Lesson Eleven: Music: more a spiral than a utility staircase, or why there's no E-sharp

In previous lessons, we've used the word "interval" to describe the gap between one musical note and another. If we take a note on the piano -- let's say F, and we find the next F going up (to the right) on the keyboard, we'll see eight keys if we count the first note and the octave note above it: F, G, A (because the letter system ends at G, we start over at A), B, C, D, E, and F. It's like a set of steps going up. 

But the octave scale is not like the set of steps in the photo above. Those steps are equally divided. If the floor you're on is like F on the piano and you go up eight steps to the next floor which is like the next F higher on the keyboard, you would find that the notes aren't an equal amount apart.  If they were musical steps like in "Big," and tuned so each note was the same interval apart, some would sound fine but you'd hit a couple clinkers. 

The music scale from F to the octave of F is more like this spiral staircase. It has order and symmetry, but the steps are sized to fit the pattern or the building plan. 

Another example of this idea is the astrological chart. This first model of a chart in astrology is made on the "equal house" system. The chart is a depiction of the sky around the Earth and the wedge-shaped sections (houses) show the movement of time. Each wedge or house stands for two hours of time, and there are twelve of them, so they cover the twenty-four hour day. 

The equal-house system is a perfectly sensible and understandable way of depicting the astrological houses, except that the cosmos is not in fact sensible and understandable. The planets are different sizes, they are clumped together in groups, and orbits aren't round but elliptical -- like a circle that's been stretched out. The heavenly bodies don't move at a perfectly even pace; that's why we have Leap Year, for instance. The clock and the calendar are measured in equal units that click along, but the universe is a whirl of rocks and gases that's expanding and contracting and whatnot.  

The chart wheel below is done in what astrologers call the Placidus system. Instead of being like a pizza cut into twelve equal pieces, some of the houses are larger and some are smaller. This more closely matches the relationship of Earth, Sun, planets, stars and orientation in space. (If you want to know more about the details on astrrological house size computation, this link takes you to a web page all about it.)

In a similar way, the intervals in the octave have larger and smaller jumps from note to note. In an earlier lesson, I mentioned Pythagoras, who studied the mathematical ratios which help us build the musical scale. That's where we get some of the notes we use. And then some of the notes we use are the result of long historic arguments about "temperament," which is which note goes where to fill in the notes we get from math. This is all in the Western music system. In other cultures, people say that there are five notes in a scale, or many more than eight notes per octave. But in these lessons, I am working with the musical systems which came to us from the ancient Greeks and then from Europe. 

We've come across the fact, in past lessons, that the black keys on the piano are patterned in groups of two, then three, then two, then three. And that these groups are separated by pairs of white keys which have no black key between them. This keyboard layout is like the Placidus astrological wheel, the one with larger and smaller wedges. And it's like the spiral staircase. The keys are symmetrical and the same size, but the "jumps" between notes are not equal. 

The jumps, or intervals, are smaller where the two white keys come together. Since you have dutifully learned to find the C and the F keys on a keyboard, you can count over and see that the two spots where the intervals are small are between the E and F key, and between the B and C key. 

There is no sharp or flat between B and C. None between E and F, So if you count up the scale chromatically, using all the sharps and flats, you skip the sharp (if you are counting up) or the flat (if you are counting down) between the B and C. Sane with E and F, The notes go up this way:  A, A-sharp, B, C, C-sharp, D, D-sharp, E, F,  F-sharp, G, G-sharp, and back to A again.  If you go down the scale using flats, the notes go A, A-flat, G, G-flat, F, E, E-flat, D, D-flat, C, B, B-flat, and A. 

NOTE: No one ever says "G-flat" or "D-flat" for reasons too dull to explain here. People always say F-sharp instead of G-flat and they say C-sharp instead of D-flat, whether they are going up or down the chromatic scale. It's A Thing. 

We'll continue with this interval spacing system next time. 

Next lesson: Major and Minor