the pattern repeats for C♭/B, the key below C on the chromatic scale.
• Lowering Locrian’s root gives us Lydian mode in the key below it on the chromatic scale.
  • In other words, lowering C Locrian’s C to C♭/B gives us C♭/B Lydian.
• The sequence repeats from there:
  • Lowering B Lydian’s E♯ gives us B Ionian.
  • Lowering B Ionian’s A♯ gives us B Mixolydian.
• This sequence will repeat for every note on the chromatic scale: C, C♭/B, B♭/A♯, A, A♭/G♯, G, G♭/F♯, F, F♭/E, E♭/D♯, D, D♭/C♯, and back to C.

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The Principles of Inverse Operations

• Now, note that we’ve effectively been subtracting 1 from a regular pattern of values in a sequence:
  • 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1…
• As a result, if we consult the mathematical properties of inverse operations, we can infer that converse of everything we’ve just done will apply if we add 1 to the same notes in the reverse order:
  • 1, 5, 2, 6, 3, 7, 4, 1, 5, 2, 6, 3, 7, 4…
• In other words:
  • Raising C♭ Lydian’s C♭ gives us C Locrian.
  • Raising C Locrian’s G♭ gives us C Phrygian.
  • Raising C Phrygian’s D♭ gives us C Aeolian.
  • And so on.

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An Audio Demonstration

• None of this will mean much to readers in isolation, so I put together a crude demonstration of the above principle ⟨aaronfreed.github.io/c_lydian_to_b_lydian.flac⟩ in Logic Pro. For the following modes, in order:
  1. C Lydian
  2. C Ionian
  3. C Mixolydian
  4. C Dorian
  5. C Aeolian
  6. C Phrygian
  7. C Locrian
  8. C♭/B Lydian
  you’ll hear the following, in order:
  1. the scale in ascending order
  2. arpeggiated and block versions of:
     1. the root (I) chord
     2. the subdominant (IV) chord
     3. the dominant (V) chord
     4. the root (I) chord
  (I could make a version going down the entire chromatic scale, but the entire thing would take thirteen and a half minutes to play through. Even I think that’s overkill, and I submit this exact page as exhibit A for what my definition of “overkill” must entail.)

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Further Notes

• It might help solidify your conception of this principle to understand that while we’re lowering (or raising) a pitch on our scale, we’re also, in a sense, jumping down (or up) a fifth. F Lydian is in the same key signature as C Ionian, not F Ionian. This is why so many of these patterns mirror the circle of fifths.
• In “Modes Descending from Lydian” above, the mode number and the pitch we lower to get it are exact mirrors of each other. This is not coincidental – it’s the direct result of a mathematical pattern.
• The modes are also always either three steps on the scale below their predecessors (usually, but not always, a perfect fourth), or four steps above (usually, but not always, a perfect fifth). In this context, they’re equivalent, since the scale repeats every octave.
  • Although this is only somewhat relevant here, one pair of intervals always constitutes an exception to the “perfect fourths” and “perfect fifths” rules: one fourth is always augmented, and one fifth is always diminished. These are actually the same interval, a tritone. The following steps on each scale are tritones in either direction:

    Diabolus in mūsicā
    #	Mode	Tritone
    4	Lydian	4	1
    1	Ionian	7	4
    5	Mixolydian	3	7
    2	Dorian	6	3
    6	Aeolian	2	6
    3	Phrygian	5	2
    7	Locrian	1	5

    Note that the third columns of “Diabolus in mūsicā” and of “Modes Descending from Lydian” are identical. This is also not coincidental.

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The Major Scale’s Modes & the Circle of Fifths

12 Major Scales × 7 Modes = 84 Combinations

• I’ve printed the C major scale’s modes in bold, blue type to make them stand out. These are the only scales played using only the white keys of the piano (this is also why music notation can express these specific keys without accidentals, either in the key signature or after it). The differences between their numbers are always multiples of twelve; this is also not a coincidence.
• Halfway between C major’s modes, I always print a mode twice, in orange, with its key signature first using six flats (G♭ major), then six sharps (F♯ major). I don’t really like either option (E♯ is F♮! C♭ is B♮!), but both avoid a repeated letter in a scale, and thus a rash of accidentals in the notation of any piece that uses them. G♭/F♯ major’s modes are also always separated by multiples of twelve, for the same reason: this progression separates all seven modes of each scale from each other by multiples of twelve.
• I’ve listed these descending by pitch so higher pitches will be, well, higher, which may confuse some people since we read from top to bottom and are used to thinking of harmony in ascending order. Someday, I plan to write a JavaScript add-on to give readers an option to reverse the order – and eventually, to give them then option to create similar tables for different scales (at the bare minimum, melodic minor, harmonic minor, and chromatic Hypolydian; possibly others as well).
• “RM” is an initialism for “Relative Major”, and “KS” is for “Key Signature”. A cheat sheet for what modes use what key signatures can be found in the section immediately following this one.
• I’ve used zero-based indexing for these tables, so they’re indexed from zero to eighty-three. This is partly because I prefer zero-based indexing, but to be honest, it was mostly to give C Ionian an index of one.

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Key Signature Cheat Sheet

• Twelve-tone equal temperament has fifteen key signatures, 𝄪/𝄫 atrocities notwithstanding:

• Legend for the above:
  • Lyd = Lydian
  • Maj = major (Ionian)
  • Mix = Mixolydian
  • Dor = Dorian (if you’re not sure why it’s grey, ask your English teacher about the picture)
  • Min = natural minor (Aeolian)
  • Phr = Phrygian
  • Loc = Locrian
  • KS = key signature (highlighted because it’s the key to the table – pun coincidental, though I definitely didn’t even try to avoid it)
• Additional notes:
  • 5♯ and 7♭ express the same pitches.
  • 6♯ and 6♭ express the same pitches.
  • 7♯ and 5♭ express the same pitches.
  • I’ve again printed C major’s modes in blue and F♯/G♭ major’s modes in orange.
• Accidentals also fall onto the circle of fifths (e.g., B♭ to E♭ and E♭ to A♭ are both perfect fifths).

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Why is this happening?

