Octave reduction: Difference between revisions
categories |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 2: | Line 2: | ||
== Practical methods == | == Practical methods == | ||
An easy way to find the reduced form of an interval is to use a specialized calculator such as [[xen-calc]]. This is especially useful when working with very complex ratios. | |||
An easy way to find the reduced form of an interval is to use a specialized calculator | |||
There are also several methods that can be followed. The choice of an appropriate method depends on the [[interval size measure]] being used: [[Interval size measure#ratio|linear measures]] (e.g. [[ratio|frequency ratios]]), or [[Interval size measure#logarithmic|logarithmic measures]] (e.g. scale steps or [[cent]]s). | There are also several methods that can be followed. The choice of an appropriate method depends on the [[interval size measure]] being used: [[Interval size measure#ratio|linear measures]] (e.g. [[ratio|frequency ratios]]), or [[Interval size measure#logarithmic|logarithmic measures]] (e.g. scale steps or [[cent]]s). | ||
=== Linear measures === | === Linear measures === | ||
Stacking intervals expressed as ratios corresponds to multiplying those ratios. For instance, going up an octave means multiplying by 2, while going down an octave means dividing by 2. | Stacking intervals expressed as ratios corresponds to multiplying those ratios. For instance, going up an octave means multiplying by 2, while going down an octave means dividing by 2. | ||
| Line 20: | Line 18: | ||
* 7/2 is greater than 2, so divide by 2 to get 7/4. | * 7/2 is greater than 2, so divide by 2 to get 7/4. | ||
* 4/1 is greater than 2, so divide by 2 to get 2/1, which is equal to 2, so divide by 2 to get 1/1. | * 4/1 is greater than 2, so divide by 2 to get 2/1, which is equal to 2, so divide by 2 to get 1/1. | ||
* Adding 4 just perfect fifths ([[3/2]] corresponds to (3/2)<sup>4</sup>, thus 81/16 (or 5.0625), which is greater than 2 octaves (2<sup>2</sup> = 4), but less than 3 octaves (2<sup>3</sup> = 8), so divide by 2 twice to get [[81/64]]. | * Adding 4 just perfect fifths ([[3/2]]) corresponds to (3/2)<sup>4</sup>, thus 81/16 (or 5.0625), which is greater than 2 octaves (2<sup>2</sup> = 4), but less than 3 octaves (2<sup>3</sup> = 8), so divide by 2 twice to get [[81/64]]. | ||
* Subtracting a just perfect fourth ([[4/3]]) from a classic minor third [[6/5]] corresponds to 6/5 divided by 4/3, thus 9/10 (or 0.9). This interval is less than a unison (2<sup>0</sup> = 1) but greater than one octave down (2<sup>-1</sup> = 1/2), so multiply by 2 once to get 9/5. | * Subtracting a just perfect fourth ([[4/3]]) from a classic minor third [[6/5]] corresponds to 6/5 divided by 4/3, thus 9/10 (or 0.9). This interval is less than a unison (2<sup>0</sup> = 1) but greater than one octave down (2<sup>-1</sup> = 1/2), so multiply by 2 once to get 9/5. | ||
=== Logarithmic measures === | === Logarithmic measures === | ||
Stacking intervals expressed with logarithmic measures corresponds to adding those measures. For instance, when working in cents, going up an octave means adding 1200 ¢, while going down an octave means subtracting 1200 ¢. | Stacking intervals expressed with logarithmic measures corresponds to adding those measures. For instance, when working in cents, going up an octave means adding 1200 ¢, while going down an octave means subtracting 1200 ¢. | ||
| Line 38: | Line 35: | ||
== General formulas == | == General formulas == | ||
=== Linear measures === | === Linear measures === | ||
For a starting interval <math>r</math> expressed as a ratio, the reduced form <math>\text{red}(r)</math> of that interval can be found using this formula: <math>\text{red}(r) = r \cdot 2^{-\left\lfloor{\log_2 r}\right\rfloor}</math>. | For a starting interval <math>r</math> expressed as a ratio, the reduced form <math>\text{red}(r)</math> of that interval can be found using this formula: <math>\text{red}(r) = r \cdot 2^{-\left\lfloor{\log_2 r}\right\rfloor}</math>. | ||
