Music almost never uses all twelve pitches equally. It picks a small set, usually seven, and treats one of them as home, and this lesson is about which seven and why the choice has the consequences it does.
The major scale is a pattern of steps
Play the white keys from C up to the next C and you have the major scale, the reference object for everything that follows. What matters is not the letters but the pattern of distances. Measured in semitones, C to D is 2, D to E is 2, E to F is 1, F to G is 2, G to A is 2, A to B is 2, B to C is 1. Writing a whole tone as T and a semitone as S, the pattern is
and the seven steps sum to , which is the octave, as any scale must.
The pattern is the definition, not the white keys. C major happens to use white keys because the keyboard was laid out around it, and any other starting note gives the same scale transposed, provided the pattern is preserved exactly. Start on G and apply T T S T T S T: G, A, B, C, D, E, and then the sixth step of a tone from E lands on F sharp, and a final semitone reaches G. So G major is G A B C D E F# G, and the sharp is not a decoration or a stylistic choice. Use F natural and the last two steps become a semitone and a tone rather than a tone and a semitone, which is a different scale.
One letter each, no exceptions
Notice that G major used every letter name once, in order, and that this is what forced the spelling F sharp rather than G flat. The rule holds for every major and minor scale: seven notes, seven consecutive letters, each appearing exactly once, with accidentals doing whatever they must.
The rule earns its keep on the flat side. Build a major scale on F: F, G, A, then a semitone to B flat, then C, D, E, F. The fourth note is a black key, and calling it A sharp would be wrong even though it is the same key on a piano, because the scale would then read F G A A# C D E and use A twice while skipping B. Spelled B flat, the scale is legible on the staff as one notehead per position, which is the whole reason the convention exists.
The extreme cases follow the rule without flinching. C sharp major is C# D# E# F# G# A# B# C#, with an E sharp and a B sharp that sound like F and C, because the letters E and B have to appear and be raised. C flat major is Cb Db Eb Fb Gb Ab Bb Cb, with an F flat. These look absurd and are perfectly correct, and the next lesson explains when a musician would rather use them than the enharmonic alternative.
Example. Write the D major scale.
Apply T T S T T S T from D. D up a tone is E, up a tone is F sharp, up a semitone is G, up a tone is A, up a tone is B, up a tone is C sharp, up a semitone is D. So D E F# G A B C# D, with each letter used once.
Now you. Write the E flat major scale.
Answer
Eb F G Ab Bb C D Eb. Checking the steps: Eb to F is 2, F to G is 2, G to Ab is 1, Ab to Bb is 2, Bb to C is 2, C to D is 2, D to Eb is 1, which is T T S T T S T.
The degrees are not interchangeable
The seven notes of a scale are its degrees, numbered 1 to 7 from the tonic, and the numbering is the most portable idea in the subject: a statement about degree 5 is true in every key at once, which is why professional discussion of harmony is conducted almost entirely in numbers.
Each degree also has a name, and the names encode a structure worth seeing rather than memorising. Degree 1 is the tonic, home. Degree 5 is the dominant, a perfect fifth above the tonic, the strongest relation there is after the octave. Degree 4 is the subdominant, and the "sub" is not "one below the dominant": it is the note a perfect fifth below the tonic, which lands on degree 4 an octave up. So the tonic sits between two fifths, one on each side, and the two nearest relatives are named for that.
The remaining names extend the scheme. Degree 3 is the mediant because it lies midway between tonic and dominant; degree 6 is the submediant, midway between tonic and subdominant on the way down. Degree 2 is the supertonic, simply the note above the tonic. Degree 7 is the leading note, or leading tone, and its name is a claim about behaviour that the next section justifies.
Example. In E flat major, name degrees 4, 5 and 7.
The scale is Eb F G Ab Bb C D. Degree 4 is A flat, the subdominant; degree 5 is B flat, the dominant; degree 7 is D, the leading note.
Now you. In B major, name the dominant, the subdominant and the leading note.
Answer
B major is B C# D# E F# G# A#. The dominant is F sharp, the subdominant is E, and the leading note is A sharp.
Why degree 7 pulls
Look at where the two semitones fall in T T S T T S T. One is between degrees 3 and 4, the other between degrees 7 and 8, that is, between the leading note and the tonic an octave up. Everything characteristic of major tonality comes from those two positions.
The semitone from 7 to 8 is the engine. A leading note sounds unfinished: play C D E F G A B and stop, and almost every listener raised on this music wants the C. The best current explanation is expectation learned from exposure rather than anything acoustic, since a semitone is the smallest available step and the note above it is the most frequent note in the style, but whatever the mechanism, the effect is reliable and composers have relied on it for four hundred years. When chords arrive, that single semitone will turn out to be half of what makes a dominant chord resolve.
