Musical Temperaments and Ratios
Local copy, original at www.phonature.com
Musical harmony is based on the fusing of sounds that are in simple ratio.
And the history of musical scale development is the more and more sophisticated
design of tone arrangements in an octave to give the best result for the different
intervals among tones.
The 2nd and 3rd harmonic : Foundation of Musical Temperaments
The first thing to happen is the octave interval. This is created by blowing to the
flute with tightened lips. The tone that goes out from the flute has a frequency
that is twice (2nd harmonic) that of the tone blow without tightening lips. For the
less experienced flute player, it is difficult to hold onto the higher pitched tone
that is an octave higher than normal and the sound switches between these two tones.
With some practice, it is even possible to create a even higher pitched note which has
a frequency that is 3 times (3rd harmonic) that of the normal tone.
After having the 2nd and 3rd harmonics, interesting things then happen. When we
create some sound, a bunch of harmonics comes out that is characteristic of the
sounding equipment which is not easy to control separately. However, if we set one
instrument to sound at 2 times of a certain frequency and another instrument to
sound at 3 times of that frequency, then by making the two instruments to sound at
different loudness, we can control their composition fairly easily. In any case,
these two sounds will combine together to appear like one.
The Pentatonic Temperament
The first structural design begins when people build on the 2:3 ratio. After
getting a tone that is 1.5 times of a beginning tone, another tone that is 1.5 times
that of the second tone is the next to follow. This being too far away, a tone that
is half that frequency brings the range back to 1.5*1.5/2 = 9/8. So we already have
tones that are with ratios 8:9:12:16.
Checking the ratio between succeeding tones, we get 9/8=1.125, 12/9=1.333 and 16/12=1.333. The second and third
gaps are too much wider than the first. A easy way out is to insert two tones that are 1.125 times
that of the second and third tones. So we get six tones with ratios 8 : 9 : 10.125 : 12 : 13.5 : 16.
Balabiliba !!! we get our first musical temperament.
The ratios between notes are either 1.125 (which is 9/8) or 1.185 (which is 32/27).
Due to the simplicity of this scale, the Pentatonic musical scale is frequently used
in folk songs, especially those for kids.
In order to give the idea of relative size for different ratios, we give them a
scale. For reasons to be explained later in the 12 Equal Temperament Chromatic
Scale, we use 1200 cents to represent the difference between two tones that are 2/1
in frequency ratio. The gaps, which in music tonality terms, are called intervals.
When represented in cents, the intervals for ratios of 9/8 and 32/27 are 204 cents
and 294 cents respectively. The ratio of these two gaps is about 2:3, which is quite
acceptable for such a simple structured musical scale.
The Pythagorean temperament
A natural extension of the Pentatonic scale is to further subdivide the wider gaps.
One tone is 8/9 times of the 12 unit tone and another is 9/8 times of the 13.5 unit
tone. If we write the ratios with the lowest tone, we get 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1
respectively.
This scale keeps the ratio for big gap to 9/8, which we call the wholetone.
But there are two not very nice small gaps with ratios of 256/243 (which is 1.0595
or 90 cents) and 2187/2048 (which is 1.0679 or 114 cents) respectively. Their
difference is 531441/524288 (which is 1.0136 or 23.5 cents). This difference that
cannot eliminate is called the comma, which represents the smallest unit of
intervals in common use.
The Just Intonation Temperament
In order to stay with simple ratios, another approach goes from the ground up to
replace complicated ratios with simple ratios.
Step 1) 
Examining the Pythagorus scale, the obviously OK tones are 1/1, 4/3,
3/2, 2/1.Then we use them to create further notes.

Step 2) 
3/2 can be rewritten as 6/4 and 2/1 being 8/4. Naturally, we would also
want to see 5/4 and 7/4. Likewise, 2/1 is 6/3 and it is good to insert 5/3.

