Duodene - Revision history

Eufalesio: Added some things

2025-12-20T20:08:41Z

Added some things

← Older revision		Revision as of 20:08, 20 December 2025
Line 21:		Line 21:

	==== Augmented diesis ====		==== Augmented diesis ====
	If the augmented diesis is tempered out (as in 15edo), the MOS scale [[3L 9s]] is obtained, where the large step represents 27/25 and 135/128, and the small step represents 16/15 and 25/24. This is one possible 12-note chromatic in [[augmented temperament]].		If the augmented diesis is tempered out (as in 15edo), the multi-period MOS scale [[3L 9s]] is obtained, where the large step represents 27/25 and 135/128, and the small step represents 16/15 and 25/24. This is one possible 12-note chromatic in [[augmented temperament]].

	==== Syntonic comma ====		==== Syntonic comma ====
	If the syntonic comma is tempered out (as in 19edo), the MOS scale [[7L 5s]] ~~is obtained~~, where the large step represents 27/25 and 16/15, and the small step represents 135/128 and 25/24. This is the 12-note chromatic of [[meantone temperament]].		If the syntonic comma is tempered out (as in 19edo), the scale becomes reachable through a chain of fifths and becomes the MOS scale [[7L 5s]], where the large step represents 27/25 and 16/15, and the small step represents 135/128 and 25/24. This is the 12-note chromatic of [[meantone temperament]].

	If both ~~chromas~~ are tempered out, the result is ~~12-tone equal temperament~~ (or an enfactoring, like [[24edo]]).		If both are tempered out, the result is [[12edo]] (or an [[enfactoring]], like [[24edo]]).

			==== Schisma ====
			If the schisma is tempered out (as in 53edo), the scale becomes reachable through a chain of fifths and will be contained within the MOS [[12L 17s]], where the large step represents the [[gothic comma]] [dd3] and the small step the [[pythagorean comma]] [-d2]. It can instead be viewed as the MODMOS scale 5L 7s; 2\|9 #1#3#8#10 or 10\|1 b4b6b9b11, where the large step is an augmented unison, the small step a minor second, and the chroma the pythagorean comma.

			Thus, ~27/25 is reached by L+3s [-dd2], ~16/15 by L+2s [A1], ~135/138 by L+s [m2], and ~25/24 by L. The sizes of the steps are equidistant, as the augmented diesis is equated to two syntonic commas.

	== Step pattern ==		== Step pattern ==
	Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis. The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo and 53edo. ~~In [[schismic]] temperament~~ (~~which equates the augmented diesis and two syntonic commas~~)~~, the sizes of the steps are equidistant~~.		Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step L (27/25), 6 medium steps M (16/15), 2 narrow steps n (135/128), and 3 small steps s (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis.

			The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo (s = 2, n = 3, M = 4, L = 5) and 53edo (s = 3, n = 4, M = 5, L = 6). 118edo is optimal (s = 7, n = 9, M = 11, L = 13).

	== Scala file ==		== Scala file ==

VectorGraphics at 21:02, 29 July 2025

2025-07-29T21:02:38Z

← Older revision		Revision as of 21:02, 29 July 2025
Line 29:		Line 29:

	== Step pattern ==		== Step pattern ==
	Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis. The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo and 53edo. In [[schismic]] temperament, the sizes of the steps are equidistant.		Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis. The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo and 53edo. In [[schismic]] temperament (which equates the augmented diesis and two syntonic commas), the sizes of the steps are equidistant.

	== Scala file ==		== Scala file ==

VectorGraphics at 21:01, 29 July 2025

2025-07-29T21:01:40Z

← Older revision		Revision as of 21:01, 29 July 2025
Line 11:		Line 11:
	As a lattice structure, it consists of a chain of three [[3/2\|perfect fifths]] ({{dash\|F, C, G, D}}) with [[5/4\|just major thirds]] above and below each of these.<ref>[http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory</ref>		As a lattice structure, it consists of a chain of three [[3/2\|perfect fifths]] ({{dash\|F, C, G, D}}) with [[5/4\|just major thirds]] above and below each of these.<ref>[http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory</ref>
	When arranged on a standard [[Halberstadt keyboard\|piano keyboard]], the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.		When arranged on a standard [[Halberstadt keyboard\|piano keyboard]], the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.

