Side note. You know each half step up increases it's cycles per second by 6 percent
Side note. You know each half step up increases it's cycles per second by 6 percent
Continuing the discussion from yesterday...
In most of the material that I've seen online, the notes are reflected across a certain tone relative to the tonic (i.e. relative to the "1", the first note in the scale/key). This tone is halfway between the tonic and the dominant, i.e. halfway between the 1st and the 5th scale tones, which unfortunately leaves it halfway between two tones on the keyboard (between the minor and major thirds). In the key of C, this is between Eb and E.
This choice of "axis" or reflection tone is not convenient. So far the only benefit I can see is that it preserves the starting chord somewhat --> C major becomes C minor when thusly reflected. But then we are in a different key, and the whole thing is just a pain. The "half tone" reflection just moves to the parallel minor, which is too arbitrary and not at all intuitive.
Aside for some math.
I looked up the MIDI numbers for the piano keyboard, so hopefully what I'm writing (i.e. the numbering system) will be somewhat universally accepted. The C's are multiples of 12, so C4 (middle C) is 60, the C above is 72, etc. each semitone is just an integer increase, so Db (C#) is 61, 73, etc
Let m be the tone across which we reflect a note x. Then f(x) = 2m - x is the image under reflection. Observe that the pitches are periodic with period 12 (because we use twelve tone equal temperance "TET" for dividing octaves).
It occurred to me that we could reflect across any tone, and it might even be more convenient to reflect across tones that are actually one the keyboard (i.e. playable with normal instruments, easy to notate). Since I already know what would be the most convenient result (i.e. that C major: C E G, reflects to A minor: A C E [though in reverse order]), I chose the half tone above the dominant.
That is, Ab (G#). The MIDI number is 68 +/- multiples of 12. So to reflect across this we take f(x) = 2m - x.
f(x) = 2(68) - x
f(x) = 136 - x
C E G (60 64 67) maps to (76 72 69). You can do a quick sanity check on this... 60 + 4 + 3... 76 - 4 - 3 (the intervals are the same). Pretty sure I didn't put that in the easiest way to digest. Oh well.
The nice, intuitive thing about this note is this. First, please look up an image of a piano keyboard. Find G# (Ab). Cut the note in half and notice - the keyboard is mirrored above and below! There is one other point on the keyboard (mod 12, of course) where this is true. Can you find it? What is significant about this other point relative to the Ab (G#)?
Application: we can now take any song (melody/chords) and reflect it into any chosen key. I'll restate some of the properties of the reflection, with a different analysis.
A major triad (e.g. C E G) is built on a major third interval (C to E) and then a minor third interval (E to G). Reflecting this will result in a minor third followed by a major third. That triad is called minor (Maj3 + min3 = major triad; min3 + Maj3 = min; Maj3 + Maj3 = augmented; min3 + min3 = diminished).
So major and minor triads reflect to each other. Diminished triads reflect to diminished; augmented reflect to augmented.
When we go beyond triads, it can be helpful to start looking at inversions. For example, a C7 triad (C E G Bb) is M3 + m3 + m3 (C to E is M3; E to G is m3; G to Bb is m3)
Stacking these in reverse order gives m3 + m3 + M3, something like A C Eb G, which isn't particularly easy to notate. If we flip the bottom note (the A) up an octave, we get C Eb G A, which is a C minor 6 (Cm6) chord. This is how I have seen dominant 7th chords (such as the C7 we started with) reflected, and I'm ok with it!
Let's look at one more 7th chord: Bm7 (B D F# A). The intervals are m3 + M3 + m3. Flipping this order doesn't change it!
Continuing the discussion from yesterday...
In most of the material that I've seen online, the notes are *reflected* across a certain tone relative to the tonic (i.e. relative to the "1", the first note in the scale/key). This tone is halfway between the tonic and the dominant, i.e. halfway between the 1st and the 5th scale tones, which unfortunately leaves it halfway between two tones on the keyboard (between the minor and major thirds). In the key of C, this is between Eb and E.
This choice of "axis" or reflection tone is not convenient. So far the only benefit I can see is that it preserves the starting chord somewhat --> C major becomes C minor when thusly reflected. But then we are in a different key, and the whole thing is just a pain. The "half tone" reflection just moves to the parallel minor, which is too arbitrary and not at all intuitive.
----
Aside for some math.
I looked up the MIDI numbers for the piano keyboard, so hopefully what I'm writing (i.e. the numbering system) will be somewhat universally accepted. The C's are multiples of 12, so C4 (middle C) is 60, the C above is 72, etc. each semitone is just an integer increase, so Db (C#) is 61, 73, etc
Let *m* be the tone across which we reflect a note *x*. Then f(x) = 2m - x is the image under reflection. Observe that the pitches are periodic with period 12 (because we use twelve tone equal temperance "TET" for dividing octaves).
----
It occurred to me that we could reflect across any tone, and it might even be more convenient to reflect across tones that are *actually one the keyboard* (i.e. playable with normal instruments, easy to notate). Since I already know what would be the most convenient *result* (i.e. that C major: C E G, reflects to A minor: A C E [though in reverse order]), I chose the half tone above the dominant.
That is, Ab (G#). The MIDI number is 68 +/- multiples of 12. So to reflect across this we take f(x) = 2m - x.
f(x) = 2(68) - x
f(x) = 136 - x
C E G (60 64 67) maps to (76 72 69). You can do a quick sanity check on this... 60 + 4 + 3... 76 - 4 - 3 (the intervals are the same). Pretty sure I didn't put that in the easiest way to digest. Oh well.
The nice, intuitive thing about this note is this. First, please look up an image of a piano keyboard. Find G# (Ab). Cut the note in half and notice - the keyboard is mirrored above and below! There is one other point on the keyboard (mod 12, of course) where this is true. Can you find it? What is significant about this other point relative to the Ab (G#)?
----
Application: we can now take any song (melody/chords) and reflect it into any chosen key. I'll restate some of the properties of the reflection, with a different analysis.
A major triad (e.g. C E G) is built on a major third interval (C to E) and then a minor third interval (E to G). Reflecting this will result in a minor third followed by a major third. *That* triad is called minor (Maj3 + min3 = major triad; min3 + Maj3 = min; Maj3 + Maj3 = augmented; min3 + min3 = diminished).
So major and minor triads reflect to each other. Diminished triads reflect to diminished; augmented reflect to augmented.
When we go beyond triads, it can be helpful to start looking at *inversions*. For example, a C7 triad (C E G Bb) is M3 + m3 + m3 (C to E is M3; E to G is m3; G to Bb is m3)
Stacking these in reverse order gives m3 + m3 + M3, something like A C Eb G, which isn't particularly easy to notate. If we flip the bottom note (the A) up an octave, we get C Eb G A, which is a C minor 6 (Cm6) chord. This is how I have seen dominant 7th chords (such as the C7 we started with) reflected, and I'm ok with it!
Let's look at one more 7th chord: Bm7 (B D F# A). The intervals are m3 + M3 + m3. Flipping this order doesn't change it!
(post is archived)