# Tetrachord

*Not to be confused with Tetrad.*

The word **tetrachord** usually refers to the interval of a perfect fourth (just or not) divided into three subintervals by the interposition of two additional notes.

John Chalmers, in Divisions of the Tetrachord, tells us:

*Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.*

## Ancient Greek genera

The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the *characteristic interval* (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.

- hyperenharmonic genus
- The CI is larger than 425 cents.
- enharmonic genus
- The CI approximates a major third, falling between 425 cents and 375 cents.
- chromatic genus
- The CI approximates a minor or neutral third, falling between 375 cents and 250 cents.
- diatonic genus
- The CI (and the other intervals) approximates a "tone", measuring less than 250 cents.

### Ptolemy's catalog

In the *Harmonics*, Ptolemy catalogs several historical tetrachords and attributes them to particular theorists.

28/27, 36/35, 5/4 | 63 + 49 + 386 | enharmonic |

28/27, 243/224, 32/27 | 63 + 141 + 294 | chromatic |

28/27, 8/7, 9/8 | 63 + 231 + 204 | diatonic |

40/39, 39/38, 19/15 | 44 + 45 + 409 | enharmonic |

20/19, 19/18, 6/5 | 89 + 94 + 316 | chromatic |

256/243, 9/8, 9/8 | 90 + 204 + 204 | diatonic |

32/31, 31/30, 5/4 | 55 + 57 + 386 | enharmonic |

16/15, 25/24, 6/5 | 112 + 74 + 316 | chromatic |

16/15, 10/9, 9/8 | 112 + 182 + 204 | diatonic |

46/45, 24/23, 5/4 | 38 + 75 + 386 | enharmonic |

28/27, 15/14, 6/5 | 63 + 119 + 316 | soft chromatic |

22/21, 12/11, 7/6 | 81 + 151 + 267 | intense chromatic |

21/20, 10/9, 8/7 | 85 + 182 + 231 | soft diatonic |

28/27, 8/7, 9/8 | 63 + 231 + 204 | diatonon toniaion |

256/243, 9/8, 9/8 | 90 + 204 + 204 | diatonon ditoniaion |

16/15, 9/8, 10/9 | 112 + 182 + 204 | intense diatonic |

12/11, 11/10, 10/9 | 151 + 165 + 182 | equable diatonic |

### Superparticular intervals

In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular.

## Jins/ajnas (tetrachords in middle-eastern music)

A concept similar to the tetrachord exists in Arabic music theory: a jins (pl. ajnas) is a set of three, four or five stepwise pitches used to build an Arabic maqam.

## Generalized tetrachords

All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals *a* & *b*, & then write our generalized tetrachord like this:

1/1, a, b, 4/3

We can build a heptatonic scale by copying this tetrachord at the perfect fifth. Thus:

1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1

Between 3/2 and 4/3, we have 9/8, so another way to write it would be:

[tetrachord], 9/8, [tetrachord]

Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord (e.g. 1/1, c, d, 4/3):

1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1

[tetrachord #1], 9/8, [tetrachord #2]

Of course, you can also put them in opposite order:

1/1, c, d, 4/3, 3/2, 3a/2, 3b/2, 2/1

[tetrachord #2], 9/8, [tetrachord #1]

### Modes of a [tetrachord], 9/8, [tetrachord] scale

mode 1 | 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1 |
---|---|

mode 2 | 1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1 |

mode 3 | 1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1 |

mode 4 | 1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1 |

mode 5 | 1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1 |

mode 6 | 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1 |

mode 7 | 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1 |

This type of scale contains not only one tetrachord, but three.

1/1, a, b, 4/3 (mode 1, mode 5)

1/1, b/a, 4/3a, 4/3 (mode 6)

1/1, 4/3b, 4a/3b, 4/3 (mode 7)

These three tetrachords are all rotations of each other (they contain the same steps in a different order).

### Tetrachord rotations

If you think of a tetrachord as three steps which total to a perfect fourth, then it makes sense that we can put those steps in any order. If we have a tetrachord with three step sizes, s, M, and L, then we have six rotations:

sML, MsL, sLM, MLs, LsM, LMs

If you have only two step sizes, s and L, then you have three possible rotations:

ssL, sLs, Lss

And, if you have only one step size (as is the case in Porcupine temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in 22edo - see 22edo tetrachords.)

## Tetrachords in equal temperaments

Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with 7edo, which has one tetrachord:

1 + 1 + 1

We can use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:

tetrachord notation | cents between steps | cents from 0 |
---|---|---|

1-1-1 | 171 + 171 + 171 | 0, 171, 343, 514 |

### Tetrachords of 10edo

Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step:

tetrachord notation | cents between | cents from 0 |
---|---|---|

1-1-2 | 120 + 120 + 240 | 0, 120, 240, 480 |

1-2-1 | 120 + 240 + 120 | 0, 120, 360, 480 |

2-1-1 | 240 + 120 + 120 | 0, 240, 360, 480 |

Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents).

### Tetrachords in other equal temperaments

- 16edo tetrachords
- 17edo tetrachords
- 22edo tetrachords
- Tricesimoprimal Tetrachordal Tesseract (tetrachords of 31edo)

If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!

## Dividing other-than-perfect fourths

A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: 6edo, 8edo, 9edo, 11edo, 13edo, 16edo. Also, one may divide a just other-than-perfect fourth, such as 21/16, 43/32, 26/19, 11/8. At what point does the concept of "tetrachord" stop being useful?

## Tetrachords and nonoctave scales

Dividing a tenth into three equal parts generates a cycle of three fourths, they resembling perfect fourths when the division is done on a minor tenth.

An example with Carlos Gamma: