Neil's replyA question from Euclid
Looking at your video on modes I wondered why there are two half-steps in each mode. In other words, who decided that there shouldn't be just one or three or more half-steps in each mode. And why were the two half-steps placed between B/C and also E/F and not, for instance, between A/B and also between C/D, if we need just two half-steps. Why were the half-steps placed where they are with two whole steps between them? And why didn't they just give each half-step a letter going from A to L instead of using sharps and flats?
Why it was decided that there should be two half-steps and only two half-steps within each mode. Who decided, and why, that there shouldn't be just one half-step or that there shouldn't be four half-steps, for example. Why couldn't a new system of music be created with seven letter notes with just one half-step and the rest all whole steps?
How appropriate, a question from Euclid, on something another Euclid may have pondered centuries ago.
When I was taking music theory classes in college, one thing the students learned was that if the instructor asked a question that we didn't know the answer to, your best bet was to reply with "the overtone series". Without going too much into the world of physics, a musical note can be measured in terms of vibrations, or frequency. We refer to this as pitch. A note vibrating at 80 cycles per second (hertz or Hz) produces overtones, or partials that are multiples of that frequency (160Hz, 240Hz, 320Hz...). On the guitar, these are what harmonics are, although I will save that for another discussion. Our ears hear notes that are multiples of each other as consonant, or pleasant combinations of sounds.
Our 12-tone system of notes has evolved as a good way to divide an octave (which is a doubling of frequencies) so that most of the consonant combinations of notes are available. The main set of notes that blend well with a given note are the ones that have become our major scale. This set of 7 notes is really derived from the partials of the overtone series.
As it turns out, in order for this to work we need 2 half steps and 5 whole steps to create a pleasant series of tones that connect notes that are an octave apart. The half steps need to be spaced so they are 2 steps and 3 steps apart (around an octave). You can see this well on a piano by how the black keys are grouped among the white keys. Our half steps of B-C and E-F fit this pattern, although there is no real reason this couldn't have been A-B and D-E, for example.
Our traditional modes (the 7 Greek ones) are all based on this arrangement of half steps, but there are other modes that space the half steps differently.
This story is bound to be continued in many chapters, if anybody is interested. Like I said, without going too much into the world of physics...
Neil