As promised in

https://news.ycombinator.com/item?id=31903188

here I give something of some first lessons in violin.

Part I

=== Music Theory 101 for Beginning Violinists

== Notes and Pitch

Like nearly all music, when a violin makes a sound, that sound as a pitch which in terms of some math, audio engineering, etc., has a pitch, that is, a fundamental frequency. Call that sound a note. Of course, commonly in music, more than one note is being played at once; such music is polyphonic.

A standard piano has 88 keys, some are white and some are black.

Near the middle of the keyboard is the key middle C, a white key. Its pitch, fundamental frequency, is 261.63 cycles per second, that is, Hertz or Hz.

Any two notes, e.g., two notes on piano, define an interval.

We will be especially interested in the intervals, we will define below, of a semi-tone, whole tone, 3rd, 4th, 5th, 6th, and octave.

Thanks to Bach and equal temperment, any two keys on a piano next to each other, two white keys or a black key and a white key, have their pitches separated by a semi-tone. The key with the higher pitch has the frequency of the lower key multiplied by quite accrately the 12th root of 2:

2^(1/12) = 1.05946309436

Well, two keys separated by two semi-tones are separated by a whole-tone. So, the ratios of frequencies should be about

1.05946309436*2 = 1.122,462,048

Two semi-tones form an interval of a whole tone or 2nd with ratio of frequencies

1.05946309436*2 = 1.122,462,048

Two whole tones form an interval of a major 3rd or just a 3rd with ratio of frequencies

1.05946309436*4 = 1.259,921,049

An interval of a 4th is 5 semi-tones so has frequency ratio

1.05946309436*5 = 1.334,839,854,2

A 5th is 7 semi-tones with ratio

1.05946309436*7 = 1.498,307,076,9

6th, 9 semi-tones, ratio

1.05946309436*9 = 1.681,792,830,5

7th, 11 semi-tones, ratio

1.05946309436*11 = 1.887,748,625,4

An 8th, octave, 12 semi-tone for ratio

1.05946309436*12 = 2

no surprise.

=== Major Keys

Suppose we pick a key on the piano and call that our tonic. If we go up whole tone, whole tone, semi-tone, whole tone, whole tone, whole tone, semi-tone, we will have gone up 12 semi-tones, an octave, and played the notes of the major scale on the note we picked as our tonic.

So, the notes of a major scale are a tonic and the notes we get going up

     tone, tone, semi-tone, tone, tone,
     tone, semi-tone

So, from the tonic we get intervals of a 2nd, 3rd, 4th, 5th, 6th, 7th, and 8th or octave.

Here is some of the importance of a major scale: A large fraction of all of Western music starts on some note selected as the tonic, plays the notes of only the major scale on that tonic, and, to end, returns to the tonic, or nearly so. The "nearly so" can be a source of variety of expression.

== Notes and Intervals for Violinists

A violin has 4 strings with names, from left to right as seen by the violinist, from lower pitch to higher pitch, G, D, A, E.

Middle C as on a piano is on a violin the first C on the G string and, thus, a 4th above the G.

The D string is a 5th above the G; the A string is a 5th above the D; and the E string is a 5th above the A. E.g., on a violin, the interval between two adjacent strings is always just a 5th. Simple tuning.

So, the A is 9 semi-tones above middle C and has frequency

     261.63 * 1.05946309436**9 =
     440.007,458,248

and we call that just 440 Hz.

So, the way we tune a violin is to get a tuning fork that vibrates at 440 Hz and use it to tune the A string to 440 Hz.

Above we saw that a 5th is 7 semi-tones so has ratio

1.05946309436*7 = 1.498,307,076

Gee, that is really close to 3/2.

Thus, for any two adjacent strings on a violin, 3 times the fundamental frequency of the string with the lower pitch is the same number as 2 times the fundamental frequency of the string with the higher pitch. That fact is a grand pillar of violin playing.

So, let's suppose we use the violin bow to play at the same time on adjacent strings D and A. Suppose we have tuned the A string to 440 Hz. Then the D string should have frequency 2/3rds of 440

     2*440 / 3 = 293.333,333,333

and two times the 440 is 880 and is the same as three time the

     2*440 / 3 = 293.333,333,333

When we bow both the D and A strings at the same time, we will be able to hear that 880 Hz.

Now if the D string frequency was off by a little, say, 294 Hz, then from the D string we will be getting

     3*294 = 882 Hz

which is high by 2 Hz.

From some basic trigonometry, what we will hear is essentially the 880 Hz sound but with its volume comming and going ~2 times a second. We will hear beats. As we adjust the tuning peg on the D string and get the D string frequency to where it belongs at

     2*440 / 3 = 293.333,333,333 Hz

the beats will go away. So, with no beats, we can tune the D string to quite accurately our desired

     2*440 / 3 = 293.333,333,333 Hz

Then working similarly, bowing the D and G strings together, we can get the G string tuned quite accurately at a 5th below the D string. Bowing the A string and the E strings together, we can the E string tuned quite accurately to a 5th above the A string.

Now our violin is tuned and ready to make music!