I'm away from my piano for 4 months. I've been sublimating my desire to improvise on that instrument by finally learning a bunch of basic harmony theory, which I practice just by singing or whistling. For example I'm getting into modes. These 7 modes are all obtained by taking the major scale and starting it at different points, but I find that's *not* the good way for me to understand the individual flavor of each one. 🧵
Much better for me is to think of each mode as the major scale (= Dorian mode) with some notes raised or lowered a half-step - since I already have an intuitive sense of what that will do to the sound. For example, anything with the third lowered a half-step (♭3) will have a minor feel. And Aeolian, which also has the 6th and 7th lowered (♭6 and ♭7), is nothing but my old friend the harmonic minor scale!
A more interesting mode is Dorian, which has just the 3rd and 7th notes lowered a half-step (3♭ and 7♭). Since this 6th is not lowered this is not as sad as minor. You can play happy tunes in minor, but it's easier to play really lugubrious tear-jerkers, which I find annoying. The major 6th of Dorian changes the sound to something more emotionally subtle. Listen to a bunch of examples here:
Some argue that the Dorian mode gets a peculiarly 'neutral' quality by being palindromic: the pattern of whole and half steps when you go *up* this mode is the same as when you go *down*:
w h w w w h w
This may seem crazily mathematical, but Pascal said "Music is the pleasure the human mind experiences from counting without being aware that it is counting."
For more, try this:
One reason I'm having fun learning music theory is that there's now a wealth of resources online, especially on YouTube with some very nice people clearly explaining things.
For something more theoretical on the Dorian mode, try this blog:
https://mynewmicrophone.com/dorian-mode/
or for something extremely practical, try this video:
@johncarlosbaez Have you heard of the strong connection between the Riemann zeta function and "good" equal temperaments? https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning
@keenanpepper - I'm suspicious, but I'll check it out. Thanks!
@johncarlosbaez @keenanpepper before reading the derivation I would first just look at a simple graph of the zeta function on the critical line!
https://www.wolframalpha.com/input?i=plot+abs%28zeta%280.5%2Bi*t*%282*pi%29%2Fln%282%29%29%29+from+t%3D0..40
You can see there are some pretty prominent peaks at some familiar looking numbers, such as 12, 19, 22, 31, 41, 53, 72 divisions of the octave, all of which have turned up pretty frequently in tuning theory (and sometimes historically, throughout the world) for having excellent representations of various just intonation ratios (relative to their size).
The derivation is not that difficult if one looks at Re[s] > 1 where the Dirichlet series is convergent, although probably best to just look at the results first! It is very easy to see manually that, for instance, 31 equal has better approximations of simple harmonic ratios than its neighbors 30 and 32 (as the zeta function correctly predicts).
@keenanpepper we should really clean this up and publish it properly sometime!
