Let x be the number to expand. Write it as x_0+x', where x_0 is the integer part and x' is the fraction. So 0≤x'< 1$. If x' is not zero, consider 1/x'=x_1 + x'', where again x_1 is the integer part and x'' is the fraction. Continue in this way, the number is written as:
{x_0, x_1, x_2, ... }
This is the continued fraction expansion of x. If at any
time x' or x'', etc, is set to zero, what remains is a
rational approximation to x, called the convergent.
The continued fraction expansion of x gives a sequence
of convergents,
x = { x0, x1, x2, x3, x4, ... }
c1 = {x0, x1}
c2 = {x0, x1, x2}
c3 = {x0, x1, x2, x3}
...
This sequence converges around x and provides the best approximations
of x by a rational for small denomiators.
To evenly divide the 12 notes of the octave, adjacent notes should be in the ratio the twelth root of two. Doing so will give 12 raisings (12 chromatic steps) equaling an octave. Tuning exactly so is called equal-temperment, because although mathematically reasonable, no interval is perfectly consonent. Each is off from harmony, where harmony is described by the ratio of two small integers.
According to a theory of harmony, sounds in the ratio of 3 to 2, 4 to 3, etc., are harmonious. Early scales were tuned exactly on these ratios, called Just Tuning. However, as just are the common intervals, the fifth (V) and the fourth (IV), so are drastically bad the less fundamental ratios. In addition, an instrument tuned just cannot play in other keys than that for which it was tuned.
So a lot of effort when into perturbing just tuning. These perturbations were called temperments, and the instrument was called tempered. Among the temperments were the mean tone, the well tempering (there were several), and the modern equal temperment.
The big question I have is what it is that violinists do. They certainly tune pure. To equal temper is to tune V'ths flat by 2 cents. This requires a tuner's ear, and may not be the most beautiful response anyway.
But here is the deal. Just temperment and equal temperment are related by continued fractions. That's the amazing thing I want to report.
I have often thought about the harmonic relationships of the major scale. I now know that Rameau wrote an important work, Treatise on Harmony, on the subject, and others, including Descartes, have commented on this.
On the one hand, one can turn to physics. For instance, that the octave and the note have a 2 to 1 frequency relationship, to explain harmony, as in sound consonence. Or one can work purely with the mathematics, and invoke a sort of mystical belief that what is good for the integers is good for nature. This, my guess, is the pythagorean outlook.
The first few intervals, the octave, the fifth, the fourth, the major third and the minor third are represented by the ratios,
2:1, 3:2, 4:3, 5:4, 6:5Working out the math, this give ratios, 1/2, 1/3, 1/4, 1/5, 1/6.
In this manner, the fourth and the fifth are division of the octave, by inversion, and hence are given the name perfect. The major third and the minor sixth divides the octave, as does the minor third and the major sixth. Since they are not both in the the major scale, they are not termed perfect, but minor and major. The minor third and minor sixth are in the (natural) minor scale.
So far so good.
As a note, Rameau seemed to want to put this all on an algebraic footing by having the ratios proceed from operations such as mean and harmonic mean. This is a third approach.
However, that this point the progression has a gap, and the remaining two notes of the major scale, the second and the seventh are not easily explained. I think that the second is 9:8, and the string division should be (therefore) 1/9. This Rameau terms the major tone, but also identifies minor tones (and several flavors of semi-tones).
The major tone is the mathematically correct gap between fourth and fifth. It is also the octave below the nineth, as created by two fifth's. This should be expected, since all derivations from the fifth would be consistent with it.
It is also a continued fraction convergent for the square root of the major third. So yet another approach is to apply continued fractions and see where this leads the theory.
So here is a possible approach. The nice ratios give the consonances,
2/1 octave 3/2 V 4/3 IV 5/4 III 6/5 iiiInverting the III gives the vi, inverting the iii gives the VI. Now calculate the tones and semitones,
Major scale
1/1 I
II
5/4 III
* 16/15 semitone
4/3 VI
* 9/8 tone+
3/2 V
* 10/9 tone-
5/3 VI
VII
2/1 octave
Minor scale
1/1 I
II
6/5 iii
* 10/9 tone-
4/3 VI
* 9/8 tone +
3/2 V
* 16/15 semitone
8/5 vi
vii
2/1 octave
The two tones form a third: (9/8)(10/9)=5/4. Set the II as a semitone
(16/15) below iii, giving tone+,
and vii as a semitone above VI. This will make
them inversions.
Set VII as a semitone below the octave, which is
the same as the inversion of the semitone,
Major scale
1/1 I
tone+
9/8 II
tone-
5/4 III
* 16/15 semitone
4/3 VI
* 9/8 tone+
3/2 V
* 10/9 tone-
5/3 VI
tone+
15/8 VII
semitone
2/1 octave
Minor scale
1/1 I
tone+
9/8 II
semitone
6/5 iii
* 10/9 tone-
4/3 VI
* 9/8 tone +
3/2 V
* 16/15 semitone
8/5 vi
tone -
16/9 vii
tone +
2/1 octave
So this looks pretty good. Everything fits more or less.