Step 1 
1/1 


4/3 
3/2 


2/1 
Step 2 
4/4 

5/4 
4/3 
6/4 
5/3 
7/4 
8/4 
Step 3 
4/4 

5/4 
4/3 
6/4 
5/3 

8/4 
Step 4 
8/8 
9/8 
10/8 
4/3 
12/8 
5/3 
15/8 
16/8 
Final 
1 
9/8 
5/4 
4/3 
3/2 
5/3 
15/8 
2 
Step 3)

So far so good except that the first gap is too wide and the gap between
5/3 and 7/4 is 21/20 (which is 1.05). This is less than half of any
other gaps. So we have to drop either 5/3 or 7/4. Obviously 5/3 is a
simpler ratio and so 7/4 is out.

Step 4)

Like the Pentatonic scale, we now have 2 big gaps. As in most other musical
scale, we repeat the wisdom of dividing of gap by treating 4/4 as 8/8 and
likewise for other tones. Which naturally lead us to the addition of 9/8 and
15/8. Do you know why 14/8 is not taken ?

Step 5)

Simplify the fractions to smallest numbers to see how small they can be.

The characteristic of the just intonation is, by way of its construction method, consist
of tones that are in simple ratios. Because of this, they have strong followings.
The spirit of this scale is elegance and perfection, regardless of the difficulties.
Indeed there are big difficulties with this scale, especially concerning the change of
key.
The Just Intonation scale consists of 3 kinds of gaps, namely 16/15, 10/9 and 9/8
respectively. The intervals are 112, 182 and 204 cents respectively. In contrast to
the Pythagorean scale, the Just Intonation has two kinds of wholetones and only one
kind of semitone.
The Quarter Comma MeanTone Temperament
If we compare the Pythagorean scale and Just Intonation scale, we find that they both
consists of eight tones in approximately equal intervals. For convenience, we give
them names unison, wholetone, third, fourth, fifth, sixth, seventh, octave respectively.
It is quite interesting that due to the existence of semitones, the intervals are not in
proportion to the suggested numbers. For example, twice the interval of a fourth is
midway between a sixth and a seventh, but not the octave !! Another notation is to name
the notes in alphabetical order from A to G. For some mysterious reasons, we often begin
in C.
The same interval name also mean differently sometimes for the Pythagorean Scale and
Just Intonation Scale. When they are different, their ratio and difference in cents is
shown.
Tone 
C 
D 
E 
F 
G 
A 
B 
C 
Name 
Unison 
Wholenote 
Third 
Fourth 
Fifth 
Sixth 
Seventh 
Octave 
Pythagorean Temperament 
1 
9/8 
81/64 
4/3 
3/2 
27/16 
243/128 
2 
Just Intonation Temperament 
1 
9/8 
5/4 
4/3 
3/2 
5/3 
15/8 
2 
Ratio 
1 
1 
81/80 
1 
1 
81/80 
81/80 
1 
In all cases when they differ, the ratio is 81/80. The interval of this ratio is about
21.5 cents. Many new musical scales are methods to tackle this difference so that the
pairs C & E can be consonant.
Note that the third is made by repreating up a fifth 4 times and then goes down 2
octaves. If we make the third of the scale exactly 5/4 as in Just Intonation, we
should compress the fifth by a quarter comma so that the compensation is distributed in
4 intervals. The new fifth, although not perfect, is fairly good. This is the Quarter
Comma MeanTone Temperament. There are other kinds of MeanTone Temperaments. But
since this is considered the best, so sometimes we omit the Quarter Comma specifier and
just call it MeanTone Temperament without confusion.
53Tone Equal Temperament
If we examine the intervals in the Just Intonation and the Pythagorean Temperament, we
find that they are very close to simple multiples of the comma.
Ratio 
531441/524288 
81/80 
256/243 
16/15 
2187/2048 
10/9 
9/8 
Interval 
23.5 cents 
21.5 cents 
90.2 cents 
112 cents 
114 cents 
182 cents 
204 cents 
Multiples of Comma 
1 
0.92 
3.85 
4.76 
4.84 
7.76 
8.70 
The use of intervals smaller than a semitone in a musical tonal arrangement is called
microtone temperament. So we are going to design our microtone Temperament. Based on
the Pythagorean Temperament, if we take semitones to be 4 units of a uniform interval, and
wholetones to be 9 such interval, we have divided an octave into 53 equal intervals.
Since the Pythagorean Temperament and the Just Intonation Temperament either agree or
differ by 81/80, this Temperament can represent both with high precision.
The difference from exact tonal values is less than 2 cents for the Pythagorean Temperament
and less than 4 cents when approximating the Just Intonation Temperament. Pianos have been
built for this 53Tone Equal Temperament. Though tonal arrangement is good, its use is
difficult.
A table for the 53Tone Equal Temperament is shown below :
53 Microtone Temperament

