Armodue harmony: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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==The supremacy of the fifth and and the seventh harmonic in Armodue== | ==The supremacy of the fifth and 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 tradion - the | 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 tradion - the cycle 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 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. | ||
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 | 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, 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, 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. | ||
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The particularity is to make the last note of one module coincide with the first note of the next group. | The particularity is to make the last note of one module coincide with the first note of the next group. | ||
For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes: | For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes (cylce of 6 eka iterated 8 times): | ||
**c** - d - eb - **e** - f# - g - **ab** - bb - b - **c** | **c** - d - eb - **e** - f# - g - **ab** - bb - b - **c** | ||
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In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts. | In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts. | ||
But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka. | But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka. | ||
If, for example, we take the interval of 6 eka (2 tones plus a quartertone in the 12edo system) we obtain the following sequence of notes: | |||
1, 4, 7#, 2, 5#, 8#, 3, 6#, 1 | 1, 4, 7#, 2, 5#, 8#, 3, 6#, 1 | ||
As we can ascertain, | As we can ascertain, over the range of three tenths (equivalent to three 12-edo octaves) the circle closes, and from the starting note 1 we arrive again at note 1 three tenths higher, by successive steps of 6 eka. | ||
In strictly | |||
Every interval of | In strictly mathematical terms, the least common multiple of 16 eka (the size of the tenth) and 6 eka (the iterated interval) is 48 eka (corresponding to the size of three tenths). | ||
Every interval of 6 eka - iterated eight times in the three tenths range - can be organized inside in various ways; for example in modules of 3 + 2 + 1 eka, in which case we obtain the following scalar system of 24 notes: | |||
1, 2#, 3#, 4; (4), 6, 7, 7#; (7#), 9, 1#, 2; (2), 3#, 5, 5#; (5#), 7, 8, 8#; (8#), 1#, 2#, 3; (3), 5, 6, 6#; (6#), 8, 9, 1. | 1, 2#, 3#, 4; (4), 6, 7, 7#; (7#), 9, 1#, 2; (2), 3#, 5, 5#; (5#), 7, 8, 8#; (8#), 1#, 2#, 3; (3), 5, 6, 6#; (6#), 8, 9, 1. | ||
[note: the repeated | [note: the repeated notes in brackets are meant to highlight how the first and the last note of every group coincide.] | ||
All the sixteen notes appear at least once inside the scale shown. | |||
So, it's essential to remember the context of membership of the notes in respect to the three tenths. | |||
So, it's | Supposing - for simplicity - to play in a three tenths register, we'll have eight notes per tenth at our disposal for the creation of the chord, melody and counterpoint texture. | ||
Supposing - for | |||
Specifically, the notes: | |||
1, 2#, 3#, 4, 6, 7, 7#, 9 in the first tenth, the lowest; | 1, 2#, 3#, 4, 6, 7, 7#, 9 in the first tenth, the lowest; | ||
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1#, 2#, 3, 5, 6, 6#, 8, 9 in the third tenth, the highest. | 1#, 2#, 3, 5, 6, 6#, 8, 9 in the third tenth, the highest. | ||
Note that if we | Note that if we iterate the original defining interval of 6 eka starting from note 1# instead of 1, we obtain the eight pivot notes missing in the sequence that starts from note 1: | ||
1#, 5, 8, 2#, 6, 9, 3#, 7, 1# | 1#, 5, 8, 2#, 6, 9, 3#, 7, 1# | ||
To the notes found that way we can apply the same division according to the 3-2-1 eka module, obtaining the same 24 notes scale, but raised 1 eka. | |||
---- | ---- | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue"></a><!-- ws:end:WikiTextHeadingRule:2 -->The supremacy of the fifth and and the seventh harmonic in Armodue</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue"></a><!-- ws:end:WikiTextHeadingRule:2 -->The supremacy of the fifth and and the seventh harmonic in Armodue</h2> | ||
<br /> | <br /> | ||
