Equal-step tuning

In an equal-step tuning, the distance between adjacent steps is of constant size. The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). Any interval, rational, Just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common leading to EDO systems. When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that root.

When a tuning is called "n-tone equal temperament" (abbreviated n-tET or n-ET), this usually means "n divisions of 2/1, the octave, or some approximation thereof" but it also implies a mindset of temperament—that is, of a harmony-centric, JI-approximation-based understanding of the scale. If you are wondering how equal divisions of the octave can become associated with temperaments, the page EDOs to ETs may help clarify.

There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, the less theory-laden term EDO (occasionally written ED2), meaning "equal divisions of the octave" (or "equal divisions of 2/1"), leaves comparison to JI out of the picture, aside from the octave itself (which is assumed to be Just). There are other less standard terms, many in the Tonalsoft Encyclopedia. More generally, the term EDn can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic).

As the steps are tuned to be equal, equal scales may be taken to close anywhere composers wish them to. Barring the convention of closing equal divisions of particular just intervals at those stated just intervals, there are infinite synonymous names for each equal scale. Barring further the large number of names which would be avoided in discourses on comparative modality and tonality, there is still a a great width to the universe of modes and keys which modal and tonal compositional art can access.

As there are infinite intervals, there are infinite equal scales. Barring technicalities, there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings sequentially or simultaneously.

Simultaneous equal divisions

What do 12ED2, 19ED3, and 28ED5 all have in common? They're all approximately the same scale. This happens because 12ED2 is an accurate temperament (for its size) that contains relatively close approximations of 3/1 and 5/1. In contrast, 11ED2 does not correspond closely to any equal division of 3/1 or 5/1.

The following plot shows equal divisions of 2/1, 3/1, 5/1, and 7/1, and points out some instances when three or more of them happen to be close together. Note that any equal division of 2/1 is automatically an equal division of 4/1; and if something is simultaneously a good equal division of both 2/1 and 3/1, then it's a good equal division of 6/1 as well.

(Unlimited resolution version: equal.svg)

For the mathematically inclined, this kind of diagram is closely related to the Riemann zeta function.

Gallery of equal divisions

…of sixths through tenths

• …of sixths (e. g. 5/3 and 11/7)
• …of sevenths (e. g. 9/5)
• ...of the Octave/Duple (2/1)
• …of ninths (e. g. 9/4)
• …of tenths (e. g. 7/3)

…of elevenths and beyond

• …of elevenths (e. g. 8/3)
  • ...of the Natave (e/1)
• …of twelfths (e. g. 3/1)
  • ...of the Tritave/Triple (3/1)
    • most famously the BP scale, but lots of others, too
• …of thirteenths (e. g. 10/3)
• …of fourteenths (e. g. 7/2)
• ...of the Double Octave (4/1)
• …of sixteenths (e. g. 9/2)
• …of seventeenths (e. g. 5/1)
  • ...of the Just Major 17th (5/1)
• …of eighteenths (e. g. 11/2)
• …of nineteenths (e. g. 6/1)
  • ...of the hexatave (6/1)
• ...of the 7th Natural (7/1)
• ...of the Triple Octave (8/1)
• ...of the Double Tritave (9/1)
• ...of the Just Major 24th (10/1)
• ...of the 11th Natural (11/1)
• ...of the Just Perfect 26th (12/1)
• ...of the 13th Natural (13/1)

...of fifths (e. g. 3/2 and 13/9)

• ...of the Perfect Fifth (3/2)
  • most famously Carlos Alpha, Beta and Gamma, but lots of others, too
• ...of the Tridecimal High Tritone (13/9)
  • 11 - Eleventh root of 13 over 9
• ...of the Septimal High Tritone (10/7)
  • 8 - Eighth root of 10 over 7

...of fourths (e. g. 4/3 and 15/11)

• ...of the Perfect Fourth (4/3)
  • 3 - Cube Root of 4/3
  • 9 - 'Noleta' Scale
• ...of the Septimal Narrow Tritone (7/5)
  • 4 - Fourth root of 7/5
  • 11 - Eleventh root of 7/5
  • 13 - Thirteenth root of 7/5
• ...of the Undecimal Semiaugmented Fourth (15/11)
  • 13 - Thirteenth root of 15/11
• ...of the Tridecimal Ultramajor Third (13/10)
  • 2 - Square root of 13 over 10

...of thirds (e. g. 9/7, 5/4, 11/9, and 6/5)

• ...of the Just Major Third (5/4)
  • 2 - Square Root of 5/4
  • 3 - Cube Root of 5/4
  • 4 - Fourth Root of 5/4
  • 5 - Fifth Root of 5/4
  • 6 - Sixth Root of 5/4
  • 7 - Seventh Root of 5/4

...of whole tones (e. g. 8/7, 9/8, and 10/9)

• ...of the Septimal Whole Tone (8/7)
  • 5 - Fifth root of 8/7
• ...of the Pythagorean Whole Tone (9/8)
  • 9 - Ninth root of 9/8

...of semitones (e. g. 15/14, 16/15, and 25/24)

• ...of the Classic Diatonic Semitone (16/15)
  • 8 - Delta scale

Equal multiplications

An equal multiplication of a rational interval can also be called an ambitonal sequence (AS). For example, "25/24s equal temperament" could also be written "AS25/24".

An equal multiplication of an irrational interval can also be called an arithmetic pitch sequence (APS). For example, "65cET" could also be written "APS65c".

...of a given cents value

• 65cET
• 88cET
• 97.5cET
• 125cET

...of various whole tones

• ...of true whole tones
  • 8/7ths equal temperament - multiples of 8/7
  • 9/8ths equal temperament - multiples of 9/8
  • 10/9ths equal temperament - multiples of 10/9
• ...of wolf whole tones
  • 11/10ths equal temperament - multiples of 11/10
  • 12/11ths equal temperament - multiples of 12/11
  • 13/12ths equal temperament - multiples of 13/12

...of various semitones

• ...of diatonic semitones
  • 14/13ths equal temperament - multiples of 14/13
  • 15/14ths equal temperament - multiples of 15/14
  • 16/15ths equal temperament - multiples of 16/15
  • 17/16ths equal temperament - multiples of 17/16
• ...of chromatic semitones
  • 18/17s equal temperament - multiples of 18/17
  • 19/18s equal temperament - multiples of 19/18
  • 20/19s equal temperament - multiples of 20/19
  • 21/20s equal temperament - multiples of 21/20
  • 22/21s equal temperament - multiples of 22/21
  • 23/22s equal temperament - multiples of 23/22
  • 24/23s equal temperament - multiples of 24/23
  • 25/24s equal temperament - multiples of 25/24

...of various microtones

• ...of third tones
  • 26/25s equal temperament - multiples of 26/25
  • 27/26s equal temperament - multiples of 27/26
  • 28/27s equal temperament - multiples of 28/27
• ...of quarter tones
  • 33/32s equal temperament - multiples of 33/32
  • 36/35s equal temperament - multiples of 36/35

See also

• Edo anatomy
• Macrotonal edos
• Maximal evenness
• Toctave
• APS (arithmetic pitch sequence)
• AS (ambitonal sequence)
• EPD (equal pitch division)
• The Riemann Zeta Function and Tuning