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!