Armodue harmony - Revision history

FloraC: Fix linking

2025-09-11T11:26:08Z

Fix linking

← Older revision		Revision as of 11:26, 11 September 2025
Line 335:		Line 335:
	A very particular sound in the Armodue environment is achieved by the symmetrical scale with the structure 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 eka or its variant 3 + 1 + 3 + 1 + 3 + 1 + 3 + 1 eka.		A very particular sound in the Armodue environment is achieved by the symmetrical scale with the structure 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 eka or its variant 3 + 1 + 3 + 1 + 3 + 1 + 3 + 1 eka.

	These two scales strongly evoke the sound of the "diminished" scales of the dodecatonic tempered system with their alternating tone-semitone pattern (widely used in jazz and blues): c - c# - d# - e - f# g - a - a#- - c (symmetrical scale of type semitone-tone) or c - d - eb - f - gb - g# - a - h - c (symmetrical scale of type tone-semitone). In Armodue, these scales are named for their special quality "neo-diminished" scales. (Actually, they are scales of [[~~Diminished~~\|diminished ~~temperament~~]], in 12edo as well as in 16edo!)		These two scales strongly evoke the sound of the "diminished" scales of the dodecatonic tempered system with their alternating tone-semitone pattern (widely used in jazz and blues): c - c# - d# - e - f# g - a - a#- - c (symmetrical scale of type semitone-tone) or c - d - eb - f - gb - g# - a - h - c (symmetrical scale of type tone-semitone). In Armodue, these scales are named for their special quality "neo-diminished" scales. (Actually, they are scales of [[diminished (temperament)\|diminished]] temperament, in 12edo as well as in 16edo!)

	{\| class="wikitable"		{\| class="wikitable"

VectorGraphics at 21:31, 7 September 2025

2025-09-07T21:31:50Z

@@ Line 38: / Line 38: @@
 eka - 15 eka (eka, major seventh)
-eka - 14 eka (neutral tone, dominant seventh)
+eka - 14 eka (neutral tone, neutral seventh)
-eka - 13 eka (whole tone, harmonic seventh)
+eka - 13 eka (whole tone, minor seventh)
 eka - 12 eka (minor third, major sixth)
@@ Line 51: / Line 51: @@
 eka (tritone)
 ==1 eka and 15 eka==
@@ Line 57: / Line 125: @@
 Because of this, the eka sounds particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, the "major 8-step", comparable to a slightly enlarged major seventh of the dodecatonic system. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal flavour.
-In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.
+In Armodue the intervals of 1 and 15 eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.
 ==2 eka and 14 eka==

VectorGraphics at 21:19, 7 September 2025

2025-09-07T21:19:06Z

← Older revision		Revision as of 21:19, 7 September 2025
Line 120:		Line 120:
	Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.		Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.

	When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval. This plays interestingly with the basic 2.5.7 chord of 4:5:7, as no matter what, any inversion of the chord will contain this unstable interval due to the [[jubilic]] temperament. Instead, it suggests that some other chord avoiding 7:5 might be a more harmonically stable base for Armodue theory, such as dyads of 4:5 and 4:7 played independently, perhaps with the flat fifth as a supporting interval.		When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval. This plays interestingly with the basic 2.5.7 chord of 4:5:7, as no matter what, any inversion of the chord will contain this unstable interval due to the [[jubilic]] temperament. Instead, it suggests that some other chord avoiding 7:5 might be a more harmonically stable base for Armodue theory, such as dyads of 4:5 and 4:7 played independently, perhaps with the flat fifth as a supporting interval to introduce an open consonance.

	The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.		The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.

VectorGraphics at 21:18, 7 September 2025

2025-09-07T21:18:45Z

← Older revision		Revision as of 21:18, 7 September 2025
Line 12:		Line 12:
	=Reasons why one would want to use 16edo=		=Reasons why one would want to use 16edo=

	==16edo as ~~based on the 5th and 7th harmonic~~==		==16edo as a 2.(3).5.7 system vs. 12edo as a 2.3.5(.7) system==
	The 12-form, which has been standard for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth [[3/2]]) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.		The 12-form, which has been standard for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth [[3/2]]) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.

