53edo

53edo, or 53 equal divisions of the octave, is the equal tuning featuring steps of (1200/53) ~= 22.64 cents, 53 of which stack to the perfect octave 2/1. Theoretical interest in this tuning system goes back to antiquity.

Theory

Unless one has a set of accidentals for the syntonic comma (see the Notation section) one is left in the unenviable position of having to label a Ptolemaic major third the same way as the Pythagorean diminished fourth, for example. Apart from that issue, 53edo is very useful for 5-limit music.

Edostep interpretations

53edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

• 65/64, the difference between the 13-limit tendoneutral third 16/13 and the classical major third 5/4
• 81/80 (the syntonic comma), the difference between 5/4 and the diatonic major third
• The Pythagorean comma, the difference between the Pythagorean diatonic and chromatic semitones
• 91/90, the difference between the 13-limit ultramajor third 13/10 and the septimal supermajor third 9/7
• 64/63, the difference between the diatonic major third and 9/7
• 512/507, the difference between the 13-limit neutral thirds

53edo tempers out the following commas:

• The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
• The vulture comma (the difference between four 320/243 intervals and the tritave)
• The amiton (the difference between a stack of five 10/9 intervals and 27/16)
• The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
• The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
• 225/224 (the difference between 15/14 and 16/15)
• 385/384 (the difference between 77/64 and 6/5)
• 121/120 (the difference between 12/11 and 11/10)
• 625/624 (the difference between 25/24 and 26/25)
• 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)

JI approximation

53edo is most usefully seen as a 2.3.5.7.13 tuning, but the 2.3.5.13 restriction is more accurate and shared with a number of its multiples, such as 159edo. Because it is not a Meantone system, there are actually multiple potential diatonic scales to use for 5-limit harmony, one of which is the Zarlino diatonic scale (LMsLMLs), tuned in 53edo as 9-8-5-9-8-9-5, though this particular scale is arguably best used for Lydian or Locrian modes. There's also the Didymic diatonic scale, tuned in 53edo as 9-8-5-9-9-8-5, which is better suited for Ionian mode and Major tonality in general. However, 53edo also features a MOS diatonic of 9-9-4-9-9-9-4, which is basically the Pythagorean diatonic scale.

Approximation of prime harmonics in 53edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-0.1	-1.4	+4.8	-7.9	-2.8	+8.3	-3.2	+5.7	-10.7	+9.7
	Relative (%)	0.0	-0.3	-6.2	+21.0	-35.0	-12.3	+36.4	-14.0	+25.1	-47.3	+42.8
Steps (reduced)		53 (0)	84 (31)	123 (17)	149 (43)	183 (24)	196 (37)	217 (5)	225 (13)	240 (28)	257 (45)	263 (51)

Thirds in 53edo

Quality	Inframinor	Subminor	Novaminor	Pentaminor	Supraminor	Submajor	Pentamajor	Novamajor	Supermajor	Ultramajor
Cents	249	272	294	317	340	362	385	408	430	453
Just interpretation	15/13	7/6, 75/64	32/27	6/5	39/32	16/13	5/4	81/64	9/7, 32/25	13/10

Diatonic thirds are bolded.

Chords

53edo has four different flavors of minor and major intervals as well as supraminor and submajor intervals. Its inframinor and ultramajor thirds approximate 15/13 and 13/10 respectively. At the same time, 53edo's subminor and supermajor intervals approximate 7/6 and 9/7. Then there's the novaminor and novamajor thirds, which are extremely close approximations of Pythagorean minor and major thirds and can be referred to as such. There are also the pentaminor and pentamajor thirds, which are very close approximations of the Ptolemaic minor and major thirds and can also be referred to as such. Finally, the supraminor and submajor thirds approximate 39/32 and 16/13. For fourth-bounded triads, there's only really five options. The first two, which involve the approximations of 9/8 and 32/27, have a marked propensity to cause crowding, and thus are dissonant. Then there's the next two, the latal triads, which involve the approximations of 8/7 and 7/6, and which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the last option, which splits the perfect fourth cleanly in half, is an ambisonance- that is, an interval that is halfway between the extremes of consonance and dissonance.

Notation

This section provides some of the options for notating 53edo.

Pythagorean notation

In 53edo, the space between each of the notes that is separated by 2 steps in 12edo is instead 9 steps; notes separated by a single step in 12edo have to be distinguished from each other as the Pythagorean diatonic semitone is 4 steps while the Pythagorean chromatic semitone is 5 steps. Furthermore, the Pythagorean comma is a single step in 53edo, unlike in 12edo where it's tempered out. It is important to understand the usage of enharmonic equivalence here; unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the multiple faces of each of 53edo's pitches (which some might consider as a downside of using the Pythagorean system).

D
^^Ebb	^D
vvEb	^^D
vEb	vvD#
Eb	vD#
^Eb	D#
^^Eb	^D#
vvE	^^D#
vE	vvDx
E

Ups and Downs

Ups and downs naturally reflect 53edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor.

Syntonic-Rastmic Subchroma notation

Syntonic-Rastmic Subchroma notation, or SRS notation for short, uses synsharp and synflat as accidentals to cover the syntonic comma. However, while SRS notation is a 2.3.5.11 notation, only the 2.3.5 portion of the notation for 53edo is shared with multiples like 159edo.

Accidentals

53edo's accidentals, as mentioned and demonstrated previously, consist of sharps and flats, as well as either up and down accidentals, or, alternatively, synsharps and synflats and their derivatives.