The Modes in “Circle of Fifths” Order

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

• 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 omitted them. (There are even monstrosities with double-sharps [𝄪] and double-flats [𝄫] … which I’ll mostly be ignoring. I can at least tolerate the double-flat symbol, but the double-sharp symbol is a monstrosity that doesn’t even bear any resemblance to the thing it’s supposed to be doubling.) I feel it was pretty extravagant just to include six-flat and six-sharp key signatures alongside each other.
• 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, ascending 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

• The twelve-note chromatic scale has fifteen key signatures (𝄪 and 𝄫 monstrosities 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.
  • As in the tables above, I’ve 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. In the introduction, I mentioned that traveling from C Lydian to C Ionian was, in a sense, traveling from G major to C major. Here’s the C table again. Note how “relative major” traverses the circle of fifths downward:

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. Almost every other pattern we’ve observed that follows the circle of fifths in some way is a direct consequence of this.

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Chord Analysis by Mode

• I’ve printed the root (I), dominant (V), and subdominant (IV) chords of each mode in bold – they are especially important to the mode’s tonality.
• I’ve printed the seventh chord in lighter text because it’s diminished in the major scale.
• Minor chords are blue; diminished chords are orange.
• Printing the chords this way makes it clearer why Ionian became the standard major scale and Aeolian became the natural minor scale: the root, dominant, and subdominant are all major in Ionian mode and all minor in Aeolian mode. 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 Scales & Tonalities

Although this page focuses on modes of the major scale, numerous possible scales (and modes thereof) don’t fit its pattern, such as:

• The 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).
• Numerous 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 diatonic major scale:

  Other Seven-Note Scales
  Scale		1	2	3	4	5	6	7	Intervals
  Chromatic Dorian	Kanakangi	C♮	D♭	E𝄫	F♮	G♮	A♭	B𝄫	½	½	1½	1	½	½	1½
  Altered diminished	Super-Locrian 𝄫2	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫	½	1	½	1	1	½	1½
  Altered dominant	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
  Jazz minor inverse		C♮	D♭	E♭	F♮	G♮	A♮	B♭	½	1	1	1	1	½	1
  Phrygian dominant		C♮	D♭	E♮	F♮	G♮	A♭	B♭	½	1½	½	1	½	1	1
  Flamenco mode	Major Phrygian	C♮	D♭	E♮	F♮	G♮	A♭	B♮	½	1½	½	1	½	1½	½
  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
  Harmonic minor		C♮	D♮	E♭	F♮	G♮	A♭	B♮	1	½	1	1	½	1½	½
  Ascending melodic minor		C♮	D♮	E♭	F♮	G♮	A♮	B♮	1	½	1	1	1	1	½
  Hungarian minor		C♮	D♮	E♭	F♯	G♮	A♭	B♮	1	½	1½	½	½	1½	½
  Ukrainian Dorian		C♮	D♮	E♭	F♯	G♮	A♮	B♭	1	½	1½	½	1	½	1
  Aeolian dominant	Mixolydian ♭6	C♮	D♮	E♮	F♮	G♮	A♭	B♭	1	1	½	1	½	1	1
  Augmented major	Ionian ♯5	C♮	D♮	E♮	F♮	G♯	A♮	B♮	1	1	½	1½	½	1	½
  Lydian augmented		C♮	D♮	E♮	F♯	G♯	A♮	B♮	1	1	1	1	½	1	½
  Lydian dominant	Mixolydian ♯4	C♮	D♮	E♮	F♯	G♮	A♮	B♭	1	1	1	½	1	½	1
  Chromatic Dorian inverse		C♮	D♯	E♮	F♮	G♮	A♯	B♮	1½	½	½	1	1½	½	½
  Chromatic Hypophrygian		C♮	D♯	E♯	F♯	G♮	A♯	B♮	1½	1	½	½	1½	½	½
  Aeolian harmonic	Lydian ♯2	C♮	D♯	E♮	F♯	G♮	A♮	B♮	1½	½	1	½	1	1	½
  Superlydian augmented		C♮	D♯	E♯	F♯	G♯	A♯	B♮	1½	1	½	1	1	½	½

  Since it would be impossible to alphabetize scales with more than one name (note that many scales have more names than I’ve printed here), I’ve applied the principles of alphabetization to the intervals, with the first interval counted as the most significant. Bold, orange text denotes raised notes and minor thirds; thin, blue text denotes lowered notes and minor seconds (doubly lowered notes are also fainter).

