Xenharmonic Wiki

Armodue theory: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>FREEZE
No edit summary
No edit summary
 
(10 intermediate revisions by 6 users not shown)
Line 1: Line 1:
__FORCETOC__
''(summary translation from the Italian language site'' [http://www.armodue.com/ricerche.htm Armodue] '')''
-----


=Armodue Theory of 16EDO=
Armodue theory is a xenharmonic notation and theory system for 16-note tuning, not only in the sense of [[16edo|16-tone equal temperament]], but also regarding half-equal tuning, [[Lou_Harrison|Lou Harrison]]'s [[JustIntonation|Just intonation]] 16 note scale, the natural octave division by [http://www.pertout.com/ Andrián Pertout], and the 16-to-31 [[OverToneSeries|overtone scale]].


''(summary translation from the italian site'' [http://www.armodue.com/ricerche.htm Armodue] )
== Notation ==
In order to make Armodue approachable, while also making its notation distinct from Western diatonic notation, the Italian creators of the [http://armodue.com/ Armodue] system named the notes using numbers from 1 to 9:


Referring not only to the [[16edo|16-edo equal temperament]], but also to half-equal and [[Lou_Harrison|Lou Harrison]]'s [[JustIntonation|Just intonation]] 16 note scale, the natural octave division by <span style="font-family: Verdana; font-size: small; line-height: normal;">[http://www.pertout.com/ Andrián Pertout] and the 16-to-31 [[OverToneSeries|overtone scale]], </span>Armodue has been proposed as a new notation and theory system.
1, 1#, 2, 2#, 3, 3#, 4, 5, 5#, 6, 6#, 7, 7#, 8, 8#, 9


Desiring to make the approach to Armodue as easy as possible, but conscious that they had to give new names to the notes that constitute the system, the Italian creators of the <span style="background-position: 100% 50%; cursor: pointer; padding-right: 10px;">[http://armodue.com/ Armodue]</span> system named them numbering from 1 to 9:
This is the recommended system for Armodue theory, though in technicality any notation system can be used.


1, 1#, 2, 2#, 3, 3#, 4, 5, 5#, 6, 6#, 7, 7#, 8, 8#, 9
Relative notation is simply done by specifying the number of edosteps, in order to support alternative scale constructions. For example, one refers not to the perfect fourth or fifth (at least in the specific sense of Armodue intervals) but to the intervals of 7-eka and 9-eka.
 
However, the interval between a note at frequency n and other at frequency 2n is called a ''tenth,'' ''decave,'' or ''decim'', as it is the tenth note of the Mavila[9] scale.


Consequently, the interval between a note at frequency n and other at frequency 2n is called a ''tenth'' or ''decave''.
The basic semitone of Armodue, whatever concrete temperament is used, is always called '''eka''' (from Sanskrit eka: one, unit). In the chromatic Armodue scale, one eka always corresponds to the interval between any two consecutive notes.


The basic semitone of Armodue, whatever concrete temperament is used, is always called ''eka'' (from Sanskrit eka: one, unit). In the chromatic Armodue scale, one eka always corresponds to the interval between any two consecutive notes.
The standard staff notation for Armodue uses 4 lines, so that 1 is always found below the first line, and 9 above the last line.  


For composing in Armodue it's useful to use a ''tetragram'' (staff with 4 lines)
[[file:armodue-TETR-1.jpg|frame|none|&copy; Armodue; used with permission.]]


{| class="wikitable"
|-
| |
{| class="wikitable"
|-
| |


<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
The notes without accidentals form a [[Mavila|Mavila]] [[7L_2s|armotonic scale (7L 2s)]].
External image: http://www.armodue.com/TETR-%5B1%5D.jpg<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
|-
| | copyright Armodue, used with permission
|}
|-
| | copyright Armodue, used with permission
|}


The notes without accidentals form a [[Mavila|Mavila]] [[7L_2s|superdiatonic scale (7L 2s)]].
This notation is similar to the KISS notation using numbers for this scale, except with the staff laid out differently (i.e. in KISS, the standard treble and bass clefs are used, and the staff has 6 lines instead of 4)


If for the execution of a musical piece we need to write on two or more tetragrams, the notes will be written in the same way for every tetragram.
When multiple staves are necessary, the notes will be written in the same way in every staff.


In other words, the "1" note will be written immediately under the first line <u>in every tenth</u>.
In other words, the "1" note will be written immediately under the first line <u>in every tenth</u>.


In Armodue we have only a numeric clef, that show us the tenth:
In Armodue we have only a numeric clef, that shows us the tenth:


{| class="wikitable"
[[file:armodue-Chiave.gif|frame|none|&copy; Armodue; used with permission.]]
|-
| |
{| class="wikitable"
|-
| |  


<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
External image: http://www.armodue.com/Chiave.gif<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
|-
| | copyright Armodue, used with permission
|}
|-
| | copyright Armodue, used with permission
|}


The clefs 1,2,3... refers to the tenths: first, second, third...
The clefs 1,2,3... refers to the tenths: first, second, third...


So, in the illustrated example above, the first tetragram (from top) refers to the 3rd tenth (central tenth, corresponding to the octave C3-C4),
So, in the illustrated example above, the first staff (from top) refers to the 3rd tenth (central tenth, corresponding to the octave C3-C4), the second tetragram to the 5th tenth and the third to the 2nd. If we need to write simultaneously on several staves, we will draw normal braces.
 
Note that you cannot connect staves like in standard notation, because the lines are offset from one decave to the next.


the second tetragram to the 5th tenth and the third to the 2nd. If we need to write simultaneously on several staves, we will draws normal braces.
This is a keyboard layout for Armodue, which can also be seen as a standard mavila keyboard layout for 16edo.  