The simple answer: whether we realized it or not, we’ve been traversing the circle of fifths this entire time. My introduction notes that traveling from C Lydian to C Ionian is, in a sense, traveling from G major to C major. Here’s the C table again. Note Relative Major’s traversal down the circle of fifths:

The explanation is that each of these modes, apart from C Ionian, has been rearranging a different major scale. Reshuffling each mode back into its Ionian form may explain the cause:

Every note of every scale in this table is, in fact, a perfect fifth below its counterpart in its predecessor. That’s the first part of the explanation. The second part has to do with interval distributions:

The Ionian scale is virtually unique among seven-note scales in 12-TET in that it is possible to swap two intervals and produce a different mode of the same scale, and the one other scale that displays this trait is actually its polar opposite in virtually every important way. In the section below on the pentatonic scale, I will go over more about why, precisely, this is. For now, the important point is that each step down the circle of fifths order changes the position of only one interval. If more intervals changed, the pattern would break.

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Chord Analysis by Mode

• I’ve printed each mode’s root (I) subdominant (IV), and dominant (V) chords in bold – they’re especially important to establishing its tonality.
• I’ve printed the seventh chord in fainter text because it’s diminished in the major scale.
• Minor chords are blue; diminished chords are orange.
• This analysis clarifies why Ionian became the standard major scale and Aeolian became the natural minor scale: Ionian’s root, subdominant, and dominant are all major; Aeolian’s are all minor. This makes them feel, respectively, especially major and especially minor.
• Since their root chords are major, Lydian and Mixolydian still sound more “major” than the other modes (except Ionian), and Dorian and Phrygian still sound more “minor” than the others (except Aeolian). Locrian, as mentioned, is the oddball because it’s the only mode of the scale whose root chord is diminished.

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Beyond the Major Scale

Other Heptatonic Scales & Tonalities

While the above analysis focuses exclusively on the Ionian scale’s modes, numerous possible scales (and modes thereof) don’t fit its pattern, many of which I’ll now analyze. Some examples:

• Numerous heptatonic (seven-note) scales that aren’t modes of the major scale. I’ve rooted the following examples (many of which are common in jazz) in C to make it clear how they change the Ionian scale:

  Other Heptatonic Scales
  Scale		1	2	3	4	5	6	7	Intervals
  Chromatic Dorian	Kanakāngi	C♮	D♭	E𝄫	F♮	G♮	A♭	B𝄫	½	½	1½	1	½	½	1½
  Mixolydian ♯1	Super-Locrian ♭7	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫	½	1	½	1	1	½	1½
  Locrian ♭4	Super-Locrian	C♮	D♭	E♭	F♭	G♭	A♭	B♭	½	1	½	1	1	1	1
  Locrian ♯6	Maqam Tarznauyn	C♮	D♭	E♭	F♮	G♭	A♮	B♭	½	1	1	½	1½	½	1
  Dorian ♭2	Jazz minor inverse	C♮	D♭	E♭	F♮	G♮	A♮	B♭	½	1	1	1	1	½	1
  Phrygian ♯3	Phrygian dominant	C♮	D♭	E♮	F♮	G♮	A♭	B♭	½	1½	½	1	½	1	1
  Major Phrygian		C♮	D♭	E♮	F♮	G♮	A♭	B♮	½	1½	½	1	½	1½	½
  Aeolian ♭5	Half-diminished	C♮	D♮	E♭	F♮	G♭	A♭	B♭	1	½	1	½	1	1	1
  Harmonic minor ♭5		C♮	D♮	E♭	F♮	G♭	A♭	B♮	1	½	1	½	1	1½	½
  Aeolian ♯7	Harmonic minor	C♮	D♮	E♭	F♮	G♮	A♭	B♮	1	½	1	1	½	1½	½
  Ionian ♭3	Melodic minor	C♮	D♮	E♭	F♮	G♮	A♮	B♮	1	½	1	1	1	1	½
  Double harmonic minor		C♮	D♮	E♭	F♯	G♮	A♭	B♮	1	½	1½	½	½	1½	½
  Dorian ♯4	Lydian diminished	C♮	D♮	E♭	F♯	G♮	A♮	B♭	1	½	1½	½	1	½	1
  Mixolydian ♭6	Aeolian dominant	C♮	D♮	E♮	F♮	G♮	A♭	B♭	1	1	½	1	½	1	1
  Ionian ♯5	Ionian augmented	C♮	D♮	E♮	F♮	G♯	A♮	B♮	1	1	½	1½	½	1	½
  Lydian ♭7	Lydian dominant	C♮	D♮	E♮	F♯	G♮	A♮	B♭	1	1	1	½	1	½	1
  Phrygian ♭1	Lydian augmented	C♮	D♮	E♮	F♯	G♯	A♮	B♮	1	1	1	1	½	1	½
  Chromatic Dorian inverse		C♮	D♯	E♮	F♮	G♮	A♯	B♮	1½	½	½	1	1½	½	½
  Lydian ♯2	Aeolian harmonic	C♮	D♯	E♮	F♯	G♮	A♮	B♮	1½	½	1	½	1	1	½
  Chromatic Hypophrygian		C♮	D♯	E♯	F♯	G♮	A♯	B♮	1½	1	½	½	1½	½	½
  Superlydian augmented		C♮	D♯	E♯	F♯	G♯	A♯	B♮	1½	1	½	1	1	½	½

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  Since many scales have up to dozens of names (if you’re on a computer, hover over dotted scale names for more information), I’ve instead used the sorting methods of Southern India’s Carnatic Numbered Mēḷakartā, which effectively alphabetize the scales themselves: sort intervals in ascending order, with the lowest intervals counting as most significant. (§9 lists the mēḷakartā in full; note that they categorically exclude scales with sharp thirds or fifths, or flat fourths or fifths.) Bold, orange text denotes raised notes and minor thirds; thin, blue text denotes lowered notes and minor seconds (doubly lowered notes are also fainter).

  This list is far from exhaustive. Note that none of these scales use B♯ because, while it’s semantically different from C, the two notes represent the same pitch in twelve-tone equal temperament. Likewise, none of these scales use D𝄫 because it’s the same pitch as C.

  In any case, one thing distinguishes these scales from the Ionian scale’s modes: we can print the latter with no accidentals outside the key signature, but every single scale in the above table requires accidentals for at least some notes. Key signatures are strictly based on the Ionian scale and the circle of fifths; using three flats that weren’t B♭, E♭, and A♭ in a key signature would just confuse readers.