  Note that this list is far from exhaustive; it doesn’t even delve into scales with different ascents and descents. Also, none of these scales use B♯ because, while it is semantically different in music theory 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 modes of diatonic scales: we can print any mode of any diatonic scale with no accidentals outside the key signature, but every single scale in the above table would require accidentals for at least some notes. Key signatures are strictly based on major scales and the circle of fifths; using three flats that weren’t B♭, E♭, and A♭ in a key signature would just confuse readers.

  One additional advantage of printing Chromatic Dorian first: It’s directly relevant to the section below on ancient Greek harmony.

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

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

  We can apply similar principles to them as we applied to the diatonic major 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.

• There are also, naturally, scales that use fewer than seven notes; pentatonic (five-note) scales are quite common in rock, blues, and jazz music, though they’re much older than that and exist in many cultures. Six-note scales are also fairly common (including whole-tone scales).
• Scales may also have more than seven notes; blues and jazz often use eight- and nine-note scales.
• And of course, all of the above is based on the twelve-tone equal temperament (12-TET) ⟨en.wikipedia.org/wiki/12_equal_temperament⟩ used in the vast majority of Western music from the last five centuries. 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⁾ typically use anywhere from four to seven svaras⁽³⁾ (roughly comparable to the notes of a Western diatonic scale) selected from the twenty-two śrútis ⟨en.wikipedia.org/wiki/Shruti_(music)⟩⁽⁴⁾ (roughly comparable to the Western twelve-note chromatic scale, except with ten more notes).
  • 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)
  • The Ancient Greek enharmonic genus, as we’ll see after our examinations of harmonic and melodic minor, also used microtonality.

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Harmonic Minor’s Modes in Detail

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

Thus, 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!

We can’t raise Mixolydian’s first degree, because it’s the first degree! Instead, we 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, harmonic minor has already raised its seventh degree for us. Let’s represent these modes again, but this time, we’ll root each mode in its respective note within its parent C minor scale:

Thus, B♭ Mixolydian is to C Aeolian as B altered diminished 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 altered diminished 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.

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 altered diminished, 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:

Finally, this table also clarifies why these charts have to use scare quotes for “circle of fifths”: once we abandon the diatonic major scale’s regularity, we’re not actually moving every note up or down by a fifth. A perfect fifth up from E♭ isn’t B; it’s B♭. A perfect fifth below F isn’t B; it’s B♭. This is very likely why the diatonic major scale uses the intervals it does: it possesses a borderline fractal level of regularity. Manifestations of the circle of fifths occur throughout it. Even bumping one of its notes a half-step up or down breaks the pattern in places. We can still approximate “circle of fifths” order, of course, but our approximations will have to carry asterisks.

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Ascending Melodic Minor’s Modes in Detail

Ascending melodic minor is perhaps better related to the modes starting with Ionian. I haven’t drawn borders this time, because… well, it’s probably better to just show the table first.

In short, it may be helpful to think of ascending melodic minor as Ionian and Dorian’s midpoint; of jazz minor inverse as Dorian and Phrygian’s midpoint; and so on. (This is, of course, an oversimplification, since it’s completely ignoring “circle of fiths” order.)

Transposed versions of the above scales:

“Circle of fifths” order makes it clear that the big note shift from Phrygian to Lydian augmented occurs in the “rooted on C” chart for the same reason the note shift between Mixolydian and altered diminished occurs with the harmonic minor scale: rooting everything to C means we can’t lower the first note and must instead raise the other notes by however much we’d have lowered the first note.

Phrygian mode corresponds to Ionian’s third degree; that’s the note ascending melodic minor lowers from Ionian. Thus, Phrygian is the mode that undergoes the note shift in the above table. Moreover, C Lydian augmented raises every note of C Phrygian except its root because its parent scale lowers its corresponding note.

For completeness, here’s “circle of fifths” order without transposition.

(At some point, I plan to make equivalents of §3’s charts for at least ascending melodic minor and harmonic minor, and perhaps for some of the Greek scales I discuss below as well… but not until I’ve written programs to automate their generation, which could take anywhere from a few days to months.)

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A Crash Course in Ancient Greek Harmony

Etymology

The names of all seven modes refer to regions in or near ancient Greece, ancient Greek tribes, or both:

However, they really don’t have anything to do with the regions or tribes they were named after, or even the ancient Greek tonoi (singular: tonos) that in many cases shared their names; it was more a case of “medieval Europeans thought it’d be cool to name their modes after aspects of Ancient Greece.”

