Golden generator - Revision history

Vector at 07:20, 5 April 2026

2026-04-05T07:20:01Z

← Older revision		Revision as of 07:20, 5 April 2026
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	As a result of this connection to the golden ratio, MOS theory has several connections to the Fibonacci sequence and other sequences generated in a similar way (which will be called golden sequences in this article).		As a result of this connection to the golden ratio, MOS theory has several connections to the Fibonacci sequence and other sequences generated in a similar way (which will be called golden sequences in this article).

	== ~~Soft MOS child~~ ==		== MOSes and golden sequences ==
	The best way to derive a generator tuning with the desired properties is by repeatedly taking the soft child of some MOS. The tuning ranges of the successive daughter MOSes will approach a value, which if used as a generator will generate soft MOSes infinitely.		The best way to derive a generator tuning with the desired properties is by repeatedly taking the soft child of some MOS. The tuning ranges of the successive daughter MOSes will approach a value, which if used as a generator will generate soft MOSes infinitely.

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	This also serves as an intuitive explanation for how the golden ratio is the "most irrational" number.		This also serves as an intuitive explanation for how the golden ratio is the "most irrational" number.

	== Root ~~MOS~~ ==		== Root MOSes ==
	But what happens when you take a hard child? The result has the step sizes flipped from what the "golden child" would be, and therefore becomes a new golden MOS chain with its own corresponding sequence. For example, if you take the Fibonacci sequence, (1, 1, 2, 3...) and stop it at 2 and 3, then flip them, you get (3, 2, 5, 7, 12, 19, 31...). Seem familiar? In fact, a common property of all hard child MOSes is that the number of large steps is smaller than the number of small steps. This means that any golden generator chain may be simply identified by an unordered pair of numbers, and allows us to unambiguously identify the root MOS of any given chain. Additionally, if we take the step sizes of any "root" MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that MOS.		But what happens when you take a hard child? The result has the step sizes flipped from what the "golden child" would be, and therefore becomes a new golden MOS chain with its own corresponding sequence. For example, if you take the Fibonacci sequence, (1, 1, 2, 3...) and stop it at 2 and 3, then flip them, you get (3, 2, 5, 7, 12, 19, 31...). Seem familiar? In fact, a common property of all hard child MOSes is that the number of large steps is smaller than the number of small steps. This means that any golden generator chain may be simply identified by an unordered pair of numbers, and allows us to unambiguously identify the root MOS of any given chain. Additionally, if we take the step sizes of any "root" MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that MOS.
	{\| class="wikitable"		{\| class="wikitable"
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	!MOS		!MOS
	\|-		\|-
	\|Aurora		\|[[Aurora]]
	\|32/21		\|32/21
	\|741.6		\|741.6
	\|1L 1s		\|1L 1s
	\|-		\|-
	\|A-team		\|[[A-team]]
	\|32/21		\|32/21
	\|734.9		\|734.9
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	\|3L 8s		\|3L 8s
	\|-		\|-
	\|Meantone		\|[[Meantone]]
	\|3/2		\|3/2
	\|696.2		\|696.2
	\|2L 3s		\|2L 3s
	\|-		\|-
	\|~~''Neogothic''~~		\|[[Leapday]]
	\| -		\| -
	\|704.1		\|704.1
	\|5L 7s		\|5L 7s
	\|-		\|-
	\|Trismegistus		\|[[Trismegistus]]
	\|28/19		\|28/19
	\|672.9		\|672.9
	\|2L 5s		\|2L 5s
	\|-		\|-
	\|Joan		\|[[Joan]]
	\|16/11		\|16/11
	\|658.6		\|658.6
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	\|1L 2s		\|1L 2s
	\|-		\|-
	\|Orgone		\|[[Orgone]]
	\|77/64		\|77/64
	\|322.3		\|322.3
	\|4L 7s		\|4L 7s
	\|-		\|-
	\|~~Rastmic~~		\|[[Rastmatic]]
	\|11/9		\|11/9
	\|354.8		\|354.8
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	\|3L 7s		\|3L 7s
	\|-		\|-
	\|Submerged		\|[[Submerged]]
	\|5/4		\|5/4
	\|373.1		\|373.1
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	\|1L 3s		\|1L 3s
	\|-		\|-
	\|Semaphore		\|[[Semaphore]]
	\|8/7		\|8/7
	\|254.0		\|254.0
	\|5L 9s		\|5L 9s
	\|-		\|-
	\|Orwell		\|[[Orwell]]
	\|7/6		\|7/6
	\|273.8		\|273.8
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	\|4L 9s		\|4L 9s
	\|-		\|-
	\|Machine		\|[[Machine]]
	\|9/8		\|9/8
	\|213.6		\|213.6
	\|1L 4s		\|1L 4s
	\|-		\|-
	\|Shoe		\|[[Shoe]]
	\|8/7		\|8/7
	\|223.0		\|223.0
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	\|1L 5s		\|1L 5s
	\|-		\|-
	\|Jugular		\|[[Jugular]]
	\|10/9		\|10/9
	\|188.0		\|188.0
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	\|1L 6s		\|1L 6s
	\|-		\|-
	\|Porcupine		\|[[Porcupine]]
	\|11/10		\|11/10
	\|162.6		\|162.6
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	\|1L 7s		\|1L 7s
	\|-		\|-
	\|Negri		\|[[Negri]]
	\|16/15		\|16/15
	\|124.8		\|124.8
	\|1L 8s		\|1L 8s
	\|-		\|-
	\|Miracle		\|[[Miracle]]
	\|16/15		\|16/15
	\|113.0		\|113.0

