### Pythag

Number of
semitones
Generic names Specific names
Quality and number Other naming conventions Pythagorean tuning 5-limit tuning 1/4-comma
meantone
Full Short
0 comma Pythagorean comma
(524288:531441)
diesis (128:125)
0 diminished second d2 (531441:524288)
1 minor second m2 semitone,
half tone,
half step
diatonic semitone,
minor semitone
limma (256:243)
1 augmented unison A1 chromatic semitone,
major semitone
apotome (2187:2048)
2 diminished third d3 tone, whole tone, whole step
2 major second M2 sesquioctavum (9:8)
3 minor third m3 semiditone (32:27) sesquiquintum (6:5)
4 major third M3 ditone (81:64) sesquiquartum (5:4)
5 perfect fourth P4 diatessaron sesquitertium (4:3)
6 diminished fifth d5 tritone
6 augmented fourth A4
7 perfect fifth P5 diapente sesquialterum (3:2)
12 (perfect) octave P8 diapason duplex (2:1)

A tuning system, or temperament, is a way to define individual pitches for music from the set of all possible high and low tones. We often talk about pitches simply by their note names (B, F♯, D♭), but what sounds do these notes actually make? We can answer this question in one of two ways: 1) we can describe the relationships between pitches in fractions or ratios, or 2) we can measure the frequencies of sounds in vibrations per second, or Hertz (Hz). Both approaches are useful, but this lesson will focus mainly on the former so that we can observe the mathematical patterns found in tuning systems.

The vast majority of western music uses a tuning system of 12 notes within each octave. The 2:1 ratio of the octave makes this a mathematically intuitive place to start. There are other tuning systems that divide an octave into a different number of parts and still others that are not based on the octave at all, but we will not explore these systems here.

If we start with octaves defined by a 2:1 ratio and we want to divide the space between the octaves into 12 parts, how do we do this?

Early tuning systems in western music divided the octave according to the simple ratios found in the harmonic series. These ratios were observed by comparing the sounds produced by objects of different sizes, like plucking or hammering a length of string and then dividing the string into smaller parts to compare the resulting frequencies.

Ratios for Intervals in the Harmonic Series
Interval Ratio
Unison $\frac{1}{1}=1.000$
Octave $\frac{2}{1}=2.000$
Perfect Fifth $\frac{3}{2}=1.500$
Perfect Fourth $\frac{4}{3}=1.333...$
Major Third $\frac{5}{4}=1.250$
Minor Third $\frac{6}{5}=1.200$

Using the ratios of the harmonic series for tuning creates beautifully pure-sounding intervals, but there is a problem with this system: the ratios don't line up with each other. This short video illustrates this problem and describes the modern solution to it: equal temperament.

# "Why It's Impossible to Tune a Piano" by MinutePhysics

## Equal Temperament

The tuning system that is standard in western music is equal temperament. In this system, the octave is divided into twelve equal parts, making the interval between each half step identical and allowing music to be transposed freely between all twelve keys. The table below shows the ratios of all the intervals in equal temperament. It also shows cents, a unit of measure used for intervals that is defined by the equal temperament system: 100 cents is equal to an equally-tempered half step.

Intervals in Equal Temperament
Interval Ratio Cents Just Intonation Just Cents Difference
Unison (C to C) ${2}^{0/12}=1.000000$ 0
Minor second (C to C♯/D♭) ${2}^{1/12}=\sqrt{2}=1.059463$ 100
Major second (C to D) ${2}^{2/12}=\sqrt{2}=1.122462$ 200
Minor third (C to D♯/E♭) ${2}^{3/12}=\sqrt{2}=1.189207$ 300
Major third (C to E) ${2}^{4/12}=\sqrt{2}=1.259921$ 400
Perfect fourth (C to F) 2 5 / 12 = 32 12 = 1.334840 500
Tritone (C to F♯/G♭) ${2}^{6/12}=\sqrt{2}=1.414214$ 600
Perfect fifth (C to G) ${2}^{7/12}=\sqrt{128}=1.498307$ 700
Minor sixth (C to G♯/A♭) ${2}^{8/12}=\sqrt{4}=1.587401$ 800
Major sixth (C to A) ${2}^{9/12}=\sqrt{8}=1.681793$ 900
Minor seventh (C to A♯/B♭) ${2}^{10/12}=\sqrt{32}=1.781797$ 1000
Major seventh (C to B) 211/12 = 12√2048 1100
Octave (C to C an octave higher) 212/12 = 2.000000 1200

