User:Eliora/Concoctic scale - Revision history

BudjarnLambeth at 01:11, 19 May 2025

2025-05-19T01:11:20Z

← Older revision		Revision as of 01:11, 19 May 2025
Line 1:		Line 1:
			{{Editable user page}}
	A '''concoctic scale''' (name proposed by Eliora) is a [[maximally even]] scale which has the same number of notes as its MOS [[generator]].		A '''concoctic scale''' (name proposed by Eliora) is a [[maximally even]] scale which has the same number of notes as its MOS [[generator]].

BudjarnLambeth: BudjarnLambeth moved page Concoctic scale to User:Eliora/Concoctic scale: Unproven but intriguing idea, covering that type of topic is the aim of user space

2025-05-19T01:11:10Z

BudjarnLambeth moved page Concoctic scale to User:Eliora/Concoctic scale: Unproven but intriguing idea, covering that type of topic is the aim of user space

← Older revision	Revision as of 01:11, 19 May 2025
(No difference)

Lériendil: Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

2025-04-17T02:30:07Z

Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

← Older revision		Revision as of 02:30, 17 April 2025
Line 184:		Line 184:
	[[Godzilla\|81/80 c.II]]		[[Godzilla\|81/80 c.II]]
	\|		\|
	\|Contorted [[Passion]], contorted [[Helmholtz temperament\|Helmholtz]] and [[Godzilla]].		\|Contorted [[Passion]], contorted [[Helmholtz (temperament)\|Helmholtz]] and [[Godzilla]].
	\|-		\|-
	\|25		\|25
Line 224:		Line 224:
	\|[[32805/32768]]		\|[[32805/32768]]
	\|		\|
	\|Forms the [[Helmholtz temperament\|Helmholtz]].		\|Forms the [[Helmholtz (temperament)\|Helmholtz]].
	\|-		\|-
	\|30		\|30

Sintel: fix helmholtz link

2025-04-16T15:37:52Z

fix helmholtz link

← Older revision		Revision as of 15:37, 16 April 2025
Line 224:		Line 224:
	\|[[32805/32768]]		\|[[32805/32768]]
	\|		\|
	\|Forms the [[Helmholtz]].		\|Forms the [[Helmholtz temperament\|Helmholtz]].
	\|-		\|-
	\|30		\|30

Sintel: fix helmholtz link

2025-04-16T15:36:56Z

fix helmholtz link

← Older revision		Revision as of 15:36, 16 April 2025
Line 184:		Line 184:
	[[Godzilla\|81/80 c.II]]		[[Godzilla\|81/80 c.II]]
	\|		\|
	\|Contorted [[Passion]], contorted [[Helmholtz]] and [[Godzilla]].		\|Contorted [[Passion]], contorted [[Helmholtz temperament\|Helmholtz]] and [[Godzilla]].
	\|-		\|-
	\|25		\|25

BudjarnLambeth: /* Concoctic scales in EDOs */ +subklei

2024-09-25T08:40:27Z

Concoctic scales in EDOs: +subklei

← Older revision		Revision as of 08:40, 25 September 2024
Line 151:		Line 151:
	\|[[25/24]] c.II		\|[[25/24]] c.II
	\|		\|
	\|Forms [[Huxley]] and [[~~Lovecraft~~]], but with a fair error.		\|Forms [[Lovecraft]], [[Huxley]] and [[Subklei]], but with a fair error.
	\|-		\|-
	\|20		\|20

BudjarnLambeth: Add links

2024-09-19T08:52:06Z

Add links

← Older revision		Revision as of 08:52, 19 September 2024
Line 91:		Line 91:
	\|16/15		\|16/15
	\|		\|
	\|Forms the Father.		\|Forms the [[Father]].
	\|-		\|-
	\|10		\|10
Line 101:		Line 101:
	\|[[25/24]]		\|[[25/24]]
	\|		\|
	\|Forms the Dicot.		\|Forms the [[Dicot]].
	\|-		\|-
	\|12		\|12
Line 151:		Line 151:
	\|[[25/24]] c.II		\|[[25/24]] c.II
	\|		\|
	\|Forms Huxley and Lovecraft, but with a fair error.		\|Forms [[Huxley]] and [[Lovecraft]], but with a fair error.
	\|-		\|-
	\|20		\|20
Line 184:		Line 184:
	[[Godzilla\|81/80 c.II]]		[[Godzilla\|81/80 c.II]]
	\|		\|
	\|Contorted Passion, contorted Helmholtz and Godzilla.		\|Contorted [[Passion]], contorted [[Helmholtz]] and [[Godzilla]].
	\|-		\|-
	\|25		\|25
Line 224:		Line 224:
	\|[[32805/32768]]		\|[[32805/32768]]
	\|		\|
	\|Forms the Helmholtz.		\|Forms the [[Helmholtz]].
	\|-		\|-
	\|30		\|30
Line 244:		Line 244:
	\|64000/59049		\|64000/59049
	\|		\|
	\|Forms the Satriyo.		\|Forms the [[Satriyo]].
	\|-		\|-
	\|33		\|33
Line 318:		Line 318:
	[[Orson\|[-21, 3, 7⟩]]		[[Orson\|[-21, 3, 7⟩]]
	\|		\|
	\|31\40 forms the [[Orwell]] or Orson.		\|31\40 forms the [[Orwell]] or [[Orson]].
	\|-		\|-
	\|41		\|41
Line 338:		Line 338:
	\|15625/15552 c.IV		\|15625/15552 c.IV
	\|		\|
	\|One step short of 53edo's perfect fifth.		\|One step short of [[53edo]]'s perfect fifth.
	\|-		\|-
	\|55		\|55