| Line 52: | Line 47: | ||
=== Logarithmic measures === | === Logarithmic measures === | ||
For a starting interval <math>l</math> and octave <math>e</math> expressed in the same units, the reduced form <math>\text{red}(l, e)</math> of that interval can be found using this formula: <math>\text{red}(l, e) = r \bmod e</math>, where <math>\bmod</math> is the modulo operation. | For a starting interval <math>l</math> and octave <math>e</math> expressed in the same units, the reduced form <math>\text{red}(l, e)</math> of that interval can be found using this formula: <math>\text{red}(l, e) = r \bmod e</math>, where <math>\bmod</math> is the modulo operation. | ||
| Line 70: | Line 64: | ||
&= 24\end{align}</math> | &= 24\end{align}</math> | ||
== | == Generalizations == | ||
=== Other equaves === | === Other equaves === | ||
Octave reduction is mainly used in the context of octave-equivalent tunings (eg. [[12edo]]), where equivalent notes are separated by octaves. However, this operation can be generalized to any [[Periodic scale|periodic tuning]] by replacing the octave by the interval of equivalence or ''[[equave]]'' of that tuning. | Octave reduction is mainly used in the context of octave-equivalent tunings (eg. [[12edo]]), where equivalent notes are separated by octaves. However, this operation can be generalized to any [[Periodic scale|periodic tuning]] by replacing the octave by the interval of equivalence or ''[[equave]]'' of that tuning. | ||
For example, '''tritave reduction''' is the analog of octave reduction in a tritave-equivalent tuning (eg. [[ | For example, '''tritave reduction''' is the analog of octave reduction in a tritave-equivalent tuning (eg. [[Bohlen–Pierce]]), where the equave is the [[tritave]]. Therefore, a tritave-reduced interval is always obtained through transposition by tritaves, and the reduced interval lies between the unison (1/1) and the tritave (3/1). | ||
The general formula for an interval <math>r</math> and an equave <math>e</math> is as follows: <math>\text{red}(r, e) = r \cdot e^{-\left\lfloor{\log_e r}\right\rfloor}</math>. Note that <math>e</math> is a variable and not Euler's number. | The general formula for an interval <math>r</math> and an equave <math>e</math> is as follows: <math>\text{red}(r, e) = r \cdot e^{-\left\lfloor{\log_e r}\right\rfloor}</math>. Note that <math>e</math> is a variable and not Euler's number. | ||
| Line 84: | Line 76: | ||
* Consider a tritave-equivalent tuning; 7/9 is less than 3, so multiply by 3 to get 7/3. | * Consider a tritave-equivalent tuning; 7/9 is less than 3, so multiply by 3 to get 7/3. | ||
* Consider a just perfect fifth-equivalent tuning; <math>\text{red}(81/64, 3/2) = 81/64 \cdot (3/2)^{-\left\lfloor{\log_{3/2} 81/64}\right\rfloor} = 1</math>. | * Consider a just perfect fifth-equivalent tuning; <math>\text{red}(81/64, 3/2) = 81/64 \cdot (3/2)^{-\left\lfloor{\log_{3/2} 81/64}\right\rfloor} = 1</math>. | ||
* In the equal-tempered | * In the equal-tempered Bohlen–Pierce tuning, a tritave can be expressed as 1300 hekts and a BP fifth down as -500 hekts. This interval is less than the unison, so add 1300 hekts to get 800 hekts. | ||
=== Balanced reduction === | === Balanced reduction === | ||
'''Balanced reduction''' is an alternate operation where the values are equally distributed around the unison instead of being situated between the unison and the octave (or the equave). | '''Balanced reduction''' is an alternate operation where the values are equally distributed around the unison instead of being situated between the unison and the octave (or the equave). | ||
| Line 101: | Line 92: | ||
== See also == | == See also == | ||
* [[Octave complement]] | * [[Octave complement]] | ||
| Line 108: | Line 98: | ||
== External links == | == External links == | ||
* https://forum.sagittal.org/viewtopic.php?p=1296 (thread on mathematical functions) | * https://forum.sagittal.org/viewtopic.php?p=1296 (thread on mathematical functions) | ||