Frequency ratio: Difference between revisions

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''For the L/s ratio, see [[Step ratio]].''{{Wikipedia|Interval ratio}}

A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).

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<math>\displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots </math>

== Extended frequency ratio (EFR) ==

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. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous ''unordered'' example, EFRs are ''ordered'' either ascending or descending. 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 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 [[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).

[[Category:Ratio| ]]

[[Category:Notation]]

[[Category:Terms]]

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 }}
 {{beginner}}
-{{Wikipedia|Interval ratio}}
+''For the L/s ratio, see [[Step ratio]].''{{Wikipedia|Interval ratio}}
 A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).
@@ Line 33: / Line 33: @@
 <math>\displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots </math>
+== Extended frequency ratio (EFR) ==
+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. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous ''unordered'' example, EFRs are ''ordered'' either ascending or descending. 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 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 [[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).
 [[Category:Ratio| ]] <!-- main article -->
 [[Category:Notation]]
 [[Category:Terms]]