  The names “Chromatic Dorian” and “Chromatic Hypophrygian” will reappear below in the section on ancient Greek harmony below. (Don’t waste time trying to discern their connections to the similarly named diatonic modes: Chromatic Dorian actually relates to Phrygian, and Chromatic Hypophrygian to Mixolydian. The section on Greek harmony explains the relationships, and why they aren’t the ones their names suggest.)

• Note that several sets of the above scales are modes of each other. In particular:

  Harmonic Minor & Melodic Minor’s Modes at a Glance
  Scale		1	2	3	4	5	6	7	Intervals
  Aeolian ♯7	Harmonic minor	C♮	D♮	E♭	F♮	G♮	A♭	B♮	1	½	1	1	½	1½	½
  Locrian ♯6	Maqam Tarznauyn	C♮	D♭	E♭	F♮	G♭	A♮	B♭	½	1	1	½	1½	½	1
  Ionian ♯5	Ionian augmented	C♮	D♮	E♮	F♮	G♯	A♮	B♮	1	1	½	1½	½	1	½
  Dorian ♯4	Lydian diminished	C♮	D♮	E♭	F♯	G♮	A♮	B♭	1	½	1½	½	1	½	1
  Phrygian ♯3	Phrygian dominant	C♮	D♭	E♮	F♮	G♮	A♭	B♭	½	1½	½	1	½	1	1
  Lydian ♯2	Aeolian harmonic	C♮	D♯	E♮	F♯	G♮	A♮	B♮	1½	½	1	½	1	1	½
  Mixolydian ♯1	Super-Locrian ♭7	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫	½	1	½	1	1	½	1½
  Ionian ♭3	Melodic minor	C♮	D♮	E♭	F♮	G♮	A♮	B♮	1	½	1	1	1	1	½
  Dorian ♭2	Jazz minor inverse	C♮	D♭	E♭	F♮	G♮	A♮	B♭	½	1	1	1	1	½	1
  Phrygian ♭1	Lydian augmented	C♮	D♮	E♮	F♯	G♯	A♮	B♮	1	1	1	1	½	1	½
  Lydian ♭7	Lydian dominant	C♮	D♮	E♮	F♯	G♮	A♮	B♭	1	1	1	½	1	½	1
  Mixolydian ♭6	Aeolian dominant	C♮	D♮	E♮	F♮	G♮	A♭	B♭	1	1	½	1	½	1	1
  Aeolian ♭5	Half-diminished	C♮	D♮	E♭	F♮	G♭	A♭	B♭	1	½	1	½	1	1	1
  Locrian ♭4	Super-Locrian	C♮	D♭	E♭	F♭	G♭	A♭	B♭	½	1	½	1	1	1	1

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  We can apply similar principles to them as we applied to the Ionian scale’s modes, though certain things may not line up as neatly with some of the scales. I leave ascertaining which other sets of scales are modes of each other as an exercise for the reader. (I didn’t list every mode of most other sets of scales.) The next two sections analyze these modes in much greater depth.

  Note that two competing shorthands exist for transformations like “Aeolian ♯7”, relative to either:

  1. The current note: “♯7” always means “raise the base scale’s seventh note half a step”. If it’s currently A♭, make it A♮. If it’s currently A, make it A♯. If it’s currently C♯, make it (ugh) C𝄪 (and Jesus wept, for there were no more worlds to conquer… wait, that’s not how it goes).
  2. The C scale’s accidental: C Aeolian 7 is B♭, so ‘Aeolian ♮7’ = ‘raise Aeolian’s seventh note half a step’.

  Not gonna front: I despise the latter with the burning passion of a thousand suns. “♯7” applies independently of the root, while in any key except C, “♮7” requires asking “wait, what accidental is on C Aeolian 7?” C♯ Aeolian 7 is already B♮, so what does “C♯ Aeolian ♮7” even tell us? F♯ Aeolian 7 is C♯, so if we read “F♯ Aeolian ♮7” literally, we’d lower the seventh note by a half-step – exactly the opposite of what it intends!

  Then again, “Lydian ♮4” (“lower Lydian’s fourth note a half-step”) and “Mixolydian ♮7” (“raise Mixolydian’s seventh note a half-step”) make for the most pretentious possible ways to say “major” this side of «διατονικός Λῡ́δῐος τόνος» (“diatonic Lydian tonos”; see the section on Ancient Greek harmony below), so I guess there is one actual upside to the C-based convention.

Beyond Pentatonic and Heptatonic Scales

Of course, scales are hardly required to contain seven notes. Due largely to twelve being divisible by neither five nor seven, pentatonic (five-note) scales provide so much to unpack that I’m giving them their own section. Let’s start with a brief overview of other scale sizes. In order of scale size:

• Hexatonic (six-note) scales are fairly common. The whole-tone scale is likely the most familiar example.
• Scales also aren’t limited to seven notes; blues and jazz often use octatonic (eight-note) and enneatonic (nine-note) scales. Decatonic (ten-note) scales are slightly less common, but still occur.
• The twelve-tone equal temperament (12-TET) ⟨en.wikipedia.org/wiki/12_equal_temperament⟩ used in the vast majority of Western music for centuries only allows for one possible hendecatonic (eleven-note) scale, though it does have eleven separate modes. Do you understand why? (Hint: There’s only a single dodecatonic scale in 12-TET for exactly the same reason.)
• The dodecatonic (twelve-note) chromatic scale (though works that use the entire twelve-note scale for non-colorative purposes are quite rare and are mostly considered avant-garde).