(Tonos comes from the ancient Greek ὁ τόνος, ho tónos, meaning cord, chord⁽⁵⁾, note, tone, or tension; its dual nominative form was τὼ τόνω, tṑ tónō, and its plural nominative form was οἱ τόνοι, hoi tónoi.)

As for the word diatonic, that comes from the Ancient Greek διατονικός (diatonikós), literally meaning two tones, in reference to the diatonic tetrachord, which… well, keep reading.

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Ancient Greek Harmony: The Cliffs Notes

This is an oversimplification by necessity, as ancient Greek music theory wasn’t unified;⁽⁶⁾ Philolaus (Φιλόλαος, Philólaos), Archytas (Ἀρχύτας), Aristoxenus (Ἀριστόξενος ὁ Ταραντῖνος, Aristóxenos ho Tarantínos), Ptolemy (Πτολεμαῖος, Ptolemaios), and others had quite different conceptions of it. I will list modern sources in an acknowledgement section below.

I should first clarify that very little music of ancient Greece actually survives to this day; the earliest known piece that can reasonably be described as complete is the Seikilos epitaph ⟨en.wikipedia.org/wiki/Seikilos_epitaph⟩ from the first or second century CE, and while we have an idea what its melody sounded like, how it would have been harmonized is a matter of conjecture. (Older pieces still survive in fragments, such as the Hurrian songs ⟨en.wikipedia.org/wiki/Hurrian_songs⟩, one of which is nearly complete.) Greek authors actually provided fairly complete descriptions of tuning practices, and in at least Archytas’ case, modern scholars believe he was describing the actual practices of his day, but we don’t really know how Greek music sounded; on some level, we’re taking the word of contemporary authors, only some of whom seem to have been especially concerned with describing actual musical practice.⁽⁷⁾

I’ll be using numbers to represent the intervals of ancient Greek harmony within 24-tone equal temperament ⟨en.wikipedia.org/wiki/Quarter_tone⟩ (24-TET), which adds an additional 12 notes exactly halfway between each note of the familiar 12-note chromatic scale. In 24-TET, an exact ratio of ²⁴√2:1 determines the spacing of the smallest interval (known as a quarter-tone, downminor second, infra second, or wide unison), thus:

Note that in scales with only whole-steps and half-steps, I’ll use H (i.e., Half) interchangeably with ½, and W (i.e., Whole) interchangeably with 1. In all other scales, I’ll only use the numbers.

24-TET is by necessity an oversimplification as well, not least since different authors defined the same interval differently. For instance, here are how Philolaus, Archytas, and 24-TET would define the intervals within a diatonic tetrachord. (A tetrachord is a major element of Greek harmony consisting of four notes. Also, I’ve taken the liberty of reversing what they called the first and third intervals – ancient Greek harmony defined scales in descending order, where we use ascending order.)

Thus, representing ancient Greek harmony using 24-TET is a substantial oversimplification, and in point of fact, we’d get closer to Archytas’ definitions in 24-TET by using a quarter tone for the low interval and 1¼-tone (that is, the interval between a major second and a minor third, known as an infra third or ultra second) for the middle interval. However, the intervals I’ve selected (a minor second and two major seconds) are closer to Philolaus’ definitions and, crucially, can be represented in modern Western music’s 12-TET.

As one further example, both Philolaus and Archytas define a diatonic tetrachord’s high and low notes as having 4:3 (1.33333…) ratios, which 24-TET would represent using a perfect fourth (≈1.33483985417). Most people’s ears are insufficiently trained to tell the difference – even trained singers with perfect pitch frequently wander off-key by larger differences than that (at least when they’re not subjected to pitch correction⁽⁸⁾).

The same ancient Greek tonos could actually use any of three genera (singular: genus). Each tonos contained two tetrachords, or sets of four notes, whose spacing differed depending on the genus; a major second would complete the octave (though in some tonoi, it occurred between the tetrachords, and in Hypodorian and Mixolydian, it occurred respectively before and after them; additionally, several tonoi split one tetrachord). All three genera used one interval once per tetrachord and another interval twice, in the following order:

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Each tonos contained two tetrachords, with what effectively reduces to a major second to complete the octave.

