159edo - Revision history

Aura at 14:46, 30 July 2026

2026-07-30T14:46:00Z

← Older revision		Revision as of 14:46, 30 July 2026
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	The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales\|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.		The interval qualities supported by 159edo are many, however, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo- resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit- the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us [[Semifourth-generated scales\|island]] temperament. Dividing the perfect fourth into three instances of 11/10 gives us [[pine]] temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us [[sextilifourths]] temperament. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.

	On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major ~~sixth~~ into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.		On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. One can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major thirteenth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on.

	== Theory ==		== Theory ==

Aura at 23:51, 7 April 2026

2026-04-07T23:51:30Z

← Older revision		Revision as of 23:51, 7 April 2026
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	=== Ups and downs ===		=== Ups and downs ===
	Ups and downs notation typically represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo. ~~You~~ may also use ups and downs to instead represent a single 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.		Ups and downs notation typically represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo. One may also use ups and downs to instead represent a single 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.

	=== Syntonic-rastmic subchroma notation ===		=== Syntonic-rastmic subchroma notation ===

Aura at 23:48, 7 April 2026

2026-04-07T23:48:01Z

← Older revision		Revision as of 23:48, 7 April 2026
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	=== Ups and downs ===		=== Ups and downs ===
	Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.		Ups and downs notation typically represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo. You may also use ups and downs to instead represent a single 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.

	(You may also use ups and downs to instead represent a 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.)

	=== Syntonic-rastmic subchroma notation ===		=== Syntonic-rastmic subchroma notation ===
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	The Functional Just System and HEJI share the same accidentals for 5 and 11 (though notated with each system's respective symbol); the 11-comma is the artodemisharp and the 5-comma is the synsharp. Neutral FJS uses the standard demisharp and half-rastma as accidentals, meaning that like with SRS notation, the artodemisharp and tendodemisharp become 11-limit accidentals. FJS, Neutral FJS, and HEJI also share an accidental for 7, which in 159edo in particular is equivalent to a rastma plus a syntonic comma.		The Functional Just System and HEJI share the same accidentals for 5 and 11 (though notated with each system's respective symbol); the 11-comma is the artodemisharp and the 5-comma is the synsharp. Neutral FJS uses the standard demisharp and half-rastma as accidentals, meaning that like with SRS notation, the artodemisharp and tendodemisharp become 11-limit accidentals. FJS, Neutral FJS, and HEJI also share an accidental for 7, which in 159edo in particular is equivalent to a rastma plus a syntonic comma.

	The ~~sagittal~~ sequence for 159edo will also be provided.		The Sagittal sequence for 159edo will also be provided.

	The simplest sequence spanning an entire 159edo chromatic semitone is thus as follows:		The simplest sequence spanning an entire 159edo chromatic semitone is thus as follows:

Vector at 20:51, 7 April 2026

2026-04-07T20:51:03Z

← Older revision		Revision as of 20:51, 7 April 2026
Line 222:		Line 222:
	=== Ups and downs ===		=== Ups and downs ===
	Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.		Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.

			(You may also use ups and downs to instead represent a 159edo step, but this makes notating the syntonic/Pythagorean comma inconvenient.)

	=== Syntonic-rastmic subchroma notation ===		=== Syntonic-rastmic subchroma notation ===

Vector at 20:49, 7 April 2026

2026-04-07T20:49:37Z

@@ Line 222: / Line 222: @@
 === Ups and downs ===
 Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.
 === Syntonic-rastmic subchroma notation ===
 Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, nor rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).
-The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
+The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.
 {{Cat|Edos}}

Aura: /* Notation */

2026-04-07T06:45:01Z

Notation

← Older revision		Revision as of 06:45, 7 April 2026
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	== Notation ==		== Notation ==
	There are multiple different ~~tuning~~ systems available for 159edo.		There are multiple different notation systems available for 159edo.

	=== Ups and downs ===		=== Ups and downs ===

Aura: /* Syntonic-rastmic subchroma notation */

2026-04-07T06:30:29Z

Syntonic-rastmic subchroma notation

← Older revision		Revision as of 06:30, 7 April 2026
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	=== Syntonic-rastmic subchroma notation ===		=== Syntonic-rastmic subchroma notation ===
	Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).		Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, nor rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

	The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}		The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Aura: Added a small section for ups and downs notation

2026-04-07T06:29:07Z

Added a small section for ups and downs notation

← Older revision		Revision as of 06:29, 7 April 2026
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	== Notation ==		== Notation ==
	There are multiple different tuning systems available for 159edo.		There are multiple different tuning systems available for 159edo.

			=== Ups and downs ===
			Ups and downs notation represents a single step of 159edo by means of lifts and drops while using the original ups and downs to represent single steps of 53edo.

	=== Sagittal notation ===		=== Sagittal notation ===
	<todo>		<todo>

	=== Syntonic-~~Rastmic Subchroma~~ notation ===		=== Syntonic-rastmic subchroma notation ===
	Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).		Syntonic-rastmic subchroma notation, or SRS for short, is an alternative to Sagittal with fewer accidentals- one which was originally designed for 159edo. In 159edo, the synsharp (81/80) and the rasharp (243/242) are 3\159 (1\53) and 1\159 respectively, treating 159edo as a subdivision of 53edo wherein three rastmas make up a syntonic comma. So far, this is functionally identical to ups and downs notation (assuming the up and down are treated as 1\53), but syntonic-rastmic subchroma notation also supports halving of all of its accidentals. While 159edo divides neither the Pythagorean, syntonic, or rastmic accidental pairs in half, this does allow for the notation of artoneutral and tendoneutral intervals via the inflection of a semisharp by half a rastma (which ends up at a full 159edo step, specifically 7 steps for the artodemisharp and 8 steps for the tendodemisharp).

	The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}		The syntonic and rastmic accidentals represent their just mappings when generalized to other EDOs.{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Aura: /* Sagittal */

2026-04-07T06:13:43Z

Sagittal

← Older revision		Revision as of 06:13, 7 April 2026
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	There are multiple different tuning systems available for 159edo.		There are multiple different tuning systems available for 159edo.

	=== Sagittal ===		=== Sagittal notation ===
	<todo>		<todo>

Aura: /* Notation = */

2026-04-07T06:12:27Z

Notation =

← Older revision		Revision as of 06:12, 7 April 2026
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	159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.		159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detempers of other, smaller tuning systems.

	== Notation ===		== Notation ==
	There are multiple different tuning systems available for 159edo.		There are multiple different tuning systems available for 159edo.