Interval arithmetic - Revision history

FloraC: Misc. cleanup: spell degrees in words, no colors, etc.

2025-04-18T12:59:37Z

Misc. cleanup: spell degrees in words, no colors, etc.

VectorGraphics: no colors allowed

2025-04-17T19:51:21Z

no colors allowed

← Older revision		Revision as of 19:51, 17 April 2025
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	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 unison]], EU for short, which changes the spelling but not the tuning.		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 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.		EUs can be essential for microtonal interval arithmetic. For example, the Porcupine 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…

Sintel: Add beginner template

2025-04-17T19:04:03Z

Add beginner template

← Older revision		Revision as of 19:04, 17 April 2025
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			{{Beginner}}
	'''Interval arithmetic''' systems are sets of rules for naming the notes or intervals produced by these operations:		'''Interval arithmetic''' systems are sets of rules for naming the notes or intervals produced by these operations:

TallKite: /* Enharmonic unisons */

2024-12-24T11:36:15Z

Enharmonic unisons

← Older revision		Revision as of 11:36, 24 December 2024
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	== Enharmonic unisons ==		== 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\|~~enharmonic unison]], EU for short, which changes the spelling but not the tuning.		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 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.		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.

TallKite: buncha fiddly little changes

2024-12-24T09:03:27Z

buncha fiddly little changes

TallKite: 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! :)

2024-12-24T08:55:10Z

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! :)

← Older revision		Revision as of 08:55, 24 December 2024
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	A notation of N notes will usually have 3 perfect intervals, N-3 major ones, N-3 minor ones, and N of everything else, distributed symmetrically around P1. For example, the pentatonic fifthchain is:		A notation of N notes will usually have 3 perfect intervals, N-3 major ones, N-3 minor ones, and N of everything else, distributed symmetrically around P1. For example, the pentatonic fifthchain is:

	...A3 A1 A4 M2 M5 P3 '''P1''' P4 m2 m5 d3 d1 d4 d2…		...A3 A1 A4 M2 M5 P3 '''P1''' P4 m2 m5 d3 d6 d4 d2…

	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).

Fredg999: /* Unconventional notations */ Fix error in pentatonic fifthchain

2024-12-24T03:17:32Z

Unconventional notations: Fix error in pentatonic fifthchain

← Older revision		Revision as of 03:17, 24 December 2024
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	A notation of N notes will usually have 3 perfect intervals, N-3 major ones, N-3 minor ones, and N of everything else, distributed symmetrically around P1. For example, the pentatonic fifthchain is:		A notation of N notes will usually have 3 perfect intervals, N-3 major ones, N-3 minor ones, and N of everything else, distributed symmetrically around P1. For example, the pentatonic fifthchain is:

	...A3 A1 A4 M2 M5 P3 '''P1''' P4 m2 m5 d3 d8 d4 d2…		...A3 A1 A4 M2 M5 P3 '''P1''' P4 m2 m5 d3 d1 d4 d2…

	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).

Fredg999: /* Qualities */ Internal link

2024-12-24T02:40:33Z

Qualities: Internal link

← Older revision		Revision as of 02:40, 24 December 2024
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	=== Qualities ===		=== Qualities ===
	The process for adding up qualities such as major or perfect is more complex. Because conventional notation is generated by fifths, the quality is determined by the fifthspan of the interval.		The process for adding up qualities such as major or perfect is more complex. Because conventional notation is generated by fifths, the quality is determined by the [[fifthspan]] of the interval.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Fredg999: /* Rules */ Fix typo

2024-12-24T02:38:53Z

Rules: Fix typo

← Older revision		Revision as of 02:38, 24 December 2024
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	The stepspan between two notes is found by counting the steps between them. The interval C-G is made up of 4 steps: C-D, D-E, E-F and F-G, making a 4step.		The stepspan between two notes is found by counting the steps between them. The interval C-G is made up of 4 steps: C-D, D-E, E-F and F-G, making a 4step.

	To add an interval to a note: for stepspans, go up the note names a certain number of steps. For degrees, go up 2 steps for s 3rd, 3 steps for a 4th, etc.		To add an interval to a note: for stepspans, go up the note names a certain number of steps. For degrees, go up 2 steps for a 3rd, 3 steps for a 4th, etc.

	=== Qualities ===		=== Qualities ===

TallKite at 11:50, 23 December 2024

2024-12-23T11:50:47Z

← Older revision		Revision as of 11:50, 23 December 2024
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	* name the intervals E–G and E–B, and identify the type of chord E–G–B forms		* name the intervals E–G and E–B, and identify 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.		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.

	== Degree vs stepspan ==		== Degree vs stepspan ==

← Older revision		Revision as of 09:03, 24 December 2024
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	Example usages of each of these operations:		Example usages of each of these operations:

	* ~~find~~ which notes make up a D major chord		* finding which notes make up a D major chord
	* ~~find~~ which scale degrees are used by a IIm chord		* finding which scale degrees are used by a IIm chord
	* ~~name~~ the intervals E–G and E–B, and ~~identify~~ the type of chord E–G–B forms		* 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 ~~pyth~~ comma is a negative dim 2nd, which has a stepspan of -1.		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~~/Diaschismic~~]] or [[Porcupine\|Triyoti~~/Porcupine~~]]. 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.		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 ~~intervals~~ ==		== 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]], EI for short, which changes the spelling but not the tuning.		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.

	~~EIs are~~ essential for microtonal interval arithmetic. For example, the 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.		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 EI for ~~Triyoti~~ is ~~vvvA1~~, vvM3 is just another way of spelling ^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 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]]