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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.




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