User:Eufalesio/Enumeration

This was actually an article for ratios, thinking that such an article didn't exist on account that they were called enumerations. So essentially this is me explaining what ratios are in my own words.

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An enumeration is a method of uniquely writing chords, scales, or more generally, groups of intervals. It is overwhelmingly used for writing JI chords, but it is not exclusively limited to JI, and can be used to write any group of intervals.

Introductory overview

An enumeration is a notation of the form a:b:c:d:e.... where a,b,c,d,e... are natural numbers, but not necessarily, which is notation shorthand for a set of ratios starting from a/a, b/a, c/a, d/a, e/a... ordered from a, all the way to the last term on the right. a is called the "root" of the chord, and it might be not be the root of the chord as defined Western music theory. It is the starting point by which all the other intervals are related. The enumeration may be spoken as "under a, b c d..."

Enumerations, much like chords in general, are abstract, that is, they need a reference note frequency in order for them to be heard.

Since enumerations are shortand for ratios, they are written on lowest terms, so 12:15:18 and 8:10:12 are equivalent to 4:5:6, which is the correct way to write it. They are also generally required to be written from smallest to biggest number, left to right, such that the ratios from the root are ordered from smallest to biggest, left to right.

When used for JI chords, enumerations are identified with a set of particular harmonics. For example, the chord 7:10:14:17 can be conceptualized as the harmonics 7, 10, 14, 17 from a ghost fundamental and having the chord be played from the 7th harmonic.

Numbers with decimals, square rots, values in cents, logarithms, and generally any nonzero positive real number, are allowed to be used inside enumerations, even if the resulting set of ratios sounds strange, so for example, sqrt(2):φ:1020.33c:e:ln(69) is a valid, albeit very unusual enumeration.

Enumerations may be used to write approximations of JI chords, such as those appearing in temperaments, for example, the meantone major chord from 31edo can be described as ~4:5:6 even if it is not exact, as it is the most plausible approximation, compared to the more exact but wildly disproportionate 840408126667725:1050985425614545:1256845381902341.

Just as enumerations can be used to model segments of the harmonic series, the reciprocal of enumerations can be used to model segments of the subharmonic series. For example, the subharmonic chord with the intervals 8/7, 4/3, 8/5 is the octave inversion of 4:5:6:7, and while it can be written as 60:70:84:105, it is more intuitive to write it as 1/(7:6:5:4) [numbers in reverse order so that the reciprocals are ordered correctly, since higher subharmonics are lower in pitch], or as a shorthand, as /7:6:5:4. The reciprocal enumeration may be spoken "over a, b c d..."

Examples

Common name	Intervals from root	Enumeration
Just major chord	5/4, 3/2	4:5:6
Pythagorean major chord	81/64, 3/2	64:81:96
Just minor chord	6/5, 3/2	/6:5:4 (10:12:15)
Just 7b5 chord	6/5, 7/5, 9/5	5:6:7:9
Septimal minor chord	7/6, 3/	6:7:9
All-odd voicing of the just major chord	3/1, 5/1	1:3:5
Harmonic ninth chord	5/4, 3/2, 7/4, 9/4	4:5:6:7:9
Subharmonic seventh chord	7/6, 7/5, 7/4	/7:6:5:4 (60:70:84:105)
Tridecimal augmented chord	5/4, 13/8	8:10:13
?	16/13, 19/13, 22/13	13:16:19:22
??	31/24, 19/12, 15/8	24:31:38:45
???	φ, φ^2	φ:φ+1:φ+2

See also

• Harmonic series
• Just Intonation
• Subharmonic series
• Chord
• List of no-twos chords in JI