Division 
Ratio To Base Note C 
3 x Ratio 
8 x Ratio 
Cents 
Closest Simple Ratio To C 
Pythaguarean Temperament Name 
Just Intonation Name 
0 
1.0000 
3.0000 
8.0000 
0.0000 
1/1 
C 
C 
1 
1.0132 
3.0395 
8.1053 
22.6415 
81/80 


2 
1.0265 
3.0795 
8.2120 
45.2830 



3 
1.0400 
3.1200 
8.3201 
67.9245 



4 
1.0537 
3.1611 
8.4296 
90.5660 



5 
1.0676 
3.2027 
8.5406 
113.2075 



6 
1.0816 
3.2449 
8.6530 
135.8491 



7 
1.0959 
3.2876 
8.7670 
158.4906 



8 
1.1103 
3.3309 
8.8824 
181.1321 
10/9 


9 
1.1249 
3.3747 
8.9993 
203.7736 
9/8 
D 
D 
10 
1.1397 
3.4192 
9.1178 
226.4151 



11 
1.1547 
3.4642 
9.2378 
249.0566 



12 
1.1699 
3.5098 
9.3594 
271.6981 



13 
1.1853 
3.5560 
9.4826 
294.3396 



14 
1.2009 
3.6028 
9.6074 
316.9811 



15 
1.2167 
3.6502 
9.7339 
339.6226 



16 
1.2328 
3.6983 
9.8620 
362.2642 



17 
1.2490 
3.7470 
9.9919 
384.9057 
10/8 

E 
18 
1.2654 
3.7963 
10.1234 
407.5472 
81/64 
E 

19 
1.2821 
3.8463 
10.2567 
430.1887 



20 
1.2990 
3.8969 
10.3917 
452.8302 



21 
1.3161 
3.9482 
10.5285 
475.4717 



22 
1.3334 
4.0002 
10.6671 
498.1132 
4/3 
F 
F 
23 
1.3509 
4.0528 
10.8075 
520.7547 



24 
1.3687 
4.1062 
10.9498 
543.3962 



25 
1.3867 
4.1602 
11.0939 
566.0377 



26 
1.4050 
4.2150 
11.2400 
588.6792 



27 
1.4235 
4.2705 
11.3879 
611.3208 



28 
1.4422 
4.3267 
11.5378 
633.9623 



29 
1.4612 
4.3836 
11.6897 
656.6038 



30 
1.4805 
4.4414 
11.8436 
679.2453 



31 
1.4999 
4.4998 
11.9995 
701.8868 
3/2 
G 
G 
32 
1.5197 
4.5591 
12.1575 
724.5283 



33 
1.5397 
4.6191 
12.3175 
747.1698 



34 
1.5600 
4.6799 
12.4797 
769.8113 



35 
1.5805 
4.7415 
12.6440 
792.4528 



36 
1.6013 
4.8039 
12.8104 
815.0943 



37 
1.6224 
4.8671 
12.9791 
837.7358 



38 
1.6437 
4.9312 
13.1499 
860.3774 



39 
1.6654 
4.9961 
13.3230 
883.0189 
5/3 

A 
40 
1.6873 
5.0619 
13.4984 
905.6604 
27/16 
A 

41 
1.7095 
5.1285 
13.6761 
928.3019 



42 
1.7320 
5.1961 
13.8561 
950.9434 



43 
1.7548 
5.2645 
14.0385 
973.5849 



44 
1.7779 
5.3338 
14.2233 
996.2264 



45 
1.8013 
5.4040 
14.4106 
1018.8679 



46 
1.8250 
5.4751 
14.6003 
1041.5094 



47 
1.8491 
5.5472 
14.7925 
1064.1509 



48 
1.8734 
5.6202 
14.9872 
1086.7925 
15/8 

B 
49 
1.8981 
5.6942 
15.1845 
1109.4340 
243/128 
B 

50 
1.9230 
5.7691 
15.3844 
1132.0755 



51 
1.9484 
5.8451 
15.5869 
1154.7170 



52 
1.9740 
5.9220 
15.7921 
1177.3585 



53 
2.0000 
6.0000 
16.0000 
1200.0000 
2/1 
C 
C 
12Tone Equal Temperament Chromatic Scale
As always noted, interval for the wholetone is approximately twice that of the semitone.
So if we take the interval of a wholetone to be twice that of a semitone, we have a 12Tone
Equal Temperament scale. Taking the unit for a semitone in this scale to be 100 cents, then
one octave has 1,200 cents. This scale has uniform interval of 100 cents. So it has no
problem changing keys. This musical tonal scale is also called Chromatic Scale.
Other Temperaments
Temperaments are proposed to solve problems in getting harmony.