Although it does not include the pure fifths, fourths, and thirds found in earlier tuning systems, equal temperament solves the problems found in tuning systems based on the ratios of the harmonic series. Together with the reference pitch A4 = 440Hz, equal temperament provides a clear standard by which all instruments can be tuned, which is essential when many instruments play together. Given these advantages, it is easy to understand why equal temperament has been the standard tuning western music for about 300 years. The table below shows the equal temperament tunings for the octave starting at middle C (C4), including A4 at 440Hz.

-->
Frequencies and Wavelengths of Pitches in Equal Temperament
Pitch Frequency (Hz) Wavelength (cm)
C0 16.35 2109.89
C♯0/D♭0 17.32 1991.47
D0 18.35 1879.69
D♯0/E♭0 19.45 1774.20
E0 20.60 1674.62
F0 21.83 1580.63
F♯0/G♭0 23.12 1491.91
G0 24.50 1408.18
G♯0/A♭0 25.96 1329.14
A0 27.50 1254.55
A♯0/B♭0 29.14 1184.13
B0 30.87 1117.67
C1 32.70 1054.94
C♯1/D♭1 34.65 995.73
D1 36.71 939.85
D♯1/E♭1 38.89 887.10
E1 41.20 837.31
F1 43.65 790.31
F♯1/G♭1 46.25 745.96
G1 49.00 704.09
G♯1/A♭1 51.91 664.57
A1 55.00 627.27
A♯1/B♭1 58.27 592.07
B1 61.74 558.84
C2 65.41 527.47
C♯2/D♭2 69.30 497.87
D2 73.42 469.92
D♯2/E♭2 77.78 443.55
E2 82.41 418.65
F2 87.31 395.16
F♯2/G♭2 92.50 372.98
G2 98.00 352.04
G♯2/A♭2 103.83 332.29
A2 110.00 313.64
A♯2/B♭2 116.54 296.03
B2 123.47 279.42
C3 130.81 263.74
C♯3/D♭3 138.59 248.93
D3 146.83 234.96
D♯3/E♭3 155.56 221.77
E3 164.81 209.33
F3 174.61 197.58
F♯3/G♭3 185.00 186.49
G3 196.00 176.02
G♯3/A♭3 207.65 166.14
A3 220.00 156.82
A♯3/B♭3 233.08 148.02
B3 246.94 139.71
C4 (middle C) 261.63 131.87
C♯4/D♭4 277.18 124.47
D4 293.66 117.48
D♯4/E♭4 311.13 110.89
E4 329.63 104.66
F4 349.23 98.79
F♯4/G♭4 369.99 93.24
G4 392.00 88.01
G♯4/A♭4 415.30 83.07
A4 (reference pitch) 440.00 78.41
A♯4/B♭4 466.16 74.01
B4 493.88 69.85
C5 523.25 65.93

## Other Tuning Systems

Before equal temperament, there were many tuning systems based on the ratios of the harmonic series, each with a unique solution for dividing up the octave and accounting for discrepancies between ratios. Some of these systems are quite complex, and much can be said about their advantages and disadvantages. For the sake of clarity and simplicity, we will look at only one earlier system: Pythagorean tuning.