(1) there are two tones and one semitone; (2) the inversions
are correct; (3) the consonances are pure; (4) it considers
simultaneously the major and (natural) minor scales, does
not subordinate one (usually the minor) as a variant of the
other.
Note that an interval in a scale whose inversion is in the scale is termed perfect. That is, IV, V, I and octave. Other intervals are paired minor and major (the tone is a II and the semitone ii)
This disagrees with Rameau in that he assigns tone- to the II. This makes for a more complicated taxonomy of semitones (ii). It is the just tuning described by Scholtz. Figure 8. It results from pure thirds and fifths for the three most important triads, I-III-V; IV-VI-I; V-VII-II.
Now consider continued factions.
II = CF[Power[2,1/6]]
= {1,8,6,...}
= 9/8, 55/49, ...
iii = CF[Power[2,1/4]]
= {1, 5, 3, ...}
= 6/5, 19/16, ...
III = CF[Power[2,1/3]]
= {1,3,1,5,...}
= 4/3, 5/4, 29/23, ...
ii* = CF[Power[2,1/12]]
= { 1,16,1,4,...}
= 17/16, 18/17, 89/84
ii = CF[Power[2,1/11]]
= {1, 15, 2}
= 16/15, 33/31, ...
So the II, iii, III all work out as convergents
of the proper root. The semitone, however, is off.
Even cooler,
vii = CF[Power[2,5/6]]
= {1,1,3,1,1,2,...}
= 2, 7/4, 9/5, 16/9, 41/23
The second possiblity, 9/5, would be the inversion of
tone-, and the last two sets in the scale would be
tone+, tone-, instead of tone-, tone+. However tone-
is among the convergents for II.
vi = CF[Power[2,2/3]]
= {1,1,1,2,2,1}
= 2, 3/2, 8/5, 19/12, ...
VI = CF[Power[2,3/4]]
= {1,1,2,7,...}
= 2, 5/3, 37/22
Arranged in a circle of fifths,
ascending fifths
1--3/2--9/8--5/3--5/4--15/8--tritone
I V II VI III VI
descending fourths
tritone--16/15--8/5--6/5--16/9--4/3--1
ii vi iii vii VI I
Moral of the story: convergents to the correct powers of the twelth root of 2 are the harmonic intervals as otherwise derived. Another moral, continued fractions, made sensible, are the harmonies of western music.
Pythagorean tuning
Pythagorean tuning uses perfect fifths up and perfect fourths down, and at the gap where the next fourth down should meet the fifth going up, or equally where the fifth up should meet the fourths down, occurs the wolf, a badly tuned interval. The ammount, 24 cents, is called the Pythagorean Comma.
Starting from the tonic, fifths are taken up to the augmented fifth, fourths down to the minor third. The wolf is therefore the supposed fifth from iii to V+. It is 24 cents flat.
The scale I propose, and I have to find its name, agrees on the I, V, II, (from ascending V-ths) and VI, vii (from descending VI'ths). The wolf between iii and V+ is 24 cents flat.
References
[1] Calculation: (4/3)/(5/4) = 20/12 = 5/3 [2] Murray Schechter, Tempered Scales and Continued Fractions, American Mathematical Monthly, Volume 87, Issue 1, January 1980 [3] Murray Schechter, Tempered Scales and Continued Fractions, American Mathematical Monthly, Volume 87, Issue 1, January 1980 [4] The addition of two pure sine waves of similar amplitude and frequency produces areas of constructive and destructive interference. [5] A. A. Goldstein, SIAM Review, Volume 19, Issue 3, Optimal Temperament, July 1977 [6] Shen Sinyan, Acoustics of Ancient Chinese Bells, Scientific American, Issue 256, Volume 94, 1987 [7] T. Blackburn, H. Stoess, Alternate Temperaments: Theory and Philosophy [8] Giovanni Maria Lanfranco, 1533, sourced from T. Blackburn, H. Stoess, Alternate Temperaments: Theory and Philosophy [9] A. A. Goldstein, SIAM Review, Volume 19, Issue 3, Optimal Temperament [10] William Pole, 1879, sourced from T. Blackburn, H. Stoess, Alternate Temperaments: Theory and Philosophy [11] H. Stoess, History Of Tuning And Temperament [12] Merope Mills, The Guardian, Saturday February 2003 http://smt.ucsb.edu/mto/issues/mto.98.4.4/mto.98.4.4.scholtz.html
10/9 is 182.404 cents (a huge 17.6 cents flat)
9/8 is 203.91 cents (4 cents sharp)
3/2 is 701.955 cents, 2 cents sharp, the 2 cents taken for tempering.
16/15 is 111.731 cents.