The above, of course, is all based on 12-TET. Plenty of other tunings have been and still are used, though; there’s nothing even restricting the number of notes per octave to twelve. For instance:

• A twenty-four-note scale ⟨en.wikipedia.org/wiki/Quarter_tone⟩ is common in Arabic and Turkic music.
• Indian rāgas ⟨en.wikipedia.org/wiki/Raga⟩⁽²⁾ (similar to scales) typically use four to seven svaras⁽³⁾ (similar to scale tones) of the twenty-two śrútis ⟨en.wikipedia.org/wiki/Shruti_(music)⟩⁽⁴⁾ (which collectively aren’t exactly 22-TET, but not entirely unlike it).
• Experimental Western musicians also occasionally use alternate temperaments. Here are a few recent rock and metal examples, in increasing order of heaviness:
  • Flying Microtonal Banana ⟨kinggizzard.bandcamp.com/album/flying-microtonal-banana⟩ by King Giz­zard & the Lizard Wizard (24-TET)
  • Nowherer ⟨votsband.bandcamp.com/album/nowherer⟩ by Victory over the Sun (17-TET)
  • any metal album by Jute Gyte ⟨jutegyte.bandcamp.com/music⟩ since Discontinuities (24-TET)
• As we’ll see when we discuss Ancient Greece, their enharmonic genus also used microtonality. (Note: enharmonic varies wildly in meaning based on context. In 12-TET, two enharmonic notes have the same pitch. In microtonal tunings, they’re about as dissonant as you can get.)

For the record:

The suffix -tonic is Greek. Friends don’t let friends mix Latin prefixes and Greek suffixes. (Latin did borrow τόνος as tonus, but only as a noun, never an adjective; it did not borrow τονικός.)

I only managed to find attestations of some of the Greek forms in this list, but it seems likely they all must have existed at some time. The ones I found are in bold; the ones I was unable to find are in fainter text.

Bolded English words, meanwhile, have attested usages for scale size in music theory contexts. Monotonic, diatonic, and tritonic are printed more faintly because they have completely different meanings that have nothing to do with the number of pitches in a scale, so using them to mean that will likely just confuse readers. The latter two are also struck through because their alternate meanings are ubiquitous in music theory contexts. You might be technically correct to use them to refer to scale size, but is that really the hill you want to die on?

1. Monotonic mostly refers to the modern Greek accent system, in contrast with the old polytonic accent system that conveyed breathing and word pitch. It does not have a widely established meaning in music theory contexts, so this is a less clear-cut case than the others.
2. Diatonic means having two interval sizes, mostly referring to what became our Ionian scale.
3. Tritonic means spanning an interval of three whole tones, i.e., a tritone.

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The Pentatonic Scale

As complement of Ionian

Pentatonic (five-note, “5t” for short) scales are ubiquitous in rock, blues, and jazz, though they’re much older than that and exist in many cultures. The most common pentatonic scale is literally Ionian’s scale complement.

What exactly is a scale complement? It’s the equivalent of a binary XOR. Say we represent a scale as a set of twelve 1s (“this tone is part of the scale”) or 0s (“this tone is not part of the scale”). Now, flip all the bits. Tones that had notes are no longer part of the scale; tones that didn’t now are. That’s the complement.

This means we can play the pentatonic scale using all the piano keys we didn’t use to play Ionian. Whenever I write “the pentatonic scale”, preceded by the definite article, I mean this pentatonic scale. To wit:

• C Ionian only uses the white keys on the piano: C, D, E, F, G, A, B, C.
• F♯ pentatonic major only uses the black keys: F♯, G♯, A♯, C♯, D♯ (or G♭, A♭, B♭, D♭, E♭, if you’re nasty prefer flats).

To a huge extent, the pentatonic and Ionian scales’ relationship even extends to their modes. For instance:

• Starting C Ionian three semitones early creates its relative minor, A Aeolian.
• Starting F♯ pentatonic major three semitones early creates its relative minor, D♯/E♭ pentatonic minor.

As truncation of Ionian

Conveniently, though, it’s not just a complement, though: it’s also a truncation. We can get the pentatonic scale simply by deleting two notes of Ionian. As a result, the two scales’ modes correspond in countless ways.

Since the pentatonic scale has two fewer notes than Ionian, our analysis must delete two modes. But which two? We can derive the pentatonic scale from Ionian using at least three different methods.

A quick warning before we proceed further: we’re about to take a quick detour into “right for the wrong reasons” land. After the third table, I’ll explain how, why, and where the first two tables go wrong.

Let’s try disregarding Phrygian and Locrian, the lowest modes in the circle of fifths progression. In this analysis:

1. Major pentatonic deletes Lydian’s fourth and seventh notes: F♯ G♯ A♯ C♯ D♯ (1, 1, 1½, 1, 1½).
2. Scottish pentatonic deletes Ionian’s third and seventh notes: C♯ D♯ F♯ G♯ A♯ (1, 1½, 1, 1, 1½).
3. Neutral pentatonic deletes Mixolydian’s third and sixth notes: G♯ A♯ C♯ D♯ F♯ (1, 1½, 1, 1½, 1). Neutral pentatonic is comparable to Dorian mode in two ways: it is a symmetrical scale, and it’s the midpoint of the pentatonic circle of fifths order (which is complex enough to merit its own section below).
4. Minor pentatonic deletes Dorian’s second and sixth notes: D♯ F♯ G♯ A♯ C♯ (1½, 1, 1, 1½, 1).
5. Blues minor deletes Aeolian’s second and fifth notes: A♯ C♯ D♯ F♯ G♯ (1½, 1, 1½, 1, 1).

Got all that? Let’s recap. (Note: “H” = half-tone, ”W” = whole tone, “M” = minor third)

I probably don’t even need to point out how many patterns recur in both scales.

We just analyzed the pentatonic modes based on notes a half-step above them, but we could just as easily have used the notes a half-step above. This means instead disregarding Lydian and Ionian. Oddly enough, we delete the same scale degrees either way:

(End warning.)

We have to go deeper (Scaleception)

But wait, there’s more! Consulting the heptatonic circle of fifths progression enables us to “average” the above two tables. Eerily, averaging them doesn’t invalidate their results in any way except a trivial one: this is the first analysis that’s actually correct. But, as the famed philosopher Nigel Tufnel put it, that’s nitpicking, innit?

So, to reiterate: The first two tables I presented are filthy half-truths: their explanations only fit due to mathematical patterns they don’t account for. Only the third is fully correct. Can you figure out why?

