# Tuning Systems

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.

## Dividing the Octave

The vast majority of western music uses a tuning system of 12 notes within each octave. The $\frac{2}{1}$ 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 $\frac{2}{1}$ 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 | $\frac{4}{3}=\mathrm{1.333...}$ | |

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 |

Unison (C to C) | ${2}^{0/12}=1.000000$ | 0 |

Minor second (C to C♯/D♭) | ${2}^{1/12}=\sqrt[12]{2}=1.059463$ | 100 |

Major second (C to D) | ${2}^{2/12}=\sqrt[6]{2}=1.122462$ | 200 |

Minor third (C to D♯/E♭) | ${2}^{3/12}=\sqrt[4]{2}=1.189207$ | 300 |

Major third (C to E) | ${2}^{4/12}=\sqrt[3]{2}=1.259921$ | 400 |

Perfect fourth (C to F) | ${2}^{5/12}=\sqrt[12]{32}=1.334840$ | 500 |

Tritone (C to F♯/G♭) | ${2}^{6/12}=\sqrt{2}=1.414214$ | 600 |

Perfect fifth (C to G) | ${2}^{7/12}=\sqrt[12]{128}=1.498307$ | 700 |

Minor sixth (C to G♯/A♭) | ${2}^{8/12}=\sqrt[3]{4}=1.587401$ | 800 |

Major sixth (C to A) | ${2}^{9/12}=\sqrt[4]{8}=1.681793$ | 900 |

Minor seventh (C to A♯/B♭) | ${2}^{10/12}=\sqrt[6]{32}=1.781797$ | 1000 |

Major seventh (C to B) | ${2}^{11/12}=\sqrt[12]{2048}=1.887749$ | 1100 |

Octave (C to C an octave higher) | ${2}^{12/12}=2.000000$ | 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) |

C_{4} (middle C) |
261.63 | 131.87 |

C♯_{4}/D♭_{4} |
277.18 | 124.47 |

D_{4} |
293.66 | 117.48 |

D♯_{4}/E♭_{4} |
311.13 | 110.89 |

E_{4} |
329.63 | 104.66 |

F_{4} |
349.23 | 98.79 |

F♯_{4}/G♭_{4} |
369.99 | 93.24 |

G_{4} |
392.00 | 88.01 |

G♯_{4}/A♭_{4} |
415.30 | 83.07 |

A_{4} (reference pitch) |
440.00 | 78.41 |

A♯_{4}/B♭_{4} |
466.16 | 74.01 |

B_{4} |
493.88 | 69.85 |

C_{5} |
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
$\frac{1}{2}$
to go down an octave) as needed. Notice how each ratio involves only powers of 2 and 3. Also notice how these
pitches compare to their equally-tempered equivalents, and how the differences are distributed symmetrically.

Pythagorean Tuning | |||
---|---|---|---|

Interval | Ratio | Cents | Difference in Cents from Equal Temperament |

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) | $\frac{4}{3}=\frac{{2}^{2}}{3}=\mathrm{1.333\dots}$ | 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♭) | $\frac{16}{9}=\frac{{2}^{4}}{{3}^{2}}=\mathrm{1.777\dots}$ | 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
$\frac{256}{243}$
(the ratio of the minor second) by
$\frac{4}{3}$
(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.