	~~The duodene can be understood as a [[detempering]] of both [[12edo]] and the [[meantone]] [[7L 5s\|chromatic scale]].~~

	It can be constructed as a [[Fokker block]] with the [[81/80\|syntonic comma]] (81/80) and the [[128/125\|enharmonic diesis]] (128/125) as chromas.		It can be constructed as a [[Fokker block]] with the [[81/80\|syntonic comma]] (81/80) and the [[128/125\|enharmonic diesis]] (128/125) as chromas.
Line 18:		Line 16:

	In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}		In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}

			=== As a detempering ===
			Duodene can be tempered to several scales, which it can itself be understood as a detempering of.

			==== Augmented diesis ====
			If the augmented diesis is tempered out (as in 15edo), the MOS scale [[3L 9s]] is obtained, where the large step represents 27/25 and 135/128, and the small step represents 16/15 and 25/24. This is one possible 12-note chromatic in [[augmented temperament]].

			==== Syntonic comma ====
			If the syntonic comma is tempered out (as in 19edo), the MOS scale [[7L 5s]] is obtained, where the large step represents 27/25 and 16/15, and the small step represents 135/128 and 25/24. This is the 12-note chromatic of [[meantone temperament]].

			If both chromas are tempered out, the result is 12-tone equal temperament (or an enfactoring, like [[24edo]]).

	== Step pattern ==		== Step pattern ==
	Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (25/24). The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo and 53edo.		Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis. The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo and 53edo. In [[schismic]] temperament, the sizes of the steps are equidistant.

	== Scala file ==		== Scala file ==

VectorGraphics at 20:40, 29 July 2025

2025-07-29T20:40:40Z

← Older revision		Revision as of 20:40, 29 July 2025
Line 18:		Line 18:

	In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}		In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}

			== Step pattern ==
			Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (25/24). The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo and 53edo.

	== Scala file ==		== Scala file ==
	<pre>		<pre>! duodene.scl
	! duodene.scl
	!		!
	Ellis's Duodene		Ellis's Duodene
Line 38:		Line 40:
	9/5		9/5
	15/8		15/8
	2/1		2/1</pre>
	</pre>

	== Music ==		== Music ==

Hkm at 18:01, 24 June 2025

2025-06-24T18:01:36Z

← Older revision		Revision as of 18:01, 24 June 2025
Line 53:		Line 53:
	[[Category:12-tone scales]]		[[Category:12-tone scales]]
	[[Category:Just intonation scales]]		[[Category:Just intonation scales]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:5-limit]]		[[Category:5-limit]]
	[[Category:Duodene]]		[[Category:Duodene]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Lériendil: Text replacement - "Category:Just intonation scales" to "Category:Just intonation scales Category:Pages with mostly numerical content"

2025-06-24T00:27:39Z

Text replacement - "Category:Just intonation scales" to "Category:Just intonation scales Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:27, 24 June 2025
Line 53:		Line 53:
	[[Category:12-tone scales]]		[[Category:12-tone scales]]
	[[Category:Just intonation scales]]		[[Category:Just intonation scales]]
			[[Category:Pages with mostly numerical content]]
	[[Category:5-limit]]		[[Category:5-limit]]
	[[Category:Duodene]]		[[Category:Duodene]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Sintel: /* Musical properties */ Typod

2025-04-17T23:56:06Z

Musical properties: Typod

← Older revision		Revision as of 23:56, 17 April 2025
Line 15:		Line 15:

	It can be constructed as a [[Fokker block]] with the [[81/80\|syntonic comma]] (81/80) and the [[128/125\|enharmonic diesis]] (128/125) as chromas.		It can be constructed as a [[Fokker block]] with the [[81/80\|syntonic comma]] (81/80) and the [[128/125\|enharmonic diesis]] (128/125) as chromas.
	It is also an [[Euler-Fokker genus]] of <math>675 = 3^3 \times 5^3</math>, meaning it comprises all divisors of 675, reduced by octave equivalence.		It is also an [[Euler-Fokker genus]] of <math>675 = 3^3 \times 5^2</math>, meaning it comprises all divisors of 675, reduced by octave equivalence.