The keyboard conceived by the Armodue authors has the same disposition as Goldsmith's one (except the curvature):
[[file:armodue-Tastiera.jpg|frame|none|&copy; Armodue; used with permission.]]


{| class="wikitable"
|-
| |
{| class="wikitable"
|-
| |


<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
External image: http://www.armodue.com/Tastiera.jpg<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]]
|-
| | copyright Armodue, used by permission
|}
|-
| | copyright Armodue, used by permission
|}
The white keys, corresponding to the notes without accidentals in the notation system, form again a [[Mavila|Mavila]] [[7L_2s|superdiatonic scale (7L 2s).]]
The white keys, corresponding to the notes without accidentals in the notation system, form again a [[Mavila|Mavila]] [[7L_2s|superdiatonic scale (7L 2s).]]


{| class="wikitable"
[[file:Armodue_Neck.PNG|frame|none|Dot-Pattern inlays, by Armodue]]
|-
| | [[File:Armodue_Neck.PNG|alt=Armodue_Neck.PNG|640x126px|Armodue_Neck.PNG]]
|-
| | Dot-Pattern inlays, by Armodue
|}
 
=Armodue just intonation - connection to Lou Harrison and Andrián Pertout=
 
Armodue Just Intonation is nothing else than the 16-note scale by[[Lou_Harrison| Lou Harrison]] based on simple ratios and pure intervals. In this scale there are practically the same twelve intervals of the "natural" or "Zarlinian" scale (the semitone 16/15, the minor wholetone 10/9, the minor third 6/5, the major third 5/4, the pure forth and fifth 4/3 and 3/2, the minor sixth 8/5, the major sixth 5/3, the minor seventh that is the complement of the minor wholetone 9/5, the major seventh 15/8) in addtion to the tritone and four other intervals that are based on the harmonic seventh (the ratios 8/7, 7/6, 12/7, 7/4).
 
The composer Andrián Pertout used a very similar scale in his composition "Sonus dulcis" ([http://web.archive.org/web/20040511172650/http://www.users.bigpond.com/apertout/Sonus.htm article on archive.org]), namely the following (according to [http://www.armodue.com/risorse.htm Armodue]):
 
{| class="wikitable"
|-
! | Armodue note
! | Interval
! | Ratio
! | Cents
|-
| | 1
| | unison
| | 1/1
| | 0
|-
| | 1#
| | major half-tone
| | 16/15
| | 112
|-
| | 2
| | minor tone
| | 10/9
| | 182
|-
| | 2#
| | major tone
| | 9/8
| | 204
|-
| | 3
| | minor third
| | 6/5
| | 316
|-
| | 3#
| | major third
| | 5/4
| | 386
|-
| | 4
| | fourth
| | 4/3
| | 498
|-
| | 5
| | harmonic tritone
| | 45/32
| | 590
|-
| | 5#
| | cyclic tritone
| | 64/45
| | 610
|-
| | 6
| | fifth
| | 3/2
| | 702
|-
| | 6#
| | diminished sixth
| | 8/5
| | 814
|-
| | 7
| | harmonic sixth
| | 5/3
| | 884
|-
| | 7#
| | harmonic minor seventh
| | 7/4
| | 969
|-
| | 8
| | minor seventh
| | 16/9
| | 996
|-
| | 8#
| | minor seventh
| | 9/5
| | 1018
|-
| | 9
| | major seventh
| | 15/8
| | 1088
|}
 
=Semi-equalized Armodue=
 
One step of 16edo (75 cents) is nearly equal to two steps (2\31) of [[31edo|31edo]] (77.42 cents). If we take the latter as a base, we get semi-equalized Armodue. In this temperament there is inevitably a smaller microtone (eka between the notes '7#' and '8'). leading to the 16 note MOS Valentine[16] of [[Starling_temperaments#Valentine temperament|valentine temperament]]. Similarly we might use three steps of [[46edo|46edo]], 3\46, 78.26 cents, or five steps of [[77edo|77edo]], 77.92 cents.
 
Semi-equalized Armodue provides a balance between the symmetry of the equalized system and the purity of natural intervals: intervals of semi-equalized Armodue are very pure, and at the same time it preserves the symmetry of the equalized system and its interval sizes almost unchanged.
 
{| class="wikitable"
|-
! | Armodue note
! | cents (16edo)
! | cents (semi-equalized Armodue based on [[31edo|31edo]])
|-
| | 1
| | 0
| | 0
|-
| | 1#
| | 75
| | 77.42
|-
| | 2
| | 150
| | 154.84
|-
| | 2#
| | 225
| | 232.26
|-
| | 3
| | 300
| | 309.68
|-
| | 3#
| | 375
| | 387.10
|-
| | 4
| | 450
| | 464.52
|-
| | 5
| | 525
| | 541.94
|-
| | 5#
| | 600
| | 619.35
|-
| | 6
| | 675
| | 696.77
|-
| | 6#
| | 750
| | 774.19
|-
| | 7
| | 825
| | 851.61
|-
| | 7#
| | 900
| | 929.03
|-
| | 8
| | 975
| | 967.74
|-
| | 8#
| | 1050
| | 1045.16
|-
| | 9
| | 1125
| | 1122.58
|}
[[Category:16edo]]
[[Category:16edo]]
[[Category:armodue]]
[[Category:Armodue]]
[[Category:notation]]
[[Category:Notation]]
[[Category:theory]]
[[Category:Valentine]]
[[Category:valentine]]
[[Category:Method]]
[[Category:Approaches to tuning systems]]
Retrieved from "https://en.xen.wiki/w/Armodue_theory"