Armodue harmony: Difference between revisions

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="font-size: 140%;"&gt;Armodue: basic elements of harmony&lt;/span&gt;

[[toc]]

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

8 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.

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.

The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) into eight equal parts, to form the [[8edo|8-equal tempered]] scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.

The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue.

The interval of 3 eka corresponds to the "wholetone" of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the eighth and ninth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.

The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.

With the intervals of 4 and 12 eka we have two intervals that are very popular and familiar to the ear translated into Armodue. Indeed 4 eka and 12 eka correspond exactly to the minor third and the major sixth of the dodecatonic system. Therefore, there is the evident possibility of evoking major and minor triads with Armodue (the minor triad is created in stacking 4 eka and 5 eka on a base tone, the major triad stacking 5 eka and 4 eka). The perfect equivalence of the two considered intervals in the dodecatonic system and Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. In fact, listening to the intervals of 4 and 12 eka, the ear will immediately associate these Armodue intervals to two already familiar ones (the minor third and the major sixth). For this reason, many of the other intervals present - in an Armodue environment - in a context to those of 4 and 12 eka are likely to be felt by the ear as abnormal and unknown. The composers will give much attention every time they use one of these two intervals, trying to predict the reactions of an ear used to the dodecatonic system.

The intervals of 4 and 12 eka belong without doubt - being of equivalent size - to the same category that includes the minor third and the major sixth: that of sweet consonances.

The interval of 5 eka is close to the natural major third that appears as the ratio between the fourth and fifth harmonic of the overtone series. The major third of the dodecatonic system (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents).

The complement of 5 eka is the interval of 11 eka, very close to the natural minor sixth that occurs between the fifth and eighth harmonic. Since the intervals of 5 and 11 eka are associated to the tempered third and sixth, the same considerations hold that were made in the previous paragraph about the intervals of 4 and 12 eka. They too are classified as sweet consonances.

The interval of 6 eka is particularly striking, with its complement of 10 eka. 6 eka is located exactly at the point of equidistance between the major third and the perfect fourth of the tempered system, while 10 eka is the average between a perfect fifth and a minor sixth.

The intervals of 6 and 10 eka are classified as neutral consonances.

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).

As for the classification, 7 and 9 eka belong to the open consonances.

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.

&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#The interval table"&gt;The interval table&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#The interval table-Qualitative categories of intervals"&gt;Qualitative categories of intervals&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#The interval table-Gradation of harmonic tensions"&gt;Gradation of harmonic tensions&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#The interval table-Arrangement of tones in the chords"&gt;Arrangement of tones in the chords&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x&amp;quot;Elastic&amp;quot; chords"&gt;&amp;quot;Elastic&amp;quot; chords&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;/div&gt;

&lt;br /&gt;

For terminology see the &lt;a class="wiki_link" href="/Armodue%20theory"&gt;Armodue overview page&lt;/a&gt;.&lt;br /&gt;

8 eka&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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.&lt;br /&gt;

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.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) into eight equal parts, to form the &lt;a class="wiki_link" href="/8edo"&gt;8-equal tempered&lt;/a&gt; scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.&lt;br /&gt;

The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The interval of 3 eka corresponds to the &amp;quot;wholetone&amp;quot; of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the eighth and ninth). If you build scales using successions of the &amp;quot;wholetone&amp;quot; of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.&lt;br /&gt;

The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

With the intervals of 4 and 12 eka we have two intervals that are very popular and familiar to the ear translated into Armodue. Indeed 4 eka and 12 eka correspond exactly to the minor third and the major sixth of the dodecatonic system. Therefore, there is the evident possibility of evoking major and minor triads with Armodue (the minor triad is created in stacking 4 eka and 5 eka on a base tone, the major triad stacking 5 eka and 4 eka). The perfect equivalence of the two considered intervals in the dodecatonic system and Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. In fact, listening to the intervals of 4 and 12 eka, the ear will immediately associate these Armodue intervals to two already familiar ones (the minor third and the major sixth). For this reason, many of the other intervals present - in an Armodue environment - in a context to those of 4 and 12 eka are likely to be felt by the ear as abnormal and unknown. The composers will give much attention every time they use one of these two intervals, trying to predict the reactions of an ear used to the dodecatonic system.&lt;br /&gt;

The intervals of 4 and 12 eka belong without doubt - being of equivalent size - to the same category that includes the minor third and the major sixth: that of sweet consonances.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The interval of 5 eka is close to the natural major third that appears as the ratio between the fourth and fifth harmonic of the overtone series. The major third of the dodecatonic system (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents).&lt;br /&gt;

The complement of 5 eka is the interval of 11 eka, very close to the natural minor sixth that occurs between the fifth and eighth harmonic. Since the intervals of 5 and 11 eka are associated to the tempered third and sixth, the same considerations hold that were made in the previous paragraph about the intervals of 4 and 12 eka. They too are classified as sweet consonances.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The interval of 6 eka is particularly striking, with its complement of 10 eka. 6 eka is located exactly at the point of equidistance between the major third and the perfect fourth of the tempered system, while 10 eka is the average between a perfect fifth and a minor sixth.&lt;br /&gt;

The intervals of 6 and 10 eka are classified as neutral consonances.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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).&lt;br /&gt;

As for the classification, 7 and 9 eka belong to the open consonances.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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.&lt;br /&gt;