Titanium: Difference between revisions
Wikispaces>MasonGreen1 **Imported revision 586865601 - Original comment: ** |
Wikispaces>MasonGreen1 **Imported revision 586918925 - 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:MasonGreen1|MasonGreen1]] and made on <tt>2016-07- | : This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-07-14 11:16:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>586918925</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<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">**Titanium** is Mason Green's proposed name for a remarkable low-complexity, though high-badness 7-limit temperament. Titanium tempers out the septimal chromatic semitone ([[21_20|21:20]]), making it a [[Septisemi temperaments|septisemi]] temperament, and the slendro diesis (49:48), making it part of the [[slendro clan]]. As such, [[6_5|6:5]], [[7_6|7:6]], and [[8_7|8:7]] are all represented by the same interval (which, in fact, is the generator). Two of these generators make a very sharp fourth (which is also a very flat [[7_5|7:5]]). Since three fifths make a minor (not major) sixth, and four make a minor (not major) third, it is also a [[pelogic]] temperament. Finally it can be also considered a sort of messed-up variant of [[orwell]] temperament as well, since the generator falls into the same range of sizes. | <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">**Titanium** is Mason Green's proposed name for a remarkable low-complexity, though high-badness 7-limit temperament. Titanium tempers out the septimal chromatic semitone ([[21_20|21:20]]), making it a [[Septisemi temperaments|septisemi]] temperament, and the slendro diesis (49:48), making it part of the [[slendro clan]]. As such, [[6_5|6:5]], [[7_6|7:6]], and [[8_7|8:7]] are all represented by the same interval (which, in fact, is the generator). Two of these generators make a very sharp fourth (which is also a very flat [[7_5|7:5]]). Since three fifths make a minor (not major) sixth, and four make a minor (not major) third, it is also a [[pelogic]] temperament. It also tempers out 27:25 and is part of the [[bug family|bug]] family, and in fact "brittle" titanium (see below) is essentially the same as bug's 7-limit extension beep; because of this, the generator also functions as [[9|10:9]] (meaning this tuning has a //negative// syntonic comma). Finally it can be also considered a sort of messed-up variant of [[orwell]] temperament as well, since the generator falls into the same range of sizes. | ||
The edos whose patent vals support titanium are [[4edo]], [[5edo]], [[9edo]], and [[14edo]]. Many other edos can be used as non-patent vals, such as 13. | The edos whose patent vals support titanium are [[4edo]], [[5edo]], [[9edo]], and [[14edo]]. Many other edos can be used as non-patent vals, such as 13. | ||
In titanium, the 7-limit tetrad has very low [[Graham complexity]] (only 3). The fact that [[7_5|7:5]] is also [[4_3|4:3]] allows a type of "tritone substitution" distinct from that which appears in [[jubilismic]] temperaments; namely, one in which the 4:3 of one chord becomes the 7:5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to //inversions// of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss). | In titanium, the 7-limit tetrad has very low [[Graham complexity]] (only 3). The fact that [[7_5|7:5]] is also [[4_3|4:3]] allows a type of "tritone substitution" distinct from that which appears in [[jubilismic]] temperaments; namely, one in which the 4:3 of one chord becomes the 7:5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). The 4:5:6:7 and 5:6:7:9 tetrads are also the same. This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to //inversions// of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss). | ||