1. If both scales have symmetrical modes (Dorian and neutral pentatonic), then:
   1. We should compare the symmetrical modes to each other.
   2. Symmetrical modes should be their circle-of-fifths-order comparisons’ center data rows.
   3. Rows whose intervals are mirrors in the base should remain mirrors in the transformation.
   4. Both circle-of-fifths-order interval comparisons should possess 180° rotational symmetry.
   All of these are false in the first two tables and true in the third.
2. Since we didn’t delete the root, our analysis must compare the base scales. Only table three does so. Not comparing the base scales is a surefire recipe for confusion.
3. Deleting the base scale’s fourth and seventh notes also deletes its fourth and seventh modes. The first table keeps Lydian. The second keeps Locrian. The third deletes both.
4. Our analysis must compare the same notes within each scale. We didn’t move notes, only remove them, so our analysis can’t either. A closer look at the “downshift” table reveals the problem here: Mixolydian’s fourth degree, Dorian’s seventh, Aeolian’s third, Phrygian’s sixth, and Locrian’s second are Ionian’s root!

   The problem isn’t as obvious in the “upshift” table, but it tells us to remove Ionian’s third degree. Major pentatonic still opens with two whole steps, just like Ionian, so we haven’t removed Ionian’s third degree! This is why our analysis should only shift notes if we delete the root. (And if you do, good luck – you’ll need it.)

I struggled with whether to call the last table “Tritone Deletion”, “Tritone Substitution”, or “Tritone Shift”; all three are correct. The notes the last table removes correspond exactly to the Ionian scale’s sole tritone; the pentatonic scales are also exactly a tritone from where its two predecessors listed them. I ultimately went with “Tritone Deletion”, though, because all the properties above are direct consequences of removing the tritone.

Things like this are one reason I self-identify as agnostic rather than atheist: part of my brain refuses to accept that these could be coincidences. (The other part of my brain replies that this is all mathematically inevitable, and I simply haven’t yet grokked all the implications of deleting Ionian’s fourth and seventh notes.)

Since pentatonic’s mode nomenclature isn’t as well established as Ionian’s, my brain’s cutesy part wants to rename them Nianoi, Niadyloxim, Niarod, Nialoea, and Niagyrhp. I’m afraid that even after the above explanation, that might confuse people, but that won’t stop me from using them as alternate names.

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Analysis of modes in scale order

The modes’ scale ordering may help further clarify why the upshift and downshift tables are wrong:

Modes are inextricably linked to their roots. We must compare the same notes in each scale:

3. We didn’t delete this note, so its mode, Ionian, becomes major pentatonic.
4. We didn’t delete this note, so its mode, Dorian, becomes neutral pentatonic.
5. We didn’t delete this note, so its mode, Phrygian, becomes blues minor.
6. Deleting this note deletes its mode. Lydian has no pentatonic equivalent.
7. We didn’t delete this note, so its mode, Mixolydian, becomes Scottish pentatonic.
1. We didn’t delete this note, so its mode, Aeolian, becomes minor pentatonic.
2. Deleting this note deletes its mode. Locrian has no pentatonic equivalent.

Put another way, recall how Ionian’s modes got their numbering:

1. Ionian starts on its first note.
2. Dorian starts on its second note.
3. Phrygian starts on its third note.
4. Lydian starts on its fourth note.
5. Mixolydian starts on its fifth note.
6. Aeolian starts on its sixth note.
7. Locrian starts on its seventh note.

So, applying the same principle to the pentatonic scale:

1. Major pentatonic starts on its first note.
2. Neutral pentatonic starts on its second note.
3. Blues minor starts on its third note.
4. Scottish pentatonic starts on its fourth note.
5. Minor pentatonic starts on its fifth note.

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Comparison of interval spacing

Why pentatonic and Ionian’s intervals are out of sync may not be obvious. Let’s revisit the above table, this time with deleted tones in red and combined intervals in purple:

It may help to emphasize that we aren’t deleting intervals; we’re deleting notes and combining intervals. For instance, deleting a scale’s second note combines its first two intervals. Also, bear in mind that:

1. Ionian has only semitones and whole tones
2. We only delete notes that:
   1. follow semitones and precede whole tones
   2. follow whole tones and precede semitones

Thus, the two intervals around every deleted note turn become a single minor third aligning exactly to the pattern of deletions. Since this pattern is out of phase with the original one, it changes, but if the deleted notes were surrounded by different intervals, the new interval pattern wouldn’t map so precisely to the deletions.

Interestingly, that relationship is vastly less obvious in scale order, even with the above highlighting:

So, let’s correct our original analysis, shall we?

1. Major pentatonic deletes Ionian’s fourth and seventh notes: C D E G A (1, 1, 1½, 1, 1½). It is the root form of the scale, and it is the pentatonic circle of fourths’ lowest mode.
2. Scottish pentatonic deletes Mixolydian’s third and seventh notes: G A C D E (1, 1½, 1, 1, 1½). It is the pentatonic circle of fourths’ second-lowest mode.
3. Neutral pentatonic deletes Dorian’s third and sixth notes: D E G A C (1, 1½, 1, 1½, 1). Like Dorian, it’s symmetrical; it’s also the pentatonic circle of fourths’ midpoint.
4. Minor pentatonic deletes Aeolian’s second and sixth notes: A C D E G (1½, 1, 1, 1½, 1). It is the pentatonic circle of fourths’ second-highest mode.
5. Blues minor deletes Phrygian’s second and fifth notes: E G A C D (1½, 1, 1½, 1, 1). It is the pentatonic circle of fourths’ highest mode.

Oh, right. Time to explain “circle of fourths”. Though I should probably explain scale rotation even before that.

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A Brief Explanation of Scale Rotation

Scale rotation is the practice of forming a different mode by moving some of a scale’s intervals either from its start to its end or from its end to its start. When I refer to moving a scale’s intervals “left”, I’m referring both to the piano keyboard and to the scale interval tables I keep using. Rotating a scale left means moving most of its intervals to a lower point in the scale. Since a scale repeats every octave, the rest move to the top.

“It might not always be ‘most’,” I hear you object. Well, true, you could rotate a seven-note scale six degrees to the right, but why would you, when that’s the same as rotating it one degree to the left?

…Oh, right. I need to explain rotation by degrees, too. No, we’re not talking angles here. Rotating a scale by a specific number of degrees moves that many intervals from the start of the scale to the end (or vice versa). Rotating Ionian one degree left gives you Dorian. Rotating Ionian three degrees left gives you Lydian. And so on.

