User:Eufalesio/Enumeration
Kite's version of this page (see also the discussion tab)
New page title: EFR (extended frequency ratio)
An extended ratio is a ratio of more than 2 numbers. For example, a recipe might combine cups of flour, milk and sugar in a 5:3:4 ratio. Xenharmonic music uses an extended ratio to indicate the various frequency ratios between 3 or more notes. Thus it is an extended frequency ratio or EFR. Unlike the previous unordered example, EFRs are ordered in either ascending or descending order. The ascending form is much more common.
Harmonic (ascending) EFRs
For example, consider a just intonation major triad on A-440 with a ratio list of 1/1 - 5/4 - 3/2. The three frequencies are 440, 550 and 660. The EFR is 440:550:660, which simplifies to 4:5:6, spoken as "four five six". (Had the root been other than A-440, the EFR would be the same.)
To convert an EFR to a ratio list, simply divide every number by the first number. For example, (4:5:6)/4 is 4/4 - 5/4 - 6/4, which simplifies to 1/1 - 5/4 - 3/2.
The EFR directly indicates the interval between any pair of notes in the chord. In the JI dom7 chord 4:5:6:7, the interval between the 3rd and the 5th is 6/5, that between the 3rd and 7th is 7/5, etc.
The EFR also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as harmonics 4, 5, 6 and 7. Thus an EFR is also a list of harmonics.
To convert a list of ratios to a list of harmonics, multiply each ratio by the LCM of the denominators. For example, 1/1 - 6/5 - 3/2 has denominators 1, 5 and 2, with an LCM of 10. Multiplying each ratio by 10 makes 10/1 - 12/1 - 15/1. Remove the ones to get 10:12:15.
Subharmonic (descending) EFRs, aka SEFRs
Consider the melodic inversion of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is. While it occurs quite high in the harmonic series, it occurs quite low in the subharmonic series as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR is a descending EFR, in this case 7:6:5:4. This list of subharmonics is spoken as "seven six five four".
To convert an SEFR to a ratio list, simply divide the first number by every number. For example, 7/(7:6:5:4) is 7/7 - 7/6 - 7/5 - 7/4, with 7/7 simplifying to 1/1.
The SEFR directly indicates the interval between any pair of notes in the chord. In 7:6:5:4, the interval between the 3rd and the 5th is 6/5, that between the 3rd and 7th is 6/4 which is 3/2, etc.
To convert a list of ratios to a list of subharmonics, divide each ratio by the LCM of the numerators. For example, 1/1 - 6/5 - 3/2 has numerators 1, 6 and 3, with an LCM of 6. Dividing each ratio by 6 makes 1/6 - 1/5 - 1/4. Remove the ones to get 6:5:4.
To convert an EFR to a SEFR or vice versa, first convert it to a ratio list.
Alternate forms
Both ratio lists and EFRs can indicate the voicing of a chord. For example, a 4:5:6 major triad in hi3add8 voicing is 1/1 - 3/2 - 2/1 - 5/2 or 2:3:4:5.
Contiguous harmonics such as n:n+1:n+2:n+3 can be written with a double colon as n::n+3. Likewise for contiguous subharmonics. This is especially common for scales like 8::16.
SEFRs are sometimes written not as a:b:c:d but as 1/(a:b:c:d).
Just as there are irrational frequency ratios like sqrt(3)/sqrt(2) (a neutral 3rd), there are irrational EFRs and SEFRs.
See also
Eufalesio's version of this page
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 |