Vector at 02:51, 11 March 2026

2026-03-11T02:51:28Z

← Older revision		Revision as of 02:51, 11 March 2026
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			''Not to be confused with [[31edo#Scales 2\|MOS scales in 31edo]].''

	A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators (such as the 5-limit with meantone) are difficult to consider in terms of any given [[form]].		A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators (such as the 5-limit with meantone) are difficult to consider in terms of any given [[form]].

Hotcrystal0 at 00:16, 11 March 2026

2026-03-11T00:16:36Z

← Older revision		Revision as of 00:16, 11 March 2026
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	\|-		\|-
	\|0L 1s		\|0L 1s
	\|1, 0, 1, 1, 2, 3, 5, 8, ~~13..~~		\|1, 0, 1, 1, 2, 3, 5, 8, 13…
	\|[[1edo\|1L 0s]], [[1L 1s]], [[2L 1s]], [[3L 2s]], [[5L 3s]], [[8L 5s]], [[13L 8s]]…		\|[[1edo\|1L 0s]], [[1L 1s]], [[2L 1s]], [[3L 2s]], [[5L 3s]], [[8L 5s]], [[13L 8s]]…
	\|458.36, 741.64		\|458.36, 741.64
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	\|-		\|-
	\|1L 2s		\|1L 2s
	\|2, 1, 3, 4, 7, 11, ~~18..~~		\|2, 1, 3, 4, 7, 11, 18…
	\|[[3L 1s]], [[4L 3s]], [[7L 4s]], [[11L 7s]], [[18L 11s]]…		\|[[3L 1s]], [[4L 3s]], [[7L 4s]], [[11L 7s]], [[18L 11s]]…
	\|331.67, 868.33		\|331.67, 868.33

Vector at 04:40, 3 March 2026

2026-03-03T04:40:01Z

← Older revision		Revision as of 04:40, 3 March 2026
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	A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators are difficult to consider in terms of any given [[form]].		A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators (such as the 5-limit with meantone) are difficult to consider in terms of any given [[form]].

	Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.		Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.

Vector at 04:39, 3 March 2026

2026-03-03T04:39:46Z

← Older revision		Revision as of 04:39, 3 March 2026
Line 1:		Line 1:
	A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size.		A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators are difficult to consider in terms of any given [[form]].

	Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.		Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.

Vector at 02:50, 22 February 2026

2026-02-22T02:50:16Z

← Older revision		Revision as of 02:50, 22 February 2026
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	Each scale actually has two silver generators, a hard one and a soft one; as such, silver generators naturally bifurcate in a somewhat similar way to how I've forced golden ones to. Soft silver [[diatonic]] is a flattone tuning.		Each scale actually has two silver generators, a hard one and a soft one; as such, silver generators naturally bifurcate in a somewhat similar way to how I've forced golden ones to. Soft silver [[diatonic]] is a flattone tuning.

			== Example, Golden meantone ==
			[[File:Meantone isomorphic table.png\|thumb\|Golden meantone intervals]]
			Golden meantone is the golden tuning of [[pentic]], being a good tuning of [[meantone]] (specifically with 11/8 as the double-diminished fifth).

Inthar: /* Golden generators for common temperaments */

2026-02-22T02:32:25Z

Golden generators for common temperaments

← Older revision		Revision as of 02:32, 22 February 2026
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	\|-		\|-
	\|Orgone		\|Orgone
	\|		\|77/64
	\|322.3		\|322.3
	\|4L 7s		\|4L 7s

Inthar: /* Golden generators for common temperaments */

2026-02-22T02:32:01Z

Golden generators for common temperaments

← Older revision		Revision as of 02:32, 22 February 2026
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	\|1L 2s		\|1L 2s
	\|-		\|-
	\|		\|Orgone
	\|		\|
	\|322.3		\|322.3

Vector at 02:27, 22 February 2026

2026-02-22T02:27:59Z

← Older revision		Revision as of 02:27, 22 February 2026
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	\|1L 9s		\|1L 9s
	\|}		\|}
			2.3.7 is often exotempered or of very high complexity, this is because of the fact that 2.3.7 just intonation itself functions as a sort of cluster temperament.

	== Argent tuning ==		== Argent tuning ==

Vector at 02:25, 22 February 2026

2026-02-22T02:25:53Z

← Older revision		Revision as of 02:25, 22 February 2026
Line 1:		Line 1:
	A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable as cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size.		A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size.

	Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.		Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.