I’ll also sometimes refer to rotation by semitones. If I rotate a scale by five semitones, that means the intervals moved from the start to the end (or from the end to the start) sum up to five semitones. Thus, rotating Ionian five semitones to the left also takes you to Lydian.

The size of a scale rotation, either in interval size or in degrees, has nothing to do with how far the notes within the scale move. Lydian may be a five-semitone leftward rotation from Ionian, but it only moves one note (the fourth degree of the scale), and that note only moves by a semitone (F to F♯).

I may also refer to scale rotations by the number of notes they move. A single-note rotation only changes the position of one note. This does not signify anything about the number of intervals moved to the front or the end of the scale, the size of those intervals, or even about the interval by which the note is moved.

So, to summarize: Ionian to Lydian is a single-note rotation; it moves the note by one semitone, but it rotates the scale by five semitones (and three degrees).

I’ll try to keep this terminology from being ambiguous, but words are an imperfect medium for discussing music at the best of times, and when we throw mathematics, geometry, and set theory into the mix, forget it. If anything feels confusingly worded, please let me know, and I’ll try to clarify.

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The Pentatonic Circle of Fourths, or, Contrary Motion Explained

The pentatonic and Ionian circle of fifths orders move their parent scales in exact opposite directions. In fact, to emphasize this, I’m not even gonna call it the pentatonic circle of fifths anymore. I’ma call it monkeydude Josh the pentatonic circle of fourths. I find this fitting for at least two reasons:

1. Scale analysis disregards octaves in such a way that moving up a perfect fifth equates to moving down a perfect fourth. The different nomenclature helps call attention to their contrary directions.
2. Since the perfect fifth (seven semitones) has been the focus of my analysis of seven-note scales, the perfect fourth (five semitones) feels like the perfect focus for analysis of five-note scales.

But why do they move in opposite directions? It’s probably easiest to analyze in terms of interval spacing.

Pentatonic changes two of Ionian’s interval pairs from “tone, semitone” to “minor third”. It so happens that one of Ionian’s two semitones closes out the scale. Thus, compare what happens when we shift pentatonic major’s intervals to the left to what happens when we shift Ionian’s.

There’s a lot to unpack there. Some analysis:

Both scales contain a repeated interval set (called a tetrachord in Ionian and a trichord in pentatonic) and a spare whole step, or synaphe. I’ve marked one possible reading within the interval list. (Two such readings exist of pentatonic, and three of Ionian; in this example, I avoided splitting tetrachords or trichords in the base scale.)

Synaphe (plural synaphai or synaphes) comes from the Attic Greek σῠνᾰφή (sŭnăphḗ, literally connection, union, junction; point or line of junction; conjunction of two tetrachords). Its Attic pronunciation is roughly suh-nuh-FAY (so, basically how a drunk person would say Santa Fe), but I think English speakers, mistakenly assuming it to be French, might say sy-NAFF. (Pro tip: If a word contains ph and doesn’t split it across two syllables, you can almost wager money that it came from Greek. Also, pronouncing a foreign word using the wrong language’s orthography is a great way to make a linguistics nerd’s blood boil. And just to prove that I’m a linguistics nerd, orthography is derived from ορθο- (ortho-, correct) and -γραφίᾱ (-graphíā, writing).)

Ionian and pentatonic have similar structures and share five notes… but for the purposes of this analysis, that’s almost where their similarities stop.

I’ve analyzed Ionian’s tetrachord as whole tone, whole tone, semitone, placing the synaphe mid-scale. Swapping the synaphe with the tetrachord above it shifts the scale down, note by note. (Remember, a scale is a repeating note pattern, so in Mixolydian, “the tetrachord above it” is intervals 1-3, and it’s split across the start and end of the scale in Aeolian and Locrian.) Only Lydian’s interval list is more front-loaded than Ionian’s – it swaps Ionian’s first semitone with the following whole tone, with the following results:

• Lydian’s fourth note is the only sharp in the entire table.
• Each mode after Ionian adds flats to its notes, one by one.
• Locrian’s first and fourth notes are the only ones that aren’t lowered from their positions in Ionian.

Note also the following patterns within each mode:

• The note it lowers is either five semitones above or seven below the one its predecessor lowered.
• Its sole tritone consists of the note it lowers and the note its successor lowers.

I’ve analyzed pentatonic’s trichord as whole tone, minor third, placing the synaphe at the start of the scale. As it happens, pentatonic major’s intervals are as back-loaded as possible: that is, every other C-rooted mode of the pentatonic scale moves at least one note up a semitone. If we want to lower any notes besides the root, we have to lower the root before them.

Thus, while Ionian is second in its circle of fifths order, major pentatonic closes out its own circle of fourths. The next transposition in this sequence will yield C♭ blues minor (or B blues minor, whichever you prefer).

A few additional observations regarding both tables:

• Note the notes in common to multiple modes of each scale:

  Land of Confusion
  Pentatonic mode	Notes					Ionian modes			Notes
  C blues minor	C♮	D♯	F♮	G♯	A♯	C Aeolian	C Phrygian	C Locrian	C♮	E♭	F♮	A♭	B♭
  C minor pentatonic	C♮	D♯	F♮	G♮	A♯	C Dorian	C Aeolian	C Phrygian	C♮	E♭	F♮	G♮	B♭
  C neutral pentatonic	C♮	D♮	F♮	G♮	A♯	C Mixolydian	C Dorian	C Aeolian	C♮	D♮	F♮	G♮	B♭
  C Scottish pentatonic	C♮	D♮	F♮	G♮	A♮	C Ionian	C Mixolydian	C Dorian	C♮	D♮	F♮	G♮	A♮
  C major pentatonic	C♮	D♮	E♮	G♮	A♮	C Lydian	C Ionian	C Mixolydian	C♮	D♮	E♮	G♮	A♮

  This may help explain a possible source of confusion about the modes’ correspondence.
• Ionian’s final semitone is why C Locrian and C Phrygian use D♭. That semitone doesn’t exist in pentatonic.
• Shifting major pentatonic’s intervals one space to the right instead changes D to D♯, E to F, and A to A♯.
• As a result, the scales have effectively opposite directions of motion.

