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(Note: this answer was given before the word "beginner" was edited into the question.)

Given a pitch, that pitch is a member of every major scale whose tonic is in the original pitch's Phrygian scale.

In other words, C is a member of every major scale whose tonic is in the C-Phrygian scale: C–D♭–E♭–F–G–A♭–B♭.

Given a pitch, that pitch is a member of every minor scale whose tonic is in the original pitches Mixolydian scale.

In other words, C is a member of every minor scale whose tonic is in the C-Mixolydian scale: C–D–E–F–G–A–B♭.

But even that's memorizing something, isn't it? :-)

If you want a method that is useful to beginners and that doesn't require additional memorization, I suggest mirroring the way the major and minor scales are constructed, which I assume a beginner will have learnt.

Iterate over the whole/half tone steps of the major and minor scale, starting from the note that the keys should contain, but go down in pitch instead of up. For the major scale the whole/half tone sequence is:

whole-whole-half-whole-whole-whole-half

and starting e.g. on C that gives you:

C → Bb → Ab → G → F → Eb → Db → C

For the minor scale the whole/half tone sequence is:

whole-half-whole-whole-half-whole-whole

and starting e.g. on C that gives you:

Cm → Bbm → Am → Gm → Fm → Em → Dm → Cm

This also works for the melodic and harmonic minor, and other scales.

The mirrored whole/half tone sequences are indeed the Phrygian and Myxolydian modes, as explained in Richard's answer. However, I assume that that would be too advanced for the beginner mentioned in the question.

1. Start with the note you want to compute against (C).


2. Go down a perfect 4th (G). Assign G a count of 1.


3. Count up in 4ths adding 1 to the count until you get to 7: G C F Bb Eb Ab Db.

Those notes and their relative minors are the major and minor scales that contain C.

Edit: I use this information all the time. It's super useful for harmonization/reharmonization, especially for same-top-note progressions (where you keep the top note steady and shift the harmony underneath, kind of like Robert Glasper).

I think the fastest way would be to memorize all the key signatures.

In your example you ask which keys contain the note C... that would be all major keys that don't have C# or Cb in the key signature: C, G, F, Bb, Eb, Ab, and Db.

Minor keys are a trickier, because there are a number of minor scales. If you're just looking at natural minors, they're the ones that share the same key signature as the majors: Am, Em, Dm, Gm, Cm, Fm, and Bbm.

For harmonic minors you'd exclude the ones where C is the subtonic (D minor) and include any where Cb is the subtonic (which would be Db minor, with eight flats).

Beginner version of Richard's answer that doesn't require knowledge of modes, based on Tim's comment.

The major keys that contain C start with the notes from the A♭ major scale (same notes as C Phrygian): A♭ B♭ C D♭ E♭ F G

The (natural) minor keys that contain C start with the notes from the F major scale (same notes as C Mixolydian). F G A B♭ C D E

In general, the major keys are from the major scale that starts with the note a major third down from the note you want (or a minor sixth up). The minor keys are from the scale that starts with the note a perfect fourth up (or a perfect fifth down).

This method requires the major scales to be memorized, and knowing intervals.

Everyone knows the major scale goes Tone Tone Semitone, Tone Tone Tone Semitone. But to see the reason why this particular scale is so common you need to examine the circle of fifths.

The most important interval in music is the octave (12 semitones.) The second most important is the perfect fifth (7 semitones.) The diatonic scale is built from a series of six stacked fifths, which is why it has a total of six chords with perfect fifths available in it (three major and three minor.)

Below is the circle of fifths opened up and arranged in a zigzag

 Gb Ab Bb C D E F# G# A# 
 Cb Db Eb F G A B C# D# E#

A couple of examples:

The notes of the key of C major are F C G D A E B or alternatively C D E on the top line and F G A B on the bottom line.

The notes of Db major are Gb Db Ab Eb Bb F C alternatively Db Eb F on the bottom line and Gb Ab Bb C on the top line.

Note: as you are looking for minimal memorization its worth noting that the above diagram can be made by writing out two whole-tone scales one above the other with an appropriate shift. This is a good way of constructing the diagram, but does not explain what is going on.

Steelpan and the circle of fifths

In order to avoid dissonant notes being close together, the notes on the Steelpan are arranged in a circle of fifths. This means all the notes of any given key are grouped together. To my knowledge it is the only instrument that uses this layout and the diagrams linked below are simpler and perhaps clearer than the wikipedia article linked above.

https://www.steeldrumshop.com/content/C%20Lead_w_range.pdf
http://www.trianglesteelbands.com/background/pan-technology.html

For example C is at the bottom (6 O'clock position) and all the notes of the key of C major are grouped from 1 O'clock to 7 O'clock. Bb is at the 8 O'clock position and the notes of Bb Major are grouped from 3 O'clock to 9 O'clock.

Go through the scale degrees. Start at 1 and go to 7 and figure out what key you'd have to be in for C to be that scale degree.

C is scale degree 1 in the key of C major.

C is scale degree 2 in the key of Bb major. Bb is a major second below C.

C is scale degree 3 in the key of Ab major. Ab is a major third below C.

C is scale degree 4 in the key of G major. G is a perfect fourth below C.

C is scale degree 5 in the key of F major. F is a perfect fifth below C.

C is scale degree 6 in the key of Eb major. Eb is a major sixth below C.

C is scale degree 7 in the key of Db major. Db is a major seventh below C.

Your example reveals the solution:

(Condition: knowledge of the circle of fifths and the relative keys!)

Then this are:

a) the tonic of the given note plus its dominante (and their relative keys) plus the 5 keys (and their relatives) “left” of the starting point of the circle of fifths. (Counterclockwise)

Notice that Db is across of G!

or with other words:

b) All keys - except the 5 keys “right” of the given (and their relatives!) starting from the secondary dominant 5 fifths clockwise in the circle of fifths.

In your example: C?

oh, you’ve already given the answer.

another example:

E?

E plus B (dominant)
A,D,G,C,F plus their relatives
mind: F is vis-a-vis of B.

or we could say:

a certain tone is contained in the keys of

the perfect cadence I-IV-V and their relatives plus the 5 fourths counterclockwise and their relatives.

The sense of these reflections could be the question of modulation.

I think using the circle of fifths is the fastest way. You have to memorize just the tonics of circle of fifths : F-C-G-D-A-E-B and Bb-Eb-Ab-Db-Gb. Then just go through the circle until you reach the note before the key. I.E you need the notes of a E Major, so you go F-C-G-D and that notes are the accidents or the Key: E, F♯, G♯, A, B, C♯,D♯.

I’m not entirely clear on what you want to do, but for memorising the accidentals, use:

Father Christmas Gave Dad An Electric Blanket
(For sharps)

And

Blanket Explodes And Dad Got Cold Feet
(For flats)