Line 120:		Line 120:
	Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.		Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.

	When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval.		When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval. This plays interestingly with the basic 2.5.7 chord of 4:5:7, as no matter what, any inversion of the chord will contain this unstable interval due to the [[jubilic]] temperament. Instead, it suggests that some other chord avoiding 7:5 might be a more harmonically stable base for Armodue theory, such as dyads of 4:5 and 4:7 played independently, perhaps with the flat fifth as a supporting interval.

	The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.		The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.

VectorGraphics at 00:29, 19 March 2025

2025-03-19T00:29:16Z

← Older revision		Revision as of 00:29, 19 March 2025
Line 1:		Line 1:
	<span style="font-size: 140%;">Armodue: basic elements of harmony</span>		<span style="font-size: 140%;">Armodue: basic elements of harmony</span>

	This is a translation of an article by Luca Attanasio, with added contributions by Xenharmonic Wiki contributors. Original page in italian: [http://www.armodue.com/armonia.htm http://www.armodue.com/armonia.htm]		This is a translation of an article by Luca Attanasio, with added contributions by Xenharmonic Wiki contributors. Original page in italian: [http://www.armodue.com/armonia.htm http://www.armodue.com/armonia.htm]. The original archived version on the wiki can be found at [[Armodue harmony/Archived version]].

	__FORCETOC__		__FORCETOC__

VectorGraphics: added 0-indexed armotonic step names (save the 1-indexed versions for Earth16 theory)

2025-03-19T00:27:24Z

added 0-indexed armotonic step names (save the 1-indexed versions for Earth16 theory)

Show changes

VectorGraphics at 10:38, 17 March 2025

2025-03-17T10:38:52Z

← Older revision		Revision as of 10:38, 17 March 2025
Line 1:		Line 1:
	<span style="font-size: 140%;">Armodue: basic elements of harmony</span>		<span style="font-size: 140%;">Armodue: basic elements of harmony</span>

			This is a translation of an article by Luca Attanasio, with added contributions by Xenharmonic Wiki contributors. Original page in italian: [http://www.armodue.com/armonia.htm http://www.armodue.com/armonia.htm]

	__FORCETOC__		__FORCETOC__
	~~This is a translation of an article by Luca Attanasio. Original page in italian: [http://www.armodue.com/armonia.htm http://www.armodue.com/armonia.htm]~~

	For terminology see the [[Armodue_theory\|Armodue overview page]].		For terminology see the [[Armodue_theory\|Armodue overview page]].
Line 8:		Line 10:
	Chapter 1:		Chapter 1:

	=~~Two theses supporting the system~~=		=Reasons why one would want to use 16edo=

	==The supremacy of the fifth and the seventh harmonic in Armodue==		==Harmonic standpoint: The supremacy of the fifth and the seventh harmonic in Armodue==
	The ~~twelve note system that~~ has been ~~ruling~~ for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.		The 12-form, which has been standard for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth [[3/2]]) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.

	~~But, if in~~ 12edo, the ~~pitch of~~ the third harmonic - hence the [[4/3\|perfect fourth]] and the [[perfect fifth]] - ~~are almost perfectly respected~~ (the tempered fourth and fifth differ by only 2 cents from JI), this cannot be said about the odd harmonics (even harmonics are not counted because they are octave-reducible to odd harmonics) immediately above the third one: the fifth and seventh harmonic.		In 12edo, the the third harmonic - hence the [[4/3\|perfect fourth]] and the [[perfect fifth]] - is extremely accurate (the tempered fourth and fifth differ by only 2 cents from JI), but this cannot be said about the odd harmonics (even harmonics are not counted because they are octave-reducible to odd harmonics) immediately above the third one: the fifth and seventh harmonic.

	In the tempered system the fifth and the seventh harmonic appear as [[major third]] and [[minor seventh]] intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the ~~sizes~~ of the ~~tempered major third and~~ minor seventh ~~do not match the sizes~~ of ~~the respective natural intervals~~.		In the tempered system the fifth and the seventh harmonic appear as [[major third]] and [[minor seventh]] intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity). However, while the major third interval is decently well-represented (but not sufficient in the eyes of some xenharmonicists), the minor seventh is "out of tune" with 7/4 enough to have a distinctly different sound.