One final note: I highlighted the extra whole step, but its different position in each scale isn’t, in and of itself, why they move in different directions. In fact, since both scales are mostly made of whole steps, multiple intervals could be considered their synaphai; the choice depends entirely on the arbitrary choice of trichord or tetrachord pattern. Two such patterns can fit for the pentatonic scale and three for Ionian; each result in different synaphai and n-chord divisions. I used my divisions because the others split one of the base scale’s n-chords. (I’ve highlighted my approaches below.)

Note that the Ancient Greeks used the final analysis for their diatonic genus, which was nearly identical to our Ionian scale in all but name. When in Greece, I shall do as the Greeks did, but in this section, I figured it was better to defer the added complexity until this part of my explanation.

In short, the extra whole step’s position per se doesn’t affect the scale’s direction; they move in different directions mostly because Ionian’s last interval is a semitone and major pentatonic’s is a minor third. However, as we’re about to see, the circle orders wouldn’t exist even without the pentatonic and Ionian scales’ atypically regular note spacing. Transforming one mode of most other scales into another requires far more work.

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An Analysis of Five-Semitone Scale Rotation

Ultimately, the pentatonic and Ionian scales are, respectively, the only pentatonic and heptatonic scales in 12-TET for which moving a single note by a semitone amounts to a scale rotation. There’s a relatively simple mathematical explanation for why, too:

• Both scales are primarily composed of two identical interval collections, summing to five semitones each.

  Ionian & Pentatonic Building Blocks
  Scale	n-chord	Notes	Intervals	Outlier
  Ionian	Tetrachord	4	3	Semitone
  Pentatonic	Trichord	3	2	Minor third

• A synaphe (additional whole tone) separates the two n-chords.
• All but one of the n-chord’s intervals are whole tones. The outlier differs in size by a semitone.
• Rotating the scale by n degrees swaps the n-chord and the synaphe.
• This has a single effect on the interval composition: a whole tone swaps positions with an outlier.
• Since the outlier differs by merely a semitone, the sole result on the scale’s harmonic composition is to move a single note by a semitone:

  Outcome of swapping n-chord with synaphe
  Outlier	Outlier moves	Note moves
  Minor third	Down	Up
  Semitone	Down	Down
  Minor third	Up	Down
  Semitone	Up	Up

• This is why the pattern depends on the notes being almost, but not quite, completely evenly distributed: rotating a less even distribution moves more notes; rotating a completely even distribution moves none.
• It’s also why the pattern is so closely linked to the circles of fourths and fifths: swapping the synaphe with the n-chord requires rotating the scale by either five or seven semitones.
• Five and seven are the magic numbers because 12 modulo 7 = 5. In other words, after a note has moved seven semitones, five semitones of the octave remain.
• For a concrete examination, let’s look again at Ionian (1, 1, ½, 1, 1, 1, ½) and Mixolydian (1, 1, ½, 1, 1, ½, 1).
  • Changing Ionian to Mixolydian causes a single change: intervals 6 and 7 swap places.
  • Ionian’s first three intervals are identical to its last three.
  • Mixolydian’s first three intervals are identical to its fourth through sixth.
  • Ionian’s first three intervals are identical to Mixolydian’s.
  • Ionian’s fifth through seventh intervals are identical to Mixolydian’s fourth through sixth.
  • Ionian’s fourth and fifth intervals are also identical to Mixolydian’s fourth and fifth.
  • Ionian’s fourth interval is identical to Mixolydian’s seventh.
  • Ionian’s sixth interval is a semitone larger than Mixolydian’s.
  • Mixolydian’s seventh interval is a semitone larger than Ionian’s.
  If any of this weren’t true, swapping those two intervals wouldn’t rotate the scale by five semitones.
• Rotating a scale with less uniform interval spacing changes its interval composition substantially more.
• Rotating a scale with completely uniform intervals does not change its interval composition at all – thus, a completely uniform scale has only a single mode.

In summary, for moving a single note of any five- or seven-note scale in 12-TET by a semitone to rotate the scale, its intervals must be almost completely uniform, with only two identical outliers that:

1. Differ in size from the remainder by only a semitone
2. Are separated by five semitones

One final question may remain for readers: why is the number of outliers so important? Actually, it isn’t; it’s just important that the outliers be identical. If a seven-note scale could be completely uniform apart from one outlier, moving that interval would also rotate the scale. And, as it turns out, it can, but not, ironically, by making its note distribution more uniform.

1. Take the temperament modulo the scale size to get the number of extra semitones to distribute: 12 modulo 7 is 5. We have five extra semitones to distribute.
2. Take the note count modulo the extra semitones to figure out the most uniform note distribution possible: 7 modulo 5 is 2.

In short, two intervals must be outliers in the most uniform heptatonic note distribution possible. The way to get a single outlier, therefore, is to go in the exact opposite direction and make the outlier as big as possible. Which brings us to our next point of analysis.

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Other Single-Note Scale Rotations

I’ve focused most of my analysis so far on pentatonic and heptatonic scales, but now that I better understand the mathematical principles explaining why this happens, I’ve expanded my scope somewhat to see if I can uncover other examples of similar patterns with scale rotations of various interval sizes. I’ve uncovered a few, which I’ll explain in a moment.

Other equal temperaments certainly have similar examples (for instance, in 24-TET, rotation by 11 or 13 quarter-tones should produce similar results for similar 11- and 13-note scales), but I haven’t finished developing tools for scale analysis outside 12-TET, so they will have to wait.

Do other scales exist in 12-TET that don’t contain the above composition for which moving a single note by a semitone will qualify as a scale rotation? As it happens, yes: I can say with complete confidence, without even having to think about it, that the hendecatonic scale must demonstrate the same principle. And I say the hendecatonic scale for a simple reason: 12-TET contains only a single hendecatonic scale. The reason may be self-explanatory, but if it isn’t, I’ll give you a hint: It’s also why it contains only a single dodecatonic scale.

In 12-TET, hendecatonic scales must contain all but one note of the chromatic scale. Thus, it must contain ten semitones and one whole tone, and swapping its whole tone with any of its semitones qualifies as a scale rotation by default. There are only eleven ways to remove notes that aren’t the root; thus, a single scale with eleven modes, and it displays similar patterns not just for the circle of fifths but for every possible interval in 12-TET.