	In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}		In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}

Sintel: add link to de Caus' book

2025-04-17T11:59:29Z

add link to de Caus' book

← Older revision		Revision as of 11:59, 17 April 2025
Line 4:		Line 4:

	== History ==		== History ==
	While Ellis formalized and named the system, it was first described by French engineer Salomon de Caus in 1615.<ref>Salomon de Caus, ''Les raisons des forces mouvantes avec diverses machines'', Francfort, 1615, Book 3, Problem III</ref>		While Ellis formalized and named the system, it was first described by French engineer Salomon de Caus in 1615.<ref>Salomon de Caus, ''Les raisons des forces mouvantes avec diverses machines'', Francfort, 1615, Book 3, Problem III.<br>Available online at: https://gallica.bnf.fr/ark:/12148/btv1b8626569p/f171.item</ref>
	[[Marin Mersenne]] mentions it in his ''Harmonie universelle (Universal Harmony)'', and among piano tuners, the system is known as "Mersenne's spinet tuning No. 1."<ref>Marin Mersenne, ''Harmonie universelle, contenant la théorie et la pratique de la musique'', Paris, 1636</ref>		[[Marin Mersenne]] mentions it in his ''Harmonie universelle (Universal Harmony)'', and among piano tuners, the system is known as "Mersenne's spinet tuning No. 1."<ref>Marin Mersenne, ''Harmonie universelle, contenant la théorie et la pratique de la musique'', Paris, 1636.</ref>
	The scale is also found in Euler's ''Tentamen novae theoriae musicae (Attempt at a New Theory of Music)'' from 1739.<ref>Leonhard Euler, ''Tentamen novae theoriae musicae'', St. Petersburg, 1739</ref><ref>David J. Benson, ''Music: a mathematical offering'', Cambridge University Press, 2006</ref>		The scale is also found in Euler's ''Tentamen novae theoriae musicae (Attempt at a New Theory of Music)'' from 1739.<ref>Leonhard Euler, ''Tentamen novae theoriae musicae'', St. Petersburg, 1739</ref><ref>David J. Benson, ''Music: a mathematical offering'', Cambridge University Press, 2006</ref>

Sintel: link Halberstadt

2025-04-16T22:49:30Z

link Halberstadt

← Older revision		Revision as of 22:49, 16 April 2025
Line 10:		Line 10:
	== Musical properties ==		== Musical properties ==
	As a lattice structure, it consists of a chain of three [[3/2\|perfect fifths]] ({{dash\|F, C, G, D}}) with [[5/4\|just major thirds]] above and below each of these.<ref>[http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory</ref>		As a lattice structure, it consists of a chain of three [[3/2\|perfect fifths]] ({{dash\|F, C, G, D}}) with [[5/4\|just major thirds]] above and below each of these.<ref>[http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory</ref>
	When arranged on a standard piano keyboard, the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.		When arranged on a standard [[Halberstadt keyboard\|piano keyboard]], the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.

	The duodene can be understood as a [[detempering]] of both [[12edo]] and the [[meantone]] [[7L 5s\|chromatic scale]].		The duodene can be understood as a [[detempering]] of both [[12edo]] and the [[meantone]] [[7L 5s\|chromatic scale]].

Sintel: minor fixes

2025-04-16T22:43:40Z

minor fixes

← Older revision		Revision as of 22:43, 16 April 2025
Line 4:		Line 4:

	== History ==		== History ==
	While Ellis formalized and named the system, ~~the scale~~ was first described by French engineer Salomon de Caus in 1615.<ref>Salomon de Caus, ''Les raisons des forces mouvantes avec diverses machines, Francfort'', 1615, Book 3, Problem III</ref>		While Ellis formalized and named the system, it was first described by French engineer Salomon de Caus in 1615.<ref>Salomon de Caus, ''Les raisons des forces mouvantes avec diverses machines'', Francfort, 1615, Book 3, Problem III</ref>
	[[Marin Mersenne]] mentions it in his ''Harmonie universelle (Universal Harmony)'', and among piano tuners, the system is known as "Mersenne's spinet tuning No. 1."<ref>Marin Mersenne, ''Harmonie universelle, contenant la théorie et la pratique de la musique'', Paris, 1636</ref>		[[Marin Mersenne]] mentions it in his ''Harmonie universelle (Universal Harmony)'', and among piano tuners, the system is known as "Mersenne's spinet tuning No. 1."<ref>Marin Mersenne, ''Harmonie universelle, contenant la théorie et la pratique de la musique'', Paris, 1636</ref>
	The scale is also found in Euler's ''Tentamen novae theoriae musicae (Attempt at a New Theory of Music)'' from 1739.<ref>Leonhard Euler, ''Tentamen novae theoriae musicae'', St. Petersburg, 1739</ref><ref>David J. Benson, ''Music: a mathematical offering'', Cambridge University Press, 2006</ref>		The scale is also found in Euler's ''Tentamen novae theoriae musicae (Attempt at a New Theory of Music)'' from 1739.<ref>Leonhard Euler, ''Tentamen novae theoriae musicae'', St. Petersburg, 1739</ref><ref>David J. Benson, ''Music: a mathematical offering'', Cambridge University Press, 2006</ref>