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!
Part II
== Checking Intonation
A violinist needs to work with the intervals within an octave, that is, less than or equal to an octave.
As he learns to hear and play these intervals he can check his intonation with beats based on ratios of small whole numbers as we saw above for a 5th.
Here is a list of all the possibly relevant small whole numbers:
Here is a list of all the frequency ratios for all intervals an octive or smaller:
Drawing from these two lists, here is a list of all the intervals with their usual names, their frequency ratios, and the best approximation from a ratio via a ratio of small whole numbers that might be used by a violinist to check intonation:
The cases of uses of ratios of small whole numbers of most interest are just the five:
== Playing a Scale
Number fingers on the left hand 1-4 with the index finger 1 and the little finger 4.
Learn to play on the A string a 4th above the A on the A string, that is, note D. Play this D with the 3rd finger -- now are playing with the left hand in the first position. That D is easy to learn because can check the pitch by playing with the open D string and listening for beats.
Then also on the A string, learn to play the 3rd above the A. Do this with the second finger. The note will be C#, and one way to check the pitch is just to have the 3rd finger on D and then have the 2nd finger about as close to the 3rd finger as you can. That is, there a semi-tone is about a finger width. Also the C# is a minor 3rd below the open E and can be checked by bowing with the open E.
Then with the 1st finger, play the 2nd, that is, B on the A string. This pitch will be a 4th below the open E string, so it is fairly easy to check the pitch with beats with the E string.
For the 4th finger, on the A string play the 5th, that is, the E, and, sure, it will have the same pitch as the open E string.
Now on the A string you can play A, B, C#, D, and E, that is, the first five notes of the A major scale. Move your left hand over to the E string and with fingers 1-3 as you had them on the A string, keep playing. Will be playing F#, G#, and A, the full A major scale. Since the pitch of the F# is a 6th above that of the open A string, can check your pitch, intonation, of the F# by bowing with the open A string and listening to beats.
Take what you did with your fingers on the A string and do the same thing with your fingers starting on the D string and play the D major scale, D, E, F#, G, A, B, C#, D.
Do the same thing starting on the G string and play the G major scale, G, A, B, C, D, E, F#, G.
Now you have a start on playing the major scale of any of the three keys G, D, and A. The D is the key of the center section of the Bach Chaconne.
== More About Fifths and Keys
If you start with the open D string and, say, on a piano go down in pitch a 5th, 7 semi-tones, then will be at a C, the first one below middle C. Then all the notes in the C major scale will be C, D, E, F, G, A, B, C, all white keys, no sharps or flats. So, go up a 5th to G and play its major scale, G, A, B, C, D, E, F#, G, and will have played one sharp, F#. Now starting on D and playing its major scale will play D, E, F#, A, B, C#, D. So, we play the F# we had with the G major scale and play one more sharp, C#.
So, with the C major scale we played no sharps; with the G major scale, up a 5th from the C, we played 1 sharp; with the D major scale, up a 5th from the G, we played 2 sharps, the F# we had before and one more, C#; with the A major scale, up a 5th from the D, we play 3 sharps, the F# and C# we had before and one more, G#. So, the pattern is, for a major scale, move the tonic up a fifth and add one more sharp. This pattern is a small part of the "circle of fifths".
As above, can use beats to pick out the notes of the A major scale and do the same with D major and G major. But should also learn to sing in the major scales. That is, should learn the basics of singing. When singing, should learn to hear the pitch before singing it, and then should apply that to getting pitches correct on violin. And, right, to help getting the pitch, and other aspects of the sound you want, from a violin, learn to hear the sound in your head just before playing it.
That's a much better introduction to violin than I had!
If you want to have fun with violin, the above is a good start.