Titanium forms enneatonic scales which may be either of the form LLsLsLsLs or ssLsLsLsL; where both step sizes are equal this simply gives [[9edo]]. An enneatonic scale has six tetrads, and just like the diatonic scale it allows a variant of the familiar "I-IV-V" type of chord progression (which, using enneatonic notation, would be I-V-VI). Titanium's enneatonic scales provide an alternative to the Erlich decatonic; they are lower in accuracy, but are also simpler and offer more freedom of modulation, and thus for some listeners they might actually be less xenharmonic. | Titanium forms enneatonic scales which may be either of the form LLsLsLsLs or ssLsLsLsL; where both step sizes are equal this simply gives [[9edo]]. An enneatonic scale has six tetrads, and just like the diatonic scale it allows a variant of the familiar "I-IV-V" type of chord progression (which, using enneatonic notation, would be I-V-VI). Titanium's enneatonic scales provide an alternative to the Erlich decatonic; they are lower in accuracy, but are also simpler and offer more freedom of modulation, and thus for some listeners they might actually be less xenharmonic. | ||
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Ductile titanium does not extend well to the 9-odd-limit since the 3:2 is already extremely flat. However, beyond the 9th harmonic it performs better, with the 11th and 13th harmonics having passable mappings (tempering out 33:32 and 65:64, respectively). While pure octaves are possible, titanium (especially the ductile version) also benefits greatly from octave stretching, since the 3rd, 7th, 11th, and 13th harmonics are all flat, while the 5th is near-just in the above example. An octave of around 1205-1207 cents is worth trying. | Ductile titanium does not extend well to the 9-odd-limit since the 3:2 is already extremely flat. However, beyond the 9th harmonic it performs better, with the 11th and 13th harmonics having passable mappings (tempering out 33:32 and 65:64, respectively). While pure octaves are possible, titanium (especially the ductile version) also benefits greatly from octave stretching, since the 3rd, 7th, 11th, and 13th harmonics are all flat, while the 5th is near-just in the above example. An octave of around 1205-1207 cents is worth trying. | ||
A good example of brittle titanium comes from using the [[23edo|23cd]] val, where the generator is about 261 cents and the fifths about 678 cents. The "5:4" is significantly sharp in this case and is actually very close to a just 14:11, making this situation akin to [[fudging]]. The 7:4 and 3:2 are both more accurate, but the 7:5 less so. The 11th harmonic is flatter than in ductile titanium, and the 13th harmonic is not matched well at all.</pre></div> | A good example of brittle titanium comes from using the [[23edo|23cd]] val, where the generator is about 261 cents and the fifths about 678 cents. The "5:4" is significantly sharp in this case and is actually very close to a just 14:11, making this situation akin to [[fudging]]. The 7:4 and 3:2 are both more accurate, but the 7:5 less so. The 11th harmonic is flatter than in ductile titanium, and the 13th harmonic is not matched well at all. [[14edo]]'s patent val gives a more expressive version of brittle titanium.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Titanium</title></head><body><strong>Titanium</strong> is Mason Green's proposed name for a remarkable low-complexity, though high-badness 7-limit temperament. Titanium tempers out the septimal chromatic semitone (<a class="wiki_link" href="/21_20">21:20</a>), making it a <a class="wiki_link" href="/Septisemi%20temperaments">septisemi</a> temperament, and the slendro diesis (49:48), making it part of the <a class="wiki_link" href="/slendro%20clan">slendro clan</a>. As such, <a class="wiki_link" href="/6_5">6:5</a>, <a class="wiki_link" href="/7_6">7:6</a>, and <a class="wiki_link" href="/8_7">8:7</a> are all represented by the same interval (which, in fact, is the generator). Two of these generators make a very sharp fourth (which is also a very flat <a class="wiki_link" href="/7_5">7:5</a>). Since three fifths make a minor (not major) sixth, and four make a minor (not major) third, it is also a <a class="wiki_link" href="/pelogic">pelogic</a> temperament. Finally it can be also considered a sort of messed-up variant of <a class="wiki_link" href="/orwell">orwell</a> temperament as well, since the generator falls into the same range of sizes.