Along the theme of the MeanTone scale, we can choose different wholetone to semitone ratio to
create different MeanTone Temperaments. Since the tonal arrangement in an octave is fixed,
this single ratio will define notes in the whole octave.
Along the theme of Equal Temperament, 19tone and 33 tone systems have been proposed. They
represent a solution midway between the 12tone scale and the 53tone scale.
It is even possible to have Temperaments with an octave that is not 2. For example, a Hong Kong
violinist proposed an octave with frequency ratio of 81/40. This is said to create better
harmony in most melodies.
The Mathematical origin of different musical temperaments
Essentially, all temperaments are based on the perfect fifth with a frequency ratio of 3/2,
which is 1.5. Calculating the LOG value of 3 in base 2 is the ratio between an interval for a
3 and 2.
LOG2(3) = 1.5849625
That means, the interval between the 3rd harmonic and the fundamental is about 1.585 times
bigger than the interval for an octave. Unfortunately, this is not easily translated into a
simple fraction. So it is impossible to create a uniform scale that covers the 3rd harmonic
and the octave. But if it is not going to be uniform, then what systematic approach can be used.
To see how we can create music temperaments that approximately covers both the 2nd and 3rd
harmonic, let's convert this number into approximate fractions. The first approximation is
obviously 1.5849625 = 1 + 1/2. But are we happy with an octave with just 2 tones ? If not,
then let's move on.
If we calculate the fraction and stop at different stages by taking the last denominator to the
nearest integer, the fractions are in sequence :
3/2 , 8/5, 19/12, 84/53
If we take away one octave to see the ratio of the fifth in an octave, we have :
1/2, 3/5, 7/12, 31/53
Here we can see the Pentatonic Temperament, 12Tone Equal Temperament Chromatic Scale, and the
almost perfect 53Tone Equal Temperament.
There is a simple way to assess how good they are. If the rounding is insignificant, the
truncated fraction expansion is a good approximation to the actual value. So seeing that 2.26
is close to 2, the 7/12 octave fifth is a good approximation, and likewise for 31/53, but not
for the other two.
So from a mathematical viewpoint, the development of the musical temperaments is a natural
consequence of having more and more precise musical instruments and better musical actueness
over many generations of cultural conditioning. If we go on with the calculation, the next
fractions will be 179/306 and 389/665. I cannot foresee one day when our ears can perceive
such fine details. Also in view that even the 53Tone Equal Temperament piano did not become
popular, it is unlikely some one would like to play a 306Tone or 665Tone Equal Temperament
piano. So we can conveniently declare here the end of search for more accurate musical
temperaments. If there are any new developments, it is bound to appear along other ideas than
better approximation of the perfect fifth.