Ancient Greek Tonoi & Modern Modes

A few notes:

• As stated above, ancient Greek harmony defined tonoi in descending order; I’ve listed them in our more familiar ascending order to keep them consistent with the other scales on this page.
• I’ve printed intervals rather than notes because the notation for microtones is extremely confusing if you’re not already familiar with it. Hopefully, the interval notation will be slightly easier to understand for the enharmonic tonoi. Below this table, I’ll present examples of the chromatic tonoi based on C.
• A thicker font denotes a larger interval size: 2, 1½, 1, ½, ¼.
• I’ve placed borders between tetrachords (again, four tonoi split one of their tetrachords) and around the extra whole step (whose background I’ve also highlighted).
• I’ve printed the modern modes with blue text and brighter borders, with a thicker border below since the next tonos will not be related.
• I’ve printed the tonoi in the order ancient Greek harmony assigned them: Mixolydian (modern Locrian’s equivalent) first, Hypodorian (modern Aeolian’s equivalent) last.
• Our Dorian, Phrygian, Lydian, and Mixolydian modes don’t match their eponymous diatonic tonoi because medieval Europeans erroneously believed the ancient Greeks described tonoi in ascending order. (Swiss poet Heinrich Glarean appears to have popularized the misconception.) I’ll explain this further below.

Here are all three genera of all seven tonoi, followed by the diatonic genus’ modern equivalent:

The above table is quite abstract, so to follow it up, here are the chromatic tonoi rooted in C and, for the sake of representing what medieval Europeans might have thought they were, their inversions. As in the previous table, I’ve separated the tetrachords in the interval listing to make it clear where they occur, and I’ve highlighted the extra whole-step (a bit more so, even, because it will become a bit less legible shortly).

Remember how I said above that Chromatic Dorian was directly relevant to this section? Well, there you go.

Scale-based transposition now. My base scales are Chromatic Lydian and Chromatic Hypophrygian inverse; this is an admittedly arbitrary choice that I made purely because they use the fewest accidentals on C. This also creates a neat pattern in the table below:

My “circle of fifths” order for the ancient Greek tonoi is:

1. Hypolydian
2. Lydian
3. Hypophrygian
4. Phrygian
5. Hypodorian
6. Dorian
7. Mixolydian

This may help explain how the Greeks got the names Hypolydian, Hypophrygian, and Hypodorian in the first place: ὑπό (hupó) is literally Ancient Greek for under, and remember, the ancient Greeks’ scales went in what we consider descending order. Note what’s going on with the intervals in the following table. (Also note that I inverted the order for the inverted scales this time around because, as the table above clearly demonstrates, they’re actually moving in the opposite direction from their namesakes.)

Greek Chromatic Tonoi & Their Inversions (mode-based roots, “circle of fifths” order)
Chromatic Tonos		1	2	3	4	5	6	7	Intervals
Hypolydian		F♮	G♭	A♮	B♮	C♮	D♭	E♮	½	1½	1	½	½	1½	½
Lydian		C♮	D♭	E♮	F♮	G♭	A♮	B♮	½	1½	½	½	1½	1	½
Hypophrygian		G♭	A♮	B♮	C♮	D♭	E♮	F♮	1½	1	½	½	1½	½	½
Phrygian		D♭	E♮	F♮	G♭	A♮	B♮	C♮	1½	½	½	1½	1	½	½
Hypodorian		A♮	B♮	C♮	D♭	E♮	F♮	G♭	1	½	½	1½	½	½	1½
Dorian		E♮	F♮	G♭	A♮	B♮	C♮	D♭	½	½	1½	1	½	½	1½
Mixolydian		B♮	C♮	D♭	E♮	F♮	G♭	A♮	½	½	1½	½	½	1½	1
Mixolydian inverse		F♮	G♮	A♯	B♮	C♮	D♯	E♮	1	1½	½	½	1½	½	½
Dorian inverse		C♮	D♯	E♮	F♮	G♮	A♯	B♮	1½	½	½	1	1½	½	½
Hypodorian inverse		G♮	A♯	B♮	C♮	D♯	E♮	F♮	1½	½	½	1½	½	½	1
Phrygian inverse		D♯	E♮	F♮	G♮	A♯	B♮	C♮	½	½	1	1½	½	½	1½
Hypophrygian inverse		A♯	B♮	C♮	D♯	E♮	F♮	G♮	½	½	1½	½	½	1	1½
Lydian inverse		E♮	F♮	G♮	A♯	B♮	C♮	D♯	½	1	1½	½	½	1½	½
Hypolydian inverse		B♮	C♮	D♯	E♮	F♮	G♮	A♯	½	1½	½	½	1	1½	½