Words from the Glossary
At the risk of boring you to death, this month we discuss tuning systems, a subject that is not easily understood, and not exactly easy to explain. But I'm going to give it a stab.
This month's terms:
tuning system,
temperament,
Pythagorean tuning
just intonation,
meantone temperament,
equal temperament.
(Note: Click on the term to view its definition in the glossary.)
Tuning Systems
Many musicians play their music without giving much thought to what tuning system we use. In our system of music, we use the equal temperament. Temperament is how we define the intervals between notes to achieve a tuning system. This usually means slightly compromising the pure intervals to produce the best overall sound.
The Greek mathematician Pythagoras is often credited with working out the first tuning system in the third century BC, but some historians say the Egyptians were doing it 2000 or 3000 years before that. Pythagoras discovered that notes with certain mathematical relationships produced sounds that were pleasing to the ear. For example, notes with the numerical ratio of frequencies of 1:2 or 2:3 are pleasant sounding. For example, two notes whose frequencies are 400 and 800 Hz have the frequency ratio of 1:2. Today we call such an interval an octave, and we give the notes the same name, but one octave apart. The interval between two notes whose frequencies are 400 and 600 Hz have a frequency ratio of 2:3 and is called a perfect fifth.
The problem is how to divide an octave into eight mostly whole steps or 12 halfstep notes. (European and western music uses 12 notes per octave, but other cultures use other numbers). Pythagoras said the intervals should be simple ratios because they sound the best. So he defined his scale using the ratios of 3:2 (the fifth) and 4:3 (the fourth). The difference between these two had the ratio of 9:8, which he defined as a whole step. He then tried to divide the octave into eight whole steps. However, to get the mathematics to add up, he wound up with two half steps with the ratio of 256:243, which we call the Pythagorean comma thath is not very melodic.
Musical Interval 
Frequency Ratio to Tonic 
Interval Ratio 
Interval in Cents 
Tonic 
1:1 
— 
— 
2nd 
9:8 
9:8 
204 
3rd 
81:64 
9.8 
204 
4th 
4:3 
256:243 
90 
5th 
3:2 
9:8 
204 
6th 
27:16 
9:8 
204 
7th 
243:128 
9:8 
204 
Octave 
2:1 
256:243 
90 
To improve upon Pythagorean tuning, just intonation was devised. Just intonation (also called Ptolemaic tuning, just temperament, pure temperament, or pure intonation) makes the major third intervals perfect by changing one of the fifths. This makes all the major sixths (except for FD) perfect. Unfortunately, the triad DFA is not very consonant, while all the others are perfectly in tune. The just intonation scale uses steps of two different sizes having the ratios 9:8 and 10:9. The results were not very satisfactory, and just intonation was not used very much.
Musical Interval 
Frequency Ratio to Tonic 
Interval Ratio 
Interval in Cents 
Tonic 
1:1 
— 
— 
2nd 
9:8 
9:8 
204 
3rd 
5:4 
10:9 
182 
4th 
4:3 
16:15 
112 
5th 
3:2 
9:8 
204 
6th 
5:3 
10:9 
182 
7th 
15:8 
9:8 
204 
Octave 
2:1 
16:15 
112 
With the meantone temperament, the major thirds are made exact. This slightly flattens the fifths, but this spreads the error of the just intonation over four fifths. This reduces the dissonance, making the fifths more acceptable. The meantone temperament has whole steps that are the same size and are equal to square root of 1.25. which equals 1.11803 (half of a major third). This Results in the meantone temperament being more melodic than just intonation.
Musical Interval 
Frequency Ratio to Tonic 
Interval in Cents 
Tonic 
1:1 
— 
2nd 
1.118:1 
193 
3rd 
5:4 
193 
4th 
1.338:1 
117 
5th 
1.495:1 
193 
6th 
1.672:1 
193 
7th 
1.869:1 
193 
Octave 
2:1 
117 
The equal temperament divides the octave into 12 equal half steps with each half step equal to the twelth root of 2, which equals about 1.059463. All the intervals are half steps. For example a fifth is 7 half steps and a third is 5 half steps. All the half steps in the equal temperament chromatic scale equal 100 cents. That means there are no perfectly tuned intervals. However, the mistuning of fifths is only 2 cents and for thirds is 14 cents. Apparently this is close enough to sound consonant to the ear.
Musical Interval 
Frequency Ratio to Tonic 
Interval in Cents 
Tonic 
1:1 
— 
2nd 
1.223:1 
200 
3rd 
1.260:1 
200 
4th 
1.335:1 
100 
5th 
1.449:1 
200 
6th 
1.682:1 
200 
7th 
1.889:1 
200 
Octave 
2:1 
100 
With Pythagorean tuning, just intonation, and meantone temperament, instruments (especially keyboard instruments, which could not adjust their tuning) would be in tune in one key, but would be dissonant in other keys. The big advantage of equal temperament is that all keys are in tune.