### Pythagorean Tuning

Like many early tuning systems, Pythagorean tuning was based on the $\frac{3}{2}$   ratio of the perfect fifth and the $\frac{4}{3}$   ratio of the perfect fourth. Pitches were found by going up (multiplying by $\frac{3}{2}$   or $\frac{4}{3}$ ) or down (multiplying by $\frac{2}{3}$   or $\frac{3}{4}$ ) by these intervals and adjusting by octaves (multiplying by $\frac{2}{1}$   to go up an octave and $\frac{1}{2}$ A Pythagorean tuning is technically a type of just intonation, in which the frequency ratios of the notes are all derived from the number ratio 3:2. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729:512, 3:2, 128:81, 27:16, 16:9, 243:128, 2:1. Also called "3-limit" because there are no prime factors other than 2 and 3, this Pythagorean system was of primary importance in Western musical development in the Medieval and Renaissance periods. As with nearly all just intonation systems, it has a wolf interval. In the example given, it is the interval between the 729:512 and the 256:243 (F♯ to D♭, if one tunes the 1/1 to C). The major and minor thirds are also impure, but at the time when this system was at its zenith, the third was considered a dissonance, so this was of no concern.

Pythagorean Tuning
Interval Ratio Cents Difference from ET
Unison (C to C) $\frac{1}{1}=1.000$ 0.00 0
Minor second (C to C♯/D♭) $\frac{256}{243}=\frac{{2}^{8}}{{3}^{5}}=1.053$ 90.225 +9.775
Major second (C to D) $\frac{9}{8}=\frac{{3}^{2}}{{2}^{3}}=1.125$ 203.910 -3.910
Minor third (C to D♯/E♭) $\frac{32}{27}=\frac{{2}^{5}}{{3}^{3}}=1.185$ 294.135 +5.865
Major third (C to E) $\frac{81}{64}=\frac{{3}^{4}}{{2}^{6}}=1.266$ 407.820 -7.820
Perfect fourth (C to F) $\frac{4}{3}=\frac{{2}^{2}}{3}=1.333\dots$ 498.045 +1.955
Tritone (C to G♭) $\frac{1024}{729}=\frac{{2}^{10}}{{3}^{6}}=1.405$ 588.270 +11.730
Tritone (C to F♯) $\frac{729}{512}=\frac{{3}^{6}}{{2}^{9}}=1.424$ 611.730 -11.730
Perfect fifth (C to G) $\frac{3}{2}=1.500$ 701.955 -1.955
Minor sixth (C to G♯/A♭) $\frac{128}{81}=\frac{{2}^{7}}{{3}^{4}}=1.580$ 792.180 +7.820
Major sixth (C to A) $\frac{27}{16}=\frac{{3}^{3}}{{2}^{4}}=1.688$ 905.865 -5.865
Minor seventh (C to A♯/B♭) $\frac{16}{9}=\frac{{2}^{4}}{{3}^{2}}=1.777\dots$ 996.090 +3.910
Major seventh (C to B) $\frac{243}{128}=\frac{{3}^{5}}{{2}^{7}}=1.898$ 1109.775 -9.775
Octave (C to C an octave higher) $\frac{2}{1}=2.000$ 1200.00 0

The discrepancy between the two tritone tunings is called the Pythagorean comma. The G♭ tuning is found by multiplying 256:243 (the ratio of the minor second) by 4:3 (an ascending fourth) resulting in $\frac{1024}{729}$ . The F♯ tuning is found by multiplying $\frac{243}{128}$   (the ratio of the major seventh) by $\frac{3}{4}$   (a descending fourth) resulting in $\frac{729}{512}$ . In practice, one of these two tunings is discarded, creating a fifth that is very out of tune. This is known as the wolf fifth and is found between F♯ and D♭ if G♭ is discarded, or B and G♭ if F♯ is discarded.

Twelve-Tone Scale in Five-limit tuning
Factor 1/9 1/3 1 3 9
5 note D A E B F
ratio 10:9 5:3 5:4 15:8 45:32
cents 182 884 386 1088 590
1 note B F C G D
ratio 16:9 4:3 1:1 3:2 9:8
cents 996 498 0 702 204
1/5 note G D A E B
ratio 64:45 16:15 8:5 6:5 9:5
cents 610 112 814 316 1018