User:Eufalesio/Enumeration: Difference between revisions

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== '''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 ratio|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 [[Harmonic|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 [https://www.mathsisfun.com/least-common-multiple.html 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 [https://en.wikipedia.org/wiki/Inversion_(music) 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 [[Kite's thoughts on hi-lo notation|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 ==

* [[Harmonic series]]

* [[Subharmonic series]]

* [[Just Intonation]]

== '''Eufalesio's version of this page''' ==

This was actually an article for [[Ratio|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 [[Just intonation|JI]] chords, but it is not exclusively limited to JI, and can be used to write any group of intervals.

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+== '''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 ratio|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 [[Harmonic|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 [https://www.mathsisfun.com/least-common-multiple.html 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 [https://en.wikipedia.org/wiki/Inversion_(music) 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 [[Kite's thoughts on hi-lo notation|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 ==
+* [[Harmonic series]]
+* [[Subharmonic series]]
+* [[Just Intonation]]
+== '''Eufalesio's version of this page''' ==
+This was actually an article for [[Ratio|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.
+––––––––––––––––––––––––––––––––––––––––––––––––––––
 An enumeration is a method of uniquely writing chords, scales, or more generally, groups of intervals. It is overwhelmingly used for writing [[Just intonation|JI]] chords, but it is not exclusively limited to JI, and can be used to write any group of intervals.