16edo: Difference between revisions
Wikispaces>igliashon **Imported revision 613256965 - Original comment: ** |
Wikispaces>igliashon **Imported revision 613257045 - Original comment: ** |
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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> | ||
: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2017-05-20 16: | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2017-05-20 16:40:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>613257045</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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 second approach is to preserve the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly". | The second approach is to preserve the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly". | ||
(Alternatively, one can use Armodue nine-nominal notation; see [[16edo#Hexadecaphonic%20Notation|below]]) | |||
||~ Degree ||~ Cents ||~ Approximate | ||~ Degree ||~ Cents ||~ Approximate | ||
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The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third. | The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third. | ||
16-EDO is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16-EDO uses 3\16. For this, there are MOS scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16-EDO supports both, but is not a very accurate tuning of either). | 16-EDO is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16-EDO uses 3\16. For this, there are MOS scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16-EDO supports both, but is not a very accurate tuning of either). | ||
16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 546875:524288, which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "**Magic family of scales**". | 16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 546875:524288, which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "**Magic family of scales**". | ||
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||= 4 ||= 300 ||= major 3rd, dim 4th ||= 3, 4bb || | ||= 4 ||= 300 ||= major 3rd, dim 4th ||= 3, 4bb || | ||
||= 5 ||= 375 ||= minor 4th ||= 4b || | ||= 5 ||= 375 ||= minor 4th ||= 4b || | ||
||= 6 ||= 450 ||= major 4th, | ||= 6 ||= 450 ||= major 4th, | ||
dim 5th ||= 4, 5b || | dim 5th ||= 4, 5b || | ||
||= 7 ||= 525 ||= aug 4th, minor 5th ||= 4#, 5 || | ||= 7 ||= 525 ||= aug 4th, minor 5th ||= 4#, 5 || | ||
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Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres. | Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres. | ||
However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16 edo approximates 7/4 well enough to use | However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16 edo approximates 7/4 well enough to use | ||
it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant. | it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant. | ||
Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads". | Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads". | ||
===MOS scales supporting Metallic Harmony in 16edo=== | ===MOS scales supporting Metallic Harmony in 16edo=== | ||
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<br /> | <br /> | ||
The second approach is to preserve the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo &quot;on the fly&quot;.<br /> | The second approach is to preserve the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo &quot;on the fly&quot;.<br /> | ||
<br /> | |||
(Alternatively, one can use Armodue nine-nominal notation; see <a class="wiki_link" href="/16edo#Hexadecaphonic%20Notation">below</a>)<br /> | |||
<br /> | <br /> | ||
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The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br /> | The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br /> | ||
<br /> | <br /> | ||
16-EDO is also a tuning for the <a class="wiki_link" href="/Jubilismic%20clan">no-threes 7-limit temperament tempering out 50/49</a>. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16-EDO uses 3\16. For this, there are MOS scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16-EDO supports both, but is not a very accurate tuning of either). <br /> | 16-EDO is also a tuning for the <a class="wiki_link" href="/Jubilismic%20clan">no-threes 7-limit temperament tempering out 50/49</a>. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16-EDO uses 3\16. For this, there are MOS scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16-EDO supports both, but is not a very accurate tuning of either).<br /> | ||
16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 546875:524288, which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under &quot;<strong>Magic family of scales</strong>&quot;.<br /> | 16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 546875:524288, which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under &quot;<strong>Magic family of scales</strong>&quot;.<br /> | ||
<br /> | <br /> | ||
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<td style="text-align: center;">450<br /> | <td style="text-align: center;">450<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">major 4th, <br /> | <td style="text-align: center;">major 4th,<br /> | ||
dim 5th<br /> | dim 5th<br /> | ||
</td> | </td> | ||
| Line 1,327: | Line 1,331: | ||
Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres.<br /> | Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres.<br /> | ||
However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16 edo approximates 7/4 well enough to use<br /> | However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16 edo approximates 7/4 well enough to use<br /> | ||
it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant. <br /> | it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant.<br /> | ||
Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads &quot;hard&quot; and &quot;soft&quot;, respectively. In addition, two other &quot;symmetrical&quot; triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to &quot;diminished&quot; and &quot;augmented&quot; triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them &quot;Metallic triads&quot;. <br /> | Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads &quot;hard&quot; and &quot;soft&quot;, respectively. In addition, two other &quot;symmetrical&quot; triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to &quot;diminished&quot; and &quot;augmented&quot; triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them &quot;Metallic triads&quot;.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:25:&lt;h3&gt; --><h3 id="toc9"><a name="Rank two temperaments-Metallic Harmony in 16 EDO-MOS scales supporting Metallic Harmony in 16edo"></a><!-- ws:end:WikiTextHeadingRule:25 -->MOS scales supporting Metallic Harmony in 16edo</h3> | <!-- ws:start:WikiTextHeadingRule:25:&lt;h3&gt; --><h3 id="toc9"><a name="Rank two temperaments-Metallic Harmony in 16 EDO-MOS scales supporting Metallic Harmony in 16edo"></a><!-- ws:end:WikiTextHeadingRule:25 -->MOS scales supporting Metallic Harmony in 16edo</h3> | ||