	In Armodue (see [[16edo\|16edo]]), in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.		In Armodue (see [[16edo\|16edo]]), in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.

	For this reason, especially important in Armodue are the intervals of 5\16edo (corresponding to the interval ratio 5/4) and 13\16edo (corresponding to the interval ratio 7/4). The circle of fifths which is the base of 12edo is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4\|seventh]] harmonic.		This can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error: by choosing a tuning with greater error on lower primes as opposed to higher ones, one can create a much more consistent feeling than if the highest errors are on higher primes. In essence, 16edo's 3/1, 5/1, and 7/1 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.

			For this reason, especially important in Armodue are the intervals of 5\16edo (corresponding to the interval ratio 5/4) and 13\16edo (corresponding to the interval ratio 7/4). The circle of fifths which is the base of 12edo is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka (or equivalently, 3 eka), emphasizing the priority of the fifth and the [[7/4\|seventh]] harmonic. Notably, neither of these intervals close the circle early like 12edo's major third does, nor do they reach each other in a small number of steps like 12edo's perfect fifth does for its major third, meaning that 16edo can be seen as a more "pure" 2.5.7 system than 12edo as a 2.3.5 system.

	==The triple mean of the double diagonal / side of the square==		==Philosophical standpoint: The triple mean of the double diagonal / side of the square==
	From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).		From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).

Line 36:		Line 40:
	Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the Tenth (interval sum of 16 eka)) in eight categories that will be to be analysed individually:		Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the Tenth (interval sum of 16 eka)) in eight categories that will be to be analysed individually:

	1 eka - 15 eka		1 eka - 15 eka (semitone, major seventh)

	2 eka - 14 eka		2 eka - 14 eka (neutral tone, dominant seventh)

	3 eka - 13 eka		3 eka - 13 eka (whole tone, harmonic seventh)

	4 eka - 12 eka		4 eka - 12 eka (minor third, major sixth)

	5 eka - 11 eka		5 eka - 11 eka (major third, minor sixth)

	6 eka - 10 eka		6 eka - 10 eka (subfourth, superfifth)

	7 eka - 9 eka		7 eka - 9 eka (perfect fourth, perfect fifth)

	8 eka		8 eka (tritone)

	==1 eka and 15 eka==		==1 eka and 15 eka==
Line 109:		Line 113:
	The intervals corresponding to the perfect fourth and and the perfect fifth in Armodue are the intervals of 7 and 9 eka, the first one quantifies in a slightly sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents).		The intervals corresponding to the perfect fourth and and the perfect fifth in Armodue are the intervals of 7 and 9 eka, the first one quantifies in a slightly sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents).

	These Armodue intervals, however, are incompatible with the concept of the cycle of the fifth that is on the base of the dodecatonic system. Especially with the intervals of 7 and 9 eka, Armodue shows an entirely new and different system, compared to the ruling dodecatonic system.		These Armodue intervals, however, are incompatible with the concept of the cycle of the fifth that is on the base of the dodecatonic system. Especially with the intervals of 7 and 9 eka, Armodue shows an entirely new and different system, compared to the ruling dodecatonic system. Specifically, these intervals generate the antidiatonic and then armotonic scale, with the latter being the main scale used in Armodue theory. If they are incorrectly treated as dodecatonic fifths, it leads to absurdities such as negative semitones and minor thirds being larger than major thirds!

	The Armodue "fourth" and "fifth" are euphonious and absolutely consonant, but in a radically different range from those of the dodecatonic system. It is suggested to use the intervals of 7 and 9 eka with caution, just to highlight the special significance that the "fourth" and "fifth" acquire in Armodue and not mislead the western ear deeply accustomed to the cycle of fifths.		The Armodue "fourth" and "fifth" are euphonious and absolutely consonant, but in a radically different range from those of the dodecatonic system. It is suggested to use the intervals of 7 and 9 eka with caution, just to highlight the special significance that the "fourth" and "fifth" acquire in Armodue and not mislead the western ear deeply accustomed to the cycle of fifths.
Line 116:		Line 120:

	==8 eka==		==8 eka==
	The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth.		The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical octave; in fact, the interval is the same as the 12edo tritone. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth.

	Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.		Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.

VectorGraphics: Work on page cleanup to use xenwiki terminology

2025-02-23T19:42:50Z

Work on page cleanup to use xenwiki terminology

VectorGraphics: VectorGraphics moved page Armodue armonia to Armodue harmony: this is the English Xenwiki

2025-02-23T19:36:21Z

VectorGraphics moved page Armodue armonia to Armodue harmony: this is the English Xenwiki

← Older revision	Revision as of 19:36, 23 February 2025
(No difference)

Lériendil: deploying new cat

2025-01-31T22:19:06Z

deploying new cat

← Older revision		Revision as of 22:19, 31 January 2025
Line 401:		Line 401:
	There arise many effective progressions of chords varied by the selection of one or two "elastic" intervals.		There arise many effective progressions of chords varied by the selection of one or two "elastic" intervals.
	[[Category:16edo]]		[[Category:16edo]]
	[[Category:~~armodue~~]]		[[Category:Armodue]]
			[[Category:Method]]
			[[Category:Approaches to tuning systems]]

← Older revision		Revision as of 19:42, 23 February 2025
Line 11:		Line 11:

	==The supremacy of the fifth and the seventh harmonic in Armodue==		==The supremacy of the fifth and the seventh harmonic in Armodue==
	The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the ~~pythagorean tradition - the cycle~~ of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.		The twelve note system that has been ruling for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.

	But, if in ~~the twelve note system~~ the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only ~~of the fiftieth part of a semitone~~ from ~~the natural fifth and fourth~~), this cannot be said about the odd harmonics (even harmonics are not counted because they are ~~simply duplicates in the~~ octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.		But, if in 12edo, the pitch of the third harmonic - hence the [[4/3\|perfect fourth]] and the [[perfect fifth]] - are almost perfectly respected (the tempered fourth and fifth differ by only 2 cents from JI), this cannot be said about the odd harmonics (even harmonics are not counted because they are octave-reducible to odd harmonics) immediately above the third one: the fifth and seventh harmonic.

	In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.		In the tempered system the fifth and the seventh harmonic appear as [[major third]] and [[minor seventh]] intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.

	In Armodue (see [[16edo\|16edo]]), in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.		In Armodue (see [[16edo\|16edo]]), in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.

	For this reason, especially important in Armodue are the intervals of ~~five eka~~ (corresponding to the interval ratio ~~given by the fifth harmonic~~) and ~~the interval of~~ 13 ~~eka~~ (corresponding to the interval ratio ~~subsisting with the seventh harmonic~~). The circle of fifths which is the base of ~~the dodecatonic system~~ is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4\|seventh]] harmonic ~~(7/4 Ratio)~~.		For this reason, especially important in Armodue are the intervals of 5\16edo (corresponding to the interval ratio 5/4) and 13\16edo (corresponding to the interval ratio 7/4). The circle of fifths which is the base of 12edo is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4\|seventh]] harmonic.

	==The triple mean of the double diagonal / side of the square==		==The triple mean of the double diagonal / side of the square==
	From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).		From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).

	Analogously, the philosophical foundation of Armodue and ~~esadecafonia~~ can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.		Analogously, the philosophical foundation of Armodue and 16edo can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.

	The arithmetic mean is exactly ~~the interval of nine eka~~, the geometric mean exactly ~~twelve eka~~ and finally the harmonic mean exactly equal to ~~fifteen eka~~.		The arithmetic mean is exactly 9\16edo, the geometric mean exactly 12\16edo and finally the harmonic mean exactly equal to 15\16edo.

	-----		-----
Line 53:		Line 53:

	==1 eka and 15 eka==		==1 eka and 15 eka==
	The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.		The interval of one eka, or 1\16edo, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.

	This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.		This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.