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 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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

An Audio Demonstration

• None of this will mean much to readers in isolation, so I put together a crude demonstration of the above principle 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.)

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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 may write a JavaScript add-on to reverse the order, but I’d have to learn JavaScript first, so no promises.
• “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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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).

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Beyond the Major Scale

Other Scales & Tonalities

• 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:

  Other Seven-Note Scales
  Scale		1	2	3	4	5	6	7
  Ascending melodic minor		C♮	D	E♭	F	G	A	B
  Dorian ♭2	Phrygian ♯6	C♮	D♭	E♭	F	G	A	B♭
  Lydian augmented		C♮	D	E	F♯	G♯	A	B
  Lydian dominant	Mixolydian ♯4	C♮	D	E	F♯	G	A	B♭
  Aeolian dominant	Mixolydian ♭6	C♮	D	E	F	G	A♭	B♭
  Aeolian ♭5	Locrian ♯2	C♮	D	E♭	F	G♭	A♭	B♭
  Super-Locrian	Altered dominant	C♮	D♭	E♭	F♭	G♭	A♭	B♭
  Harmonic minor	Aeolian ♯7	C♮	D	E♭	F	G	A♭	B
  Locrian ♮6		C♮	D♭	E♭	F	G♭	A	B♭
  Augmented major	Ionian ♯5	C♮	D	E	F	G♯	A	B
  Hungarian minor		C♮	D	E♭	F♯	G	A♭	B
  Ukrainian Dorian		C♮	D	E♭	F♯	G	A	B♭
  Phrygian dominant		C♮	D♭	E	F	G	A♭	B♭
  Lydian ♯2		C♮	D♯	E	F♯	G	A	B
  Altered diminished	Super-Locrian 𝄫2	C♮	D♭	E♭	F♭	G♭	A♭	B𝄫

  I’ve again printed augmented intervals in bold, orange text and minor or diminished intervals in thin, blue text. Note that this list is far from exhaustive; it doesn’t even delve into scales with different ascents and descents. (I also didn’t list every name of some scales on this list; some have at least four.)
  However, 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.
• 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; eight- and nine-note scales are common in blues and jazz.
• And of course, all of the above is based on the twelve-tone 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 is common in Arabic and Turkic music.
  • Indian rāgas⁽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⁽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 by King Gizzard & the Lizard Wizard (24-tone scale)
    • Nowherer by Victory over the Sun (17-tone scale)
    • any metal album by Jute Gyte since Discontinuities (also 24-tone scale)
  • The Ancient Greek enharmonic genus, as we shall see shortly, also used microtonality.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

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.

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Ancient Greek Tonoi & Modern Modes

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. Wikipedia has much, much more on the subject; the Xenharmonic wiki, Midicode, and Feel Your Sound have also been helpful. I consulted several other resources researching this section, but most were too technical to be of interest to non-specialists.

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 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, 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 what Greek music sounded like; 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 (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:

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 in between a major second and a minor third, sometimes called 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 the ratio of a diatonic tetrachord’s high and low notes as 4:3 (1.33333…), 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:

Back to top · My discography · Marathon soundtracks · Contact me · Website index

Each tonos contained two tetrachords, with what effectively reduces to a major second to complete the octave. 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 intervals are slightly easier to understand.
• 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.
• Apparently, our Dorian, Phrygian, Lydian, and Mixolydian modes don’t match the 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.)

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

Acknowledgements to Marty O’Donnell – yes, that Marty O’Donnell – for pointing out that it wasn’t sufficiently clear what parts of this section I was oversimplifying; his feedback resulted in a much clearer explanation. However, the opinions presented in this section are entirely my own (Marty believes modern scholars have extrapolated more from ancient Greek texts than is merited by their contents).

Those seeking more detailed technical analysis of ancient Greek tuning systems may find Robert Erickson’s analysis of Archytas (who provided a detailed and apparently accurate description of what modern scholars believe to have been the actual musical practices of his era) to be of interest. Our knowledge of Archytas’ musical writings apparently comes from Ptolemy’s Harmonics, whose author comments in depth on the former’s writings⁽⁷⁾; large fragments of Aristoxenus’ Elements of Harmony and smaller fragments of Philolaus’ musical writings survive to this day. Here, via Cris Forester’s book on the subject, is an excerpt from Philolaus, whom I quote less for his comprehensibility than for his technical detail:

The magnitude of harmonia is syllaba and di’oxeian. The di’oxeian is greater than the syllaba in epogdoic ratio. From hypate [E] to mese [A] is a syllaba, from mese [A] to neate [or nete, E¹] is a di’oxeian, from neate [E¹] to trite [later paramese, B] is a syllaba, and from trite [B] to hypate [E] is a di’oxeian. The interval between trite [B] and mese [A] is epogdoic [9:8], the syllaba is epitritic [4:3], the di’oxeian hemiolic [3:2], and the dia pason is duple [2:1]. Thus harmonia consists of five epogdoics and two dieses; di’oxeian is three epogdoics and a diesis, and syllaba is two epogdoics and a diesis. [Text and ratios in brackets are Cris Forester’s.]

–Philolaus, translated by Andrew Barker, Greek Musical Writings, Vol. 2 (1989: Cambridge University Press).

Difficult as this is to parse, a close reading reveals that Philolaus has just described the diatonic major scale:

• 3:2 is 1.5; our perfect fifth rounds to 1.49830707688.
• 4:3 is 1.333…; our perfect fourth rounds to 1.33483985417.
• 9:8 is 1.125; our whole step rounds to 1.12246204831.
• Philolaus’ dieses are therefore 256:243, or 1.05349794239; our half step rounds to 1.05946309436.

In other words:

• A dia pason (or harmonia) is exactly an octave.
• A di’oxean is almost exactly a perfect fifth.
• A syllaba is almost exactly a perfect fourth.
• An epogdoic is almost exactly a whole step.
• A diesis is almost exactly a half-step.

Plugging those in gives us:

The magnitude of an octave is a perfect fourth and a perfect fifth. The perfect fifth is greater than the perfect fourth in whole-step ratio. From hypate [E] to mese [A] is a perfect fourth, from mese [A] to neate [or nete, E¹] is a perfect fifth, from neate [E¹] to trite [later paramese, B] is a perfect fourth, and from trite [B] to hypate [E] is a perfect fifth. The interval between trite [B] and mese [A] is a whole step [9:8], the perfect fourth is epitritic [4:3], the perfect fifth hemiolic [3:2], and the octave is duple [2:1]. Thus an octave consists of five whole steps and two half-steps; a perfect fifth is three whole steps and a half-step, and a perfect fourth is two whole steps and a half-step.

Plus ça change, plus c’est la même chose.

In closing, I should reiterate that “ancient Greek harmony” was not uniform; various authors described other tonoi and indeed entirely different tuning systems. I’ve focused on the tonoi described above because they were the clear inspiration for the modern modes’ names.

(Keep an eye on this page – I intend to add information on the medieval church modes that served as the precursors to our modern modes someday.)

Back to top · My discography · Marathon soundtracks · Contact me · Website index