Intonation and the Violin

Tuning by Harmonics

Once we have agreed on the A (440 Hz ), division of this string into 3 half-waves gives us an octave above E-string pitch. The frequency of the E-string (natural scale with tonic A)is therefore
440x3/2 = 660 Hz. In practice we listen for unison of the harmonic at the mid-point of the E-string with that found one third of the way along the A-string. Tuning the D-string in the same manner, it will be seen that the pitch of this D is 440 x 2/3 i.e. 293.33Hz. Moving to the G-string , the pitch of G is 440 x 2/3 x 2/3 =
195.55 Hz. It is interesting to calculate the corresponding frequencies on the "equal tempered" scale i.e. on the piano scale. In this scale, the pitch of a note is 1.05946 times that of its nearest neighbor below . Seven semitone steps in succession produce a piano fifth interval, so we need to multiply 1.05946 by itself 7 times to get the number which replaces 3/2 in the earlier arithmetic. The results are as follows:-
Piano E........ 659.25 Hz.
Piano A.........440 Hz. (fixed by convention)
Piano D.........293.66Hz
Piano G.........195.99Hz. Close enough for practical puposes? Perhaps....

The Natural Scale from the fiddler`s viewpoint.

We can get harmonics by lightly touching the open string at a point 1/2, 1/3, 1/4 etc. of the way from the nut end. These points are nodes for 2, 3, 4 half-waves. Calling the open string frequency f, the harmonics give us 2f,3f,4f,5f etc. Division by 2 or 4 in the usual way generates 3/2,5/4,7/4 as ratios lying in the octave interval 1 to 2. So far we have:- doh mi soh doh` as 1, 5/4, 3/2, 2. i.e. the relative pitches of the major triad are 4;5;6.Therefore the triad based on "soh" will be 3/2, 15/8, 9/4. i.e. soh, te, re`."Fah" and "lah" are identified by building a triad with doh` at the top, namely 4/3,5/3, 2. The complete scale is
doh, re, mi, fah, soh, lah, te, doh`
1, 9/8, 5/4,4/3, 3/2, 5/3, 15/8, 2
Many players are unaware of these arithmetical relationships and what follows from them, although "tuning by harmonics" is quite common.
I list the frequency ratos from one step of the scale of D major to its neighbour:-

(D,E)(E,F#)(F#,G)(G,A)(A,B)(B,C#)(C#,D)
9/8...10/9..16/15...9/8..10/9..9/8..16/15

Observe the two kinds of "whole tone"; 9/8 and 10/9.Playing this scale starting on open D and progressing on to the A-string, we are tempted to interchange (A,B) with (B,C#) because finger patterns on adjacent strings then match.

0......1/10....1/5..1/4... stopped distance
1......9/10....4/5..3/4.....fraction of string in vibration
1......10/9.....5/4..4/3...interval from open A
3/2....5/3......15/8..2....freq. multiples of tonic D
o--------o-------o---o-- A string stoppings
o---------o------o---o----o D string stoppings
1.......9/8.....5/4..4/3 3/2 freq. multiples (tonic D)
1.......8/9.....4/5...3/4..2/3......fraction of string in vibration
0.......1/9.....1/5...1/4...1/3...stopped distance

Note that the interval (E,B) = 40/27, not 3/2 (the natural fifth). Of course, in a change of key to A-major, B is pitched 9/8 times that of A. The discrepancy between the B`s 5/3~27/16 translates to a stopping adjustment of approx. 3.5mm on the A-string! Practical advice ? "try to be aware of the key you are in and(as always) use your ears."
Questions such as "what is the difference between C# and Db?" can be answered by calculating pitches of notes linking D-major(contains C#) through the flat keys to Ab-major(contains Db).Unfortunately the answer may be route-dependent.

The interval lah to doh` is called the minor third
and its ratio is 2 divided by 5/3 i.e. 6/5. We need the minor sixth and the minor seventh intervals too. The former is mi to doh` i.e. 8/5
and the latter is re to doh`, i.e. 16/9. From these we can construct the melodic and harmonic
minor scales.
We have ignored the interval 7/4 (among many others) --- the subject of "Just Intonation" is rather extensive. It might be useful to mention the Pythagoras scale here; he stepped up (and down) from the tonic in a sequence of 5ths. This "cycle" of fiftha does not close, but Mr.P detested the idea of the irrational number "twelth root of 2", and so missed inventing equal temperament by 2000 years. His scale ratios are:-
1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2.

In this system, the stopped fifths problem mentioned earlier may be avoided, but see what has become of our natural triad ratios;
4:5:6 (natl.)---> 4: 81/64 :6 (Pyth.)
Violin teachers who stick strips of paper across the fingerboards of their pupils` instruments are (unknowingly) vindicated by Pythagoras---however, I know which scale system I prefer.

Mathematical comments.

The symbol ^ is used in absence of the power index , thus
p^q means p multiplied by itself q times.
e.g. 5^3 = 5 x 5 x 5 = 125.
2^1/12 = 1.05946.. or, 1.05946..^12 = 2
The frequency of notes on the piano are:-

f(n) = 440 x (2 ^ n/12 )....... n=0 is concert A
Non- closure of Pythagoras "cycle":-
(3/2)^12=2^m .....for integer m . Has no solution.
To see this, rewrite equation as 3^12=2^(m+12)
The left hand side is odd, but the right is even.
Reductio ad absurdum. However the best compromise is m=7.
Since 2^19 < 3^12 ......( ie, 524288 < 531441) it follows that Db is flatter than C#, Gb is flatter than F# , etc...The relative discrepancy in all cases being approx. 1.3%.
...............................BVT May 2002

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