Frequency ratio: Difference between revisions

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 three or more notes. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous unordered example, EFRs are ordered, with harmonic EFR typically written in an ascending order and subharmonic EFR typically written in a descending order.  

=== Harmonic EFR ===

For example, consider a [[just intonation]] major triad on {{w|A440}} 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 A440, the EFR would be the same.

To convert an EFR to a ratio list, simply divide every harmonic by the first harmonic. 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 harmonic seventh chord 4:5:6:7, the interval between the third and the fifth is 6/5, that between the third and seventh is 7/5, etc.  

The EFR also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as [[harmonic]]s 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 EFR, a.k.a. SEFR ===

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 because 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 represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Confusingly, some writers tend to omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.  

To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(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 1/(7:6:5:4), the interval between the third and the fifth is 6/5, that between the third and seventh 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. Extract the ones to get 1/(6:5:4).

=== Alternative 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.