Fredg999: Markup

2023-08-26T20:51:02Z

Markup

← Older revision		Revision as of 20:51, 26 August 2023
Line 1:		Line 1:
	~~Concoctic~~ scale (name proposed by Eliora) is a [[~~Maximal evenness\|~~maximally even]] scale which has the same number of notes as its MOS generator.		A '''concoctic scale''' (name proposed by Eliora) is a [[maximally even]] scale which has the same number of notes as its MOS [[generator]].

	12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.		12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.

Inthar at 13:27, 26 August 2023

2023-08-26T13:27:07Z

← Older revision		Revision as of 13:27, 26 August 2023
Line 1:		Line 1:
	Concoctic scale (name proposed by Eliora) is a [[Maximal evenness\|~~maximum eveness~~]] scale which has the same number of notes as its MOS generator.		Concoctic scale (name proposed by Eliora) is a [[Maximal evenness\|maximally even]] scale which has the same number of notes as its MOS generator.

	12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.		12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.

	== Mathematical derivation ==		== Mathematical derivation ==
	The length of a ~~maximum evenness~~ scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.		The length of a maximally even scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.

	<math>ax \equiv 1\mod N</math>,		<math>ax \equiv 1\mod N</math>,
Line 25:		Line 25:
	12edo keyboard layout predominantly in use in the world today features 7 white keys and 5 black keys. In direction-conscious manner, the diatonic scale of 7 keys is obtained by stacking the generator, 7\12 fifth 7 times. Likewise, the pentatonic of black keys is obtained by stacking the 5\12 perfect fourth 5 times. And such scale is generated with the first formula.		12edo keyboard layout predominantly in use in the world today features 7 white keys and 5 black keys. In direction-conscious manner, the diatonic scale of 7 keys is obtained by stacking the generator, 7\12 fifth 7 times. Likewise, the pentatonic of black keys is obtained by stacking the 5\12 perfect fourth 5 times. And such scale is generated with the first formula.

	On the other hand, in [[25edo]], stacking 18\25 will lead to ~~maximum evenness~~ scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.		On the other hand, in [[25edo]], stacking 18\25 will lead to maximally even scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.

	=== Observations ===		=== Observations ===

Eliora: /* Mathematical definition */ added more precise explanation of ortho- and para-

2022-12-12T22:15:13Z

Mathematical definition: added more precise explanation of ortho- and para-

← Older revision		Revision as of 22:15, 12 December 2022
Line 3:		Line 3:
	12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.		12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.

	== Mathematical ~~definition~~ ==		== Mathematical derivation ==
	The length of a maximum evenness scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>:		The length of a maximum evenness scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.

	<math>ax \equiv 1\mod N</math>,		<math>ax \equiv 1\mod N</math>,
Line 16:		Line 16:
	<math>a^2 \equiv 1\mod N \hspace{4cm} (1)</math>.		<math>a^2 \equiv 1\mod N \hspace{4cm} (1)</math>.

	There are also paraconcoctic scales, or chroma-negative concoctic scales. The formula for such a scale is		A scale is called '''orthoconcoctic''', if the generator corresponding to note amount is the chroma-positive generator, for example - the 12edo diatonic scale is. There are also '''paraconcoctic''' scales, or chroma-negative concoctic scales. The formula for such a scale is

	<math>a^2 \equiv -1\mod N \hspace{4cm} (2)</math>.		<math>a^2 \equiv -1\mod N \hspace{4cm} (2)</math>.

	Since octave-inverting the MOS generator has no impact on the scale, paraconcoctic scales are identical to their usual, orthoconcoctic counterparts. However, the difference is pronounced ~~in keyboard making -~~ in terms of ~~chroma direction, the white keys' generator will be the amount of black keys and vice versa~~.		Since octave-inverting the MOS generator has no impact on the scale, paraconcoctic scales are identical to their usual, orthoconcoctic counterparts. However, the difference is pronounced in terms of modal brightness.

	=== Example ===		=== Example ===
Line 131:		Line 131:
	\|[[15625/15552]]*		\|[[15625/15552]]*
	\|		\|
	\|Forms the [[Hanson]] (11b & 15)		\|*Forms the [[Hanson]] (11b & 15)
	\|-		\|-
	\|16		\|16