Of course, the very fact that only one hendecatonic scale exists in 12-TET somehow makes this feel vastly less impressive, even though it has exactly the same cause as Ionian’s circle of fifths pattern. Funny how that works.

Applying this same principle, we can determine that one other heptatonic scale exists in 12-TET for which moving a series of single notes by a constant interval size will rotate the scale. However, you don’t move its notes by a semitone; you move them by a perfect fourth. And, ironically, this doesn’t rotate it by five semitones; it rotates it by one. And its root won’t progress through the chromatic scale by semitones: it’ll progress through it by perfect fourths. (This will still take it all the way around the chromatic scale, just in a different order.) It’s the heptatonic chromatic scale, which has the following interval spacing:

½ ½ ½ ½ ½ ½ 3

Which is to say:

semitone, semitone, semitone, semitone, semitone, semitone, tritone

The notation for this is absolutely cursed, so I won’t bother displaying charts for its modes. But by following the above explanation, you may be able to understand the inevitable mathematical result of this scale composition.

Similar corollaries apply to some other scale sizes: moving successive notes of the pentatonic chromatic scale (½ ½ ½ ½ 4, or semitone, semitone, semitone, semitone, minor sixth) by seven semitones will inevitably take the scale through all its modes, moving the scale root up by a perfect fifth each time.

However, this only applies to a few scale sizes. Why? To rotate the hexatonic chromatic scale (½ ½ ½ ½ 3½, or semitone, semitone, semitone, perfect fifth), we must move notes by a tritone each. So when we move the root up, we move it a tritone. Two tritones amount to an octave, so we skip five-sixths of the chromatic scale.

The only non-semitone interval of any such truncation of the chromatic scale is (13 - n) semitones, where n is the scale’s note count. Thus, scale rotation moves notes by (12 - n) semitones. For rotation of such a scale to take us through all of its modes across the entire chromatic scale, this interval cannot be a factor of 12. In fact, in any n-TET, for scale rotations that move single notes by more than a chromatic step to cover the entire chromatic scale, n modulo the interval size (in units of 1/n octave) must not be 0. Thus, in 12-TET:

• Moving notes a semitone runs through the entire chromatic scale in linear order.
• Moving notes two semitones (major second) skips half of the scale.
• Moving notes three semitones (minor third) skips two-thirds of the scale.
• Moving notes four semitones (major third) skips three-quarters of the scale.
• Moving notes five semitones (perfect fourth) runs through all notes in circle of fourths order.
• Moving notes six semitones (tritone) skip five-sixths of the scale.

Anything beyond a tritone can be evaluated using the first six parameters: moving a note up by seven semitones equates to moving it down by five. Thus, in 12-TET, the only truncations of the chromatic scale for which this method of scale rotation will work contain 1, 5, 7, and 11 notes, and their rotations will respectively move single notes by a semitone, a perfect fourth, a perfect fourths, and a semitone.

Thus, the Ionian and heptatonic chromatic scales are the only heptatonic scales in 12-TET where single-note rotations will take us through the entire chromatic scale. As far as I can ascertain, only two other heptatonic scales exist in 12-TET that have single-note rotations. They respectively have the following interval spacing:

Hungarian major: minor third, semitone, whole tone, semitone, whole tone, semitone, whole tone

Lydian ♯23: minor third, whole tone, semitone, whole tone, semitone, whole tone, semitone

In case it isn’t obvious why these won’t take us through the entire chromatic scale: in both cases, the single-note rotation moves a note by a minor third. Four minor thirds make up an octave. Thus, cannot cycle through the entire chromatic scale using these kinds of transformations.

Note that in both these scales, there are three pairs of whole tones and semitones, and one additional minor third. A whole tone and a semitone, of course, add up to a minor third. Thus, although this is in some ways the least regular interval distribution we’ve examined, the fact that it can be divided into four three-semitone regions necessitates a rotation that lines up exactly with the parent tonality.

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Transformations of the Ionian Scale

Harmonic Minor

Since learning harmonic and melodic minor’s modes is absolutely essential for anyone who wants to play jazz, I’ve created several sets of tables to help people visualize them better.

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Thus, as the first set of mode names suggests, harmonic minor’s modes respectively raise by a half-step:

• Aeolian’s seventh degree,
• Locrian’s sixth degree,
• Ionian’s fifth degree,
• Dorian’s fourth degree,
• Phrygian’s third degree,
• Lydian’s second degree,
• C-C-C-C-COMBO BREAKER!

Since we’re transposing every mode to C, we can’t raise Mixolydian’s first degree, because it’s the first degree! Instead, we must lower every other degree by a half-step. Say wha?

Somehow, it’s actually both even weirder than that, and not weird at all: what we do in the above table is the equivalent of raising the first degree. Since we’re constraing ourselves to a root of C, raising the first note of a scale by a half-step requires us to lower every note of that scale by a half-step. This results in the first note being the only scale degree we don’t lower: ½ − ½ = 0.

In practice, though, it’s usually already raised for us: it’s harmonic minor’s seventh degree! Let’s see what happens when we root these modes on the corresponding notes of their respective parent C minor scales:

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Thus, B♭ Mixolydian is to C Aeolian as B Mixolydian ♯1 is to C harmonic minor: each starts from its parent scale’s seventh note. There’s actually nothing odd going on here at all; it’s exactly how modes are supposed to behave. C Mixolydian ♯1 equates to lowering every note of C Mixolydian except C by a half-step – and to raising only the B in B Mixolydian by a half-step.

Observant readers may have noticed that the “rooted on C” table above actually appears to contain several shifts. The missing puzzle piece is that it lists the modes in ascending order rather than “circle of fifths” order, which I did to make the scales’ intervals easier to relate to each other. So let’s return to “circle of fiths” order.

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We now see that, broadly speaking, in all except one case, each scale has one fewer sharp or one more flat than its predecessor two entries above. The clear outlier is Mixolydian ♯1, and this table may further clarify why the mode corresponding to Mixolydian is the one thus affected. Mixolydian corresponds to Ionian’s fifth scale degree, Dorian’s fourth scale degree, Phrygian’s third scale degree… and that’s the degree that harmonic minor’s modes raise. For completeness, here are the modes in “circle of fifths” order, rooted to their respective notes within their parent C minor scales: