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:

2/1 = 2
3/2 = 1.500,000,000,0
4/3 = 1.333,333,333,3
5/2 = 2.500,000,000,0
5/3 = 1.666,666,666,7
5/4 = 1.250,000,000,0
6/5 = 1.200,000,000,0
7/2 = 3.500,000,000,0
7/3 = 2.333,333,333,3
7/4 = 1.750,000,000,0
7/5 = 1.400,000,000,0
7/6 = 1.166,666,666,7
8/3 = 2.666,666,666,7
8/5 = 1.600,000,000,0
8/7 = 1.142,857,142,9

Here is a list of all the frequency ratios
for all intervals an octive or smaller:

1.05946309436**1 = 1.059,463,094,4
1.05946309436**2 = 1.122,462,048,3
1.05946309436**3 = 1.189,207,115,0
1.05946309436**4 = 1.259,921,049,9
1.05946309436**5 = 1.334,839,854,2
1.05946309436**6 = 1.414,213,562,4
1.05946309436**7 = 1.498,307,076,9
1.05946309436**8 = 1.587,401,052,0
1.05946309436**9 = 1.681,792,830,5
1.05946309436**10 = 1.781,797,436,3
1.05946309436**11 = 1.887,748,625,4
1.05946309436**12 = 2

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:

2nd
1.05946309436**2 = 1.122,462,048,3
7/6 = 1.166,666,666,7

minor 3rd
1.05946309436**3 = 1.189,207,115,0
6/5 = 1.200,000,000,0

major 3rd
1.05946309436**4 = 1.259,921,049,9
5/4 = 1.250,000,000,0

4th
1.05946309436**5 = 1.334,839,854,2
4/3 = 1.333,333,333,3

1.05946309436**6 = 1.414,213,562,4
7/5 = 1.400,000,000,0

5th
1.05946309436**7 = 1.498,307,076,9
3/2 = 1.500,000,000,0

1.05946309436**8 = 1.587,401,052,0

6th
1.05946309436**9 = 1.681,792,830,5
5/3 = 1.666,666,666,7

1.05946309436**10 = 1.781,797,436,3
7/4 = 1.750,000,000,0

7th
1.05946309436**11 = 1.887,748,625,4

octave
1.05946309436**12 = 2
2/1 = 2

The cases of uses of ratios of small whole
numbers of most interest are just the
five:

major 3rd
1.05946309436**4 = 1.259,921,049,9
5/4 = 1.250,000,000,0

4th
1.05946309436**5 = 1.334,839,854,2
4/3 = 1.333,333,333,3

5th
1.05946309436**7 = 1.498,307,076,9
3/2 = 1.500,000,000,0

6th
1.05946309436**9 = 1.681,792,830,5
5/3 = 1.666,666,666,7

octave
1.05946309436**12 = 2
2/1 = 2

== 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.