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 tradion - the | 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 tradion - the cycle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.<br /> | ||
<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
<br /> | <br /> | ||
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 | 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.<br /> | ||
<br /> | <br /> | ||
In Armodue, 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.<br /> | In Armodue, 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.<br /> | ||
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The particularity is to make the last note of one module coincide with the first note of the next group.<br /> | The particularity is to make the last note of one module coincide with the first note of the next group.<br /> | ||
<br /> | <br /> | ||
For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes:<br /> | For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes (cylce of 6 eka iterated 8 times):<br /> | ||
<br /> | <br /> | ||
<strong>c</strong> - d - eb - <strong>e</strong> - f# - g - <strong>ab</strong> - bb - b - <strong>c</strong><br /> | <strong>c</strong> - d - eb - <strong>e</strong> - f# - g - <strong>ab</strong> - bb - b - <strong>c</strong><br /> | ||
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In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts.<br /> | In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts.<br /> | ||
<br /> | <br /> | ||
But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.<br /> | But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.<br /> | ||
<br /> | <br /> | ||
If, for example, we take the interval of 6 eka (2 tones plus a quartertone in the 12edo system) we obtain the following sequence of notes:<br /> | |||
<br /> | <br /> | ||
1, 4, 7#, 2, 5#, 8#, 3, 6#, 1<br /> | 1, 4, 7#, 2, 5#, 8#, 3, 6#, 1<br /> | ||
<br /> | <br /> | ||
As we can ascertain, | As we can ascertain, over the range of three tenths (equivalent to three 12-edo octaves) the circle closes, and from the starting note 1 we arrive again at note 1 three tenths higher, by successive steps of 6 eka.<br /> | ||
In strictly | <br /> | ||
Every interval of | In strictly mathematical terms, the least common multiple of 16 eka (the size of the tenth) and 6 eka (the iterated interval) is 48 eka (corresponding to the size of three tenths).<br /> | ||
<br /> | |||
Every interval of 6 eka - iterated eight times in the three tenths range - can be organized inside in various ways; for example in modules of 3 + 2 + 1 eka, in which case we obtain the following scalar system of 24 notes:<br /> | |||
<br /> | <br /> | ||
1, 2#, 3#, 4; (4), 6, 7, 7#; (7#), 9, 1#, 2; (2), 3#, 5, 5#; (5#), 7, 8, 8#; (8#), 1#, 2#, 3; (3), 5, 6, 6#; (6#), 8, 9, 1.<br /> | 1, 2#, 3#, 4; (4), 6, 7, 7#; (7#), 9, 1#, 2; (2), 3#, 5, 5#; (5#), 7, 8, 8#; (8#), 1#, 2#, 3; (3), 5, 6, 6#; (6#), 8, 9, 1.<br /> | ||
<br /> | <br /> | ||
[note: the repeated | [note: the repeated notes in brackets are meant to highlight how the first and the last note of every group coincide.]<br /> | ||
<br /> | |||
All the sixteen notes appear at least once inside the scale shown.<br /> | |||
<br /> | <br /> | ||
So, it's essential to remember the context of membership of the notes in respect to the three tenths. <br /> | |||
So, it's | Supposing - for simplicity - to play in a three tenths register, we'll have eight notes per tenth at our disposal for the creation of the chord, melody and counterpoint texture. <br /> | ||
Supposing - for | <br /> | ||
Specifically, the notes:<br /> | |||
<br /> | <br /> | ||
1, 2#, 3#, 4, 6, 7, 7#, 9 in the first tenth, the lowest;<br /> | 1, 2#, 3#, 4, 6, 7, 7#, 9 in the first tenth, the lowest;<br /> | ||
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1#, 2#, 3, 5, 6, 6#, 8, 9 in the third tenth, the highest.<br /> | 1#, 2#, 3, 5, 6, 6#, 8, 9 in the third tenth, the highest.<br /> | ||
<br /> | <br /> | ||
Note that if we | Note that if we iterate the original defining interval of 6 eka starting from note 1# instead of 1, we obtain the eight pivot notes missing in the sequence that starts from note 1:<br /> | ||
<br /> | <br /> | ||
1#, 5, 8, 2#, 6, 9, 3#, 7, 1#<br /> | 1#, 5, 8, 2#, 6, 9, 3#, 7, 1#<br /> | ||
<br /> | <br /> | ||
To the notes found that way we can apply the same division according to the 3-2-1 eka module, obtaining the same 24 notes scale, but raised 1 eka.<br /> | |||
<br /> | |||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
Chapter 4: <br /> | Chapter 4: <br /> | ||