Interval arithmetic: Difference between revisions
Undo revision 172778 by Fredg999 (talk) When we stack 5ths, the ratios in the fifthchain go 3/2, 9/8 (not 9/4), 27/16 (not 27/32), etc. We sometimes add 5ths and sometimes subtract 4ths, whichever one keeps everything between 1/1 and 2/1. The diminished penta-1sn is 243/256. We want the dim penta-hexave 243/128. Just like heptatonically we'd want the major 7th. BTW originally I wrote d8 when I meant d6. Thx for helping me find my typo! :) Tag: Undo |
buncha fiddly little changes |
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Example usages of each of these operations: | Example usages of each of these operations: | ||
* | * finding which notes make up a D major chord | ||
* | * finding which scale degrees are used by a IIm chord | ||
* | * naming the intervals E–G and E–B, and identifying the type of chord E–G–B forms | ||
A consistent interval arithmetic will produce 2nds larger than 3rds, 3rds larger than 4ths, etc. For example, 7/5 is logically a 5th because it's the sum of two 3rds, 7/6 and 6/5. This makes 10/7 a 4th. But since 10/7 is larger than 7/5, the 50/49 interval between them must be a negative 2nd. Some microtonalists notate 7/5 as a 4th, avoiding negative 2nds but abandoning consistency. Some notations allow one to rename 7/5 as a 4th via accidentals that raise/lower by a pythagorean comma, providing both freedom and consistency. | A consistent interval arithmetic will produce 2nds larger than 3rds, 3rds larger than 4ths, etc. For example, 7/5 is logically a 5th because it's the sum of two 3rds, 7/6 and 6/5. This makes 10/7 a 4th. But since 10/7 is larger than 7/5, the 50/49 interval between them must be a negative 2nd. Some microtonalists notate 7/5 as a 4th, avoiding negative 2nds but abandoning consistency. Some notations allow one to rename 7/5 as a 4th via accidentals that raise/lower by a pythagorean comma, providing both freedom and consistency. | ||
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The [[stepspan]] of an interval is the number of steps it spans. The stepspan is 1 less than the degree: the stepspan of any type of 3rd (major, minor, etc.) is 2, the stepspan of any type of 4th is 3, and so on. Interval names based on stepspans are 2step, 3step, etc. The unison is a 0step. | The [[stepspan]] of an interval is the number of steps it spans. The stepspan is 1 less than the degree: the stepspan of any type of 3rd (major, minor, etc.) is 2, the stepspan of any type of 4th is 3, and so on. Interval names based on stepspans are 2step, 3step, etc. The unison is a 0step. | ||
For negative intervals, interpret "1 less than the degree" as "1 step closer to zero". For example the | For negative intervals, interpret "1 less than the degree" as "1 step closer to zero". For example the [[pythagorean comma]] is a negative dim 2nd, which has a stepspan of -1. | ||
While degrees use 1-based indexing, stepspans use 0-based indexing. Many microtonalists prefer the latter because it makes interval arithmetic easier. This can be useful when working with a less-familiar framework like octotonic or nonotonic. This article mostly discusses heptatonic notations, and mostly uses 1-based indexing. | While degrees use 1-based indexing, stepspans use 0-based indexing. Many microtonalists prefer the latter because it makes interval arithmetic easier. This can be useful when working with a less-familiar framework like octotonic or nonotonic. This article mostly discusses heptatonic notations, and mostly uses 1-based indexing. | ||
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The minor penta-2nd has a fifthspan of +2, and is 9/8. The major penta-2nd (fifthspan -3) is 32/27. The latter is major not minor simply because it is larger than 9/8 (the original meaning of major). | The minor penta-2nd has a fifthspan of +2, and is 9/8. The major penta-2nd (fifthspan -3) is 32/27. The latter is major not minor simply because it is larger than 9/8 (the original meaning of major). | ||
However, some notations have multiple fifthchains, such as [[Diaschismic|Saguguti | However, some notations have multiple fifthchains, such as [[Diaschismic|Diaschismic/Saguguti]] or [[Porcupine|Porcupine/Triyoti]]. Notations for tunings of rank 3 or higher, including just intonation, has a theoretically infinite number of fifthchains. These fifthchains are usually distinguished by microtonal accidentals. Within each fifthchain, there will be 3 perfect intervals, N-3 major ones, etc. | ||
The concept of fifthspan can be generalized to genspan, which also adds up directly. We can assign qualities in non-fifth-generated notations using a genchain. For example, Triyoti's heptatonic genchain of perfect 2nds is: | The concept of fifthspan can be generalized to genspan, which also adds up directly. We can assign qualities in non-fifth-generated notations using a genchain. For example, Triyoti's heptatonic genchain of perfect 2nds is: | ||
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The rules for interval arithmetic can be applied to these unconventional fifthchains and genchains as well. For example, fifthspans and genspans always add up directly, and the quality of an interval is always related directly to its fifthspan or genspan. | The rules for interval arithmetic can be applied to these unconventional fifthchains and genchains as well. For example, fifthspans and genspans always add up directly, and the quality of an interval is always related directly to its fifthspan or genspan. | ||
== Enharmonic | == Enharmonic unisons == | ||
Adding together two intervals can make awkward names, e.g m3 + m2 = dim 4th. To make them less awkward, one can add or subtract an [[enharmonic interval]], | Adding together two intervals can make awkward names, e.g m3 + m2 = dim 4th. To make them less awkward, one can add or subtract an [[enharmonic interval|enharmonic unison]], EU for short, which changes the spelling but not the tuning. | ||
EUs can be essential for microtonal interval arithmetic. For example, the Porcupine/Triyoti genchain can instead be named using [[Pergen|ups and downs]]. This preserves the three foundational properties of conventional notation: octave-equivalent, fifth-generated, and heptatonic. | |||
…M6 vM7 ^1 M2 vM3 ^4 P5 vM6 ^m7 '''P1''' vM2 ^m3 P4 v5 ^m6 m7 v8 ^m2 m3… | …M6 vM7 ^1 M2 vM3 ^4 P5 vM6 ^m7 '''P1''' vM2 ^m3 P4 v5 ^m6 m7 v8 ^m2 m3… | ||
Because these three properties are preserved, interval arithmetic with such a genchain is exactly as usual, but with the ups and downs added in. But the genchain looks inconsistent. Shouldn't two steps from P1 be vM2 + vM2 = vvM3? It is, but since the | Because these three properties are preserved, interval arithmetic with such a genchain is exactly as usual, but with the ups and downs added in. But the genchain looks inconsistent. Shouldn't two steps from P1 be vM2 + vM2 = vvM3? It is, but since the EU for this notation is v<sup>3</sup>A1, vvM3 is just another way of spelling ^m3. | ||
[[File:Diatonic generator step markers.png|thumb|Figure 1: diatonic generator step markers]] | [[File:Diatonic generator step markers.png|thumb|Figure 1: diatonic generator step markers]] | ||