The consequence to hold on to is that a scale is not a set of pitches. It is a set of pitches plus a hierarchy, and moving the same seven notes around a different centre gives different music, which is the point of the final section of this lesson.
Minor comes in three versions
The natural minor scale has the step pattern
which on A gives the white keys again: A B C D E F G A. Compared with A major it flattens degrees 3, 6 and 7, and the flattened third is most of what makes it sound the way it does.
There is a problem, and it is precisely the leading note. In A natural minor, degree 7 is G, a whole tone below the tonic rather than a semitone, so it does not pull. Composers writing minor-key music in the tonal style wanted that pull, and bought it back by raising degree 7, giving the harmonic minor: A B C D E F G# A, with steps T S T T S A2 S where that A2 is three semitones.
The cost shows up immediately. Raising G to G sharp opens a gap of three semitones between F and G sharp, an interval of an augmented second, since it spans two letters. It is awkward to sing and instantly recognisable, and it is why harmonic minor has a colouring that most listeners associate with music from far outside the classical tradition even though it was invented inside it.
The melodic minor is the patch for the patch. Raise degree 6 as well and the gap closes: A B C D E F# G# A, which is a smooth line with a working leading note. Classical practice raises 6 and 7 only when ascending and uses the natural minor coming down, on the argument that the leading note is only needed when the line is heading for the tonic. Jazz practice keeps the raised form in both directions and calls it the melodic minor scale full stop. Both are in current use, which is a good reminder that these three scales are not three facts about nature but three solutions to one compositional problem.
Example. Write C harmonic minor and identify the augmented second.
C natural minor is C D Eb F G Ab Bb C. Raise degree 7 from B flat to B natural: C D Eb F G Ab B C. The augmented second is A flat up to B natural, three semitones across two letters.
Now you. Write F sharp harmonic minor.
Answer
F sharp natural minor is F# G# A B C# D E F#. Raising degree 7 gives F# G# A B C# D E# F#. The seventh must be spelled E sharp, not F natural, since the letter E has to appear once. The augmented second is D to E sharp.
Scales with fewer notes, and with more
Not every scale has seven degrees. The major pentatonic takes degrees 1, 2, 3, 5 and 6, dropping exactly the two notes that were adjacent to a semitone, so C major pentatonic is C D E G A. With no semitones left, no two notes in it clash, which is why it is nearly impossible to play something ugly on it. This is also why the five black keys work as a group: F# G# A# C# D# is the pentatonic of G flat major, and improvising on the black keys alone is the standard party demonstration of that fact.
The minor pentatonic is its rotation, taking degrees 1, flat 3, 4, 5 and flat 7, so A minor pentatonic is A C D E G, the same five pitches as C major pentatonic around a different home. Add a flattened fifth as a passing note, E flat here, and you have the blues scale.
In the other direction, the chromatic scale uses all twelve semitones and so has no hierarchy at all; it is a supply of notes rather than a key, and appears inside tonal music as decoration. The whole tone scale takes six equal whole-tone steps, C D E F# G# A#, and being perfectly symmetrical it has no semitone, no leading note and no obvious tonic. Debussy used it for exactly that quality of floating, and there are only two distinct whole tone scales, since the third one you try turns out to be the first transposed.
The modes are rotations, but a rotation is not enough
Play the white keys from D to D and the step pattern is T S T T T S T. That is not major and not natural minor; it is a third arrangement, called Dorian. Rotating the major scale to start on each of its degrees gives seven such patterns: Ionian on 1 (major), Dorian on 2, Phrygian on 3, Lydian on 4, Mixolydian on 5, Aeolian on 6 (natural minor), Locrian on 7.
The useful way to hold them is as alterations of a scale you already know rather than as rotations you have to count. Against major, Lydian raises degree 4 and Mixolydian flattens degree 7. Against natural minor, Dorian raises degree 6 and Phrygian flattens degree 2. Locrian flattens degree 5 as well as 2, 3, 6 and 7, which leaves its tonic triad diminished and is why it is nearly unusable as a key.
The honest limit is the one people skate over. Playing the white keys from D does not make music Dorian, because a set of pitches carries no centre with it. Whether a passage is in D Dorian or in C major depends entirely on what the music does: which note the phrases land on, which chord ends them, which note the bass insists on. Modal writing gets its centre by repetition and by drone or pedal rather than by the leading-note pull that major and minor use, and a modal passage with a strong V chord in it usually stops sounding modal and starts sounding like ordinary minor with a decorated sixth.
Writing any of these out in full is tedious, because the accidentals have to be printed on every affected note in every bar. The shorthand that fixes this is the key signature, and the order in which its sharps and flats appear turns out not to be arbitrary at all.