Aristoxenian Tetrachord Calculator

How do I use this thing 😕?

Input the two intervals: Hypate to Parhypate, and Hypate to Lichanus (that's scale degrees 1 to 3, not scale degrees 2 to 3), into the fields below, then click Recalculate to generate a tetrachord.

This is an Enharmonic tetrachord with an Enharmonic diesis

281:273

281:273

5:4

1/4 tone

1/4 tone

2 tones

50

50

400

16:15

1/2 tone

100

Pycnum

281:273

2 + 1/4 tones

450

4:3

2 + 1/2 tones

500

Click a button to generate one of the tetrachords Aristoxenus explicitly mentions in Elementa Harmonica

What's the genus?

Aristoxenus would consider this tetrachord to be of the Enharmonic genus. This is because the interval from Lichanus to Mese is between 367 and 400 cents, or 1 + 5/6 and 2 tones.

What's the diesis?

Aristoxenus would describe this tetrachord's diesis as Enharmonic. defines the diesis as the smallest interval in the tetrachord. This interval will always be the interval between Hypate and Parhypate, and will always be between 50 and 100 cents, or 1/4 and 1/2 tone.

Is this is a special tetrachord?

In his Elementa Harmonica, Artistoxenus provided 11 example tetrachords, 9 rule-following, and 2 rule-breaking (on pdf pages 1 and 2) and this isn't one of them.

Here's the essentials

In his chapter on differences of the genera in the Elementa Harmonica, Artistoxenus summarizes the tetrachords (four note pitch collections that span a perfect Fourth) based on their intervals, genus, and shade.

The degrees of the tetrachord from lowest to highest are:
  1. Hypate
  2. Parhypate
  3. Lichanus
  4. Mese
Guidelines for tetrachord structure
  1. The outer interval (Hypate to Mese) is a perfect Fourth (500 cents).
  2. The interval from Hypate to Parhypate will always be between 1/4 (50 cents) and 1/2 of a tone (100 cents). It will also be the smallest interval in the tetrachord.
  3. The interval from Hypate to Lichanus will always be between 1/2 tone (50 cents) and 1 and 1/2 tones (300 cents).
  4. The interval from Lichanus to Mese will always be the remaining interval that completes the perfect Fourth, which is between (inclusive) 1 (200 cents) and 2 tones (400 cents).
Tetrachord Genera

A tetrachord's genus (plural, genera) is defined by the distance between Lichanus and Mese. The genus also has a particular shade based on where in the range it falls. There are three different genera (or flavors) of tetrachord:

  • Enharmonic - greater than 1 and 5/6 tones (about 367 cents) to equal to 2 tones (400 cents)
  • Chromatic - greater than 1 and 1/4 tones (250 cents) to less than or equal to 1 and 5/6 tones (about 367 cents)
  • Diatonic - greater than or equal to 1 tone (200 cents) to 1 and 1/4 tones (50 cents)

This means there are, technically, infinite tetrachord possibilities. However, Aristoxenus describes 6 noteworthy shades of tetrachord, 1 Enharmonic, 3 Chromatic, and 2 Diatonic:

  • Enharmonic - 1/4 + 1/4 + 2 tones (50, 50, and 400 cents)
  • Soft Chromatic - 1/3 + 1/3 + 1 and 5/6 tones (all approximations 67, 67, and 367 cents)
  • Hemiolic Chromatic - 3/8 + 3/8 + 1 and 3/4 tones (75, 75, and 350 cents)
  • Tonic Chromatic - 1/2 + 1/2 + 1 and 1/2 tones (100, 100, and 300 cents)
  • Soft Diatonic - 1/2 + 3/4 + 1 and 1/4 tones (100, 150, and 250 cents)
  • Sharp Diatonic - 1/2 + 1 + 1 tones (100, 200, 200 cents)
Dieses

The term diesis (plural, dieses) refers to the smallest interval in the tetrachord when it's less than 1/2 tone (you can read more about it in the Book 1 section).

  • An Enharmonic diesis (or the smallest Enharmonic diesis) is 1/4 tone (50 cents)
  • A smallest Chromatic diesis is 1/3 tone (about 67 cents)
  • A Hemiolic Chromatic diesis is 3/8 tone (about 75 cents)

Keeping this in mind, we could create a series of ranges that describe Enharmonic and Chromatic dieses:

  • Enharmonic - greater than or equal to 1/4 tone and less than 1/3 tone
  • Chromatic - greater than or equal to 1/3 tone and less than 1/2 tone