BAIN MUSC 726T

*Tuning Theory*

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726T

Dividing a string of length
*L* and fundamental frequency *f* into parts such as

1/2
*L,* 1/3* L,* 1/4
*L,* 1/5 *L,* etc.,

produces the following *
*frequencies, respectively:

2/1*
f*, 3/1 *f*,
4/1* f*, 5/1*
f*, etc.

For more information about the relationship between string length and
pitch interval, see the Wikipedia article Mersenne's
Laws.

**Reciprocal
relationship**

Be sure to notice the *reciprocal relationship* between *L*
and *f*, where the* reciprocal *of *x*
is defined to be 1/*x*.

**Example**.
The reciprocal of 4/3 is calculated: 1 /
(4/3) = 1 * (3/4) = 3/4

For more information, see Wikipedia: Multiplicative Inverse.

In mathematics, ratios are typically reduced to their

For more information, see Reduced
fraction in MathWorld, or Simplifying
(reducing) fractions in the Wikipedia article Fraction.

** Octave
reduction
**In tuning theory, pitches and intervals in scales are typically
expressed under

**Examples**.
The frequency ratios 2/1, 3/1,
4/1 & 5/1
are equivalent to the following ratios under octave reduction,
respectively:

2/1*
f*, 3/2 *f*,
2/1* f*, 5/4*
f*, etc.

- Harmonic series {WP}
- 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, ...
- Under octave reduction
- 1/1, 2/1, 3/2, 2/1, 5/4, 3/2, 7/4, 2/1, 9/8, 5/4, ...
- Tetractys {WP}: 1, 2, 3, 4
- PU: 1/1 {XW}
- P8: 2/1 {XW}
- P5: 3/2 {XW}
- P4: 4/3 {XW}
- Complementary intervals, also called octave complement

e.g., (3/2) **x*= 2/1;*x*= (2/1) * (2/3) = 4/3

- Pythagorean (3-limit)
{WP;
XW}

2* 3^{p}, where^{q}*p*and*q*are integers

Diatonic scale: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128 - 1/1 PU
- 2/1 P8
- 3/2 P5
- 4/3 P4
- 9/8 M2 {XW}
- 27/16 M6 {XW}
- 81/64 M3 {XW};
- 243/128: M7 {XW}
- Other intervals
- m7: (4/3) * (4/3) = 16/9
- m3: (4/3) / (9/8) = (4/3) * (8/9) = 32/27 {XW}
- m6: (2/1) / (81/64) = (2/1) * (64/81) = 128/81 {XW}
- m2: (4/3) / (81/64) = (4/3) * (64/81) = 256/243 {XW}
- etc.
- Just (5-limit) {WP;
XW}

2* 3^{p}* 5^{q}, where^{r}*p*,*q*, and*r*are integers

Diatonic scale: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 - 1/1 PU
- 2/1 P8
- 3/2 P5
- 4/3 P4
- 5/4 M3
- 6/5 m3
- 5/3 M6
- 8/5 m6
- 9/5 m7
- 15/8 M7
- 16/15 m2
- Other intervals
- Small M2: (5/4) / (9/8) = (5/4) * (8/9) = 10/9
- A4: (9/8) * (5/4) = 45/32

To compare the size of two
interval frequency ratios, we will typically convert ratios
to cents (¢).
The *cent* (*c*) is a logarithmic
unit of pitch interval proposed by the nineteenth-century English
mathematician A.J.
Ellis (1814-90). By definition, there are 1200 cents in an octave
and 100 cents in a 12-tone equal-tempered (12-tet)
semitone, which may be notated:

*c* = 1\1200 octave

The following formula may be
used to convert an interval *frequency ratio *(*f*_{1}/*f*_{2})
to cents:

*c* = 1200 log_{2} (*f*_{1}/*f*_{2})

For example, the ratio 3/2 is equivalent to 702 cents (rounding to the nearest cent), or more precisely:

1200 log_{2} (3/2) ≈ 701.995
¢

The following mobile app can perform this calculation:

**Calculator**: Bain, Ratio
to Cents

For a multi-purpose interval
calculator/converter, see Matthew Yacavone's Xen-calc.
For other interval size units, see *Interval size measure* {XW}.

**Pythagorean comma**(PC) {WP; XW}

- (3/2)^12 / (2/1)^7 = 531441/524288 ≈ 23.5 ¢
- (9/8)^6 / (2/1) ≈ 23.5 ¢
- 1.5^12 / 2^7 ≈ 129.746/128 ≈ 1.014
**Syntonic comma**(SC)- (81/64) / (5/4) = (81/64) * (4/5) = 81/80 ≈ 21.5 ¢
- 81/80 ≈ 1.013

**Other commas** {WP}
(Gann 2019, pp. 35-36)

**Diesis**{WP; XW}- (2/1) / (5/4)^3 = (128/64) / (125/64) = (128/64) x (64/125) = 128/125 ≈ 41.1 ¢
**Schisma**{WP; XW}- PC/SC = (531441/524288) / (81/80) = 32805/32768 ≈ 1.95 ¢
**Septimal comma**{WP; XW}- (16/9) / (7/4) = (16/9) * (4/7) = 64/63 ≈ 27.3 ¢
**Limma**{WP}**Kleisma**{WP; XW}

**Interval Galleries**

The following interval galleries will be
useful when we compare various tuning systems:

- Kyle Gann,
*Anatomy of an Octave*– A list of over 1000 intervals within the octave {Gann 2019} (Gann 2019) - Manuel Op de Coul, Scala: List of intervals {HFF} (See also: Scala)
- Xenharmonic Wiki

**Links**

Huygens-Fokker Foundation {HFF}

Wolfram Mathworld {Mathworld}

Xenharmonic Wiki {XW}

Wikipedia {WP}

**Reference**

Gann,
Kyle. 2019. *The Arithmetic of Listening: Tuning Theory and History
for the Impractical Musician*. Urbana:
University of Illinois Press. {GB}

Updated: February 27, 2024

Reginald
Bain | University of South Carolina
| School of
Music

https://reginaldbain.com/vc/musc726t/

https://reginaldbain.com/vc/musc726t/