<br /> | <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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Titanium</title></head><body><strong>Titanium</strong> is Mason Green's proposed name for a remarkable low-complexity, though high-badness 7-limit temperament. Titanium tempers out the septimal chromatic semitone (<a class="wiki_link" href="/21_20">21:20</a>), making it a <a class="wiki_link" href="/Septisemi%20temperaments">septisemi</a> temperament, and the slendro diesis (49:48), making it part of the <a class="wiki_link" href="/slendro%20clan">slendro clan</a>. As such, <a class="wiki_link" href="/6_5">6:5</a>, <a class="wiki_link" href="/7_6">7:6</a>, and <a class="wiki_link" href="/8_7">8:7</a> are all represented by the same interval (which, in fact, is the generator). Two of these generators make a very sharp fourth (which is also a very flat <a class="wiki_link" href="/7_5">7:5</a>). Since three fifths make a minor (not major) sixth, and four make a minor (not major) third, it is also a <a class="wiki_link" href="/pelogic">pelogic</a> temperament. It also tempers out 27:25 and is part of the <a class="wiki_link" href="/bug%20family">bug</a> family, and in fact &quot;brittle&quot; titanium (see below) is essentially the same as bug's 7-limit extension beep; because of this, the generator also functions as <a class="wiki_link" href="/9">10:9</a> (meaning this tuning has a <em>negative</em> syntonic comma). Finally it can be also considered a sort of messed-up variant of <a class="wiki_link" href="/orwell">orwell</a> temperament as well, since the generator falls into the same range of sizes.<br /> | ||
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The edos whose patent vals support titanium are <a class="wiki_link" href="/4edo">4edo</a>, <a class="wiki_link" href="/5edo">5edo</a>, <a class="wiki_link" href="/9edo">9edo</a>, and <a class="wiki_link" href="/14edo">14edo</a>. Many other edos can be used as non-patent vals, such as 13.<br /> | The edos whose patent vals support titanium are <a class="wiki_link" href="/4edo">4edo</a>, <a class="wiki_link" href="/5edo">5edo</a>, <a class="wiki_link" href="/9edo">9edo</a>, and <a class="wiki_link" href="/14edo">14edo</a>. Many other edos can be used as non-patent vals, such as 13.<br /> | ||
<br /> | <br /> | ||
In titanium, the 7-limit tetrad has very low <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> (only 3). The fact that <a class="wiki_link" href="/7_5">7:5</a> is also <a class="wiki_link" href="/4_3">4:3</a> allows a type of &quot;tritone substitution&quot; distinct from that which appears in <a class="wiki_link" href="/jubilismic">jubilismic</a> temperaments; namely, one in which the 4:3 of one chord becomes the 7:5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to <em>inversions</em> of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss).<br /> | In titanium, the 7-limit tetrad has very low <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> (only 3). The fact that <a class="wiki_link" href="/7_5">7:5</a> is also <a class="wiki_link" href="/4_3">4:3</a> allows a type of &quot;tritone substitution&quot; distinct from that which appears in <a class="wiki_link" href="/jubilismic">jubilismic</a> temperaments; namely, one in which the 4:3 of one chord becomes the 7:5 of the next or vice versa. This is equivalent to modulating upward or downward by a generator. The more usual type of modulation (upward or downward by fifths) is also easy since two generators make a fifth. These tetrads, despite being relatively inaccurate, are still easily recognizable and not necessarily unpleasant-sounding (though of course it depends on the timbre of the instrument they are played on). Another thing to watch out for is that due to tempering, the tetrads are of the form Lsss, which means they are their own inverses (i.e. the utonal and otonal tetrads are the same). The 4:5:6:7 and 5:6:7:9 tetrads are also the same. This gives them a chameleonic quality, akin to power chords; they can sound either major or minor, which strongly depends on the voicing used. The terms major and minor can still be used to refer to <em>inversions</em> of the basic tetrad (the major inversion being Lsss, and the minor inversion being sLss).<br /> | ||
<br /> | <br /> | ||
Titanium forms enneatonic scales which may be either of the form LLsLsLsLs or ssLsLsLsL; where both step sizes are equal this simply gives <a class="wiki_link" href="/9edo">9edo</a>. An enneatonic scale has six tetrads, and just like the diatonic scale it allows a variant of the familiar &quot;I-IV-V&quot; type of chord progression (which, using enneatonic notation, would be I-V-VI). Titanium's enneatonic scales provide an alternative to the Erlich decatonic; they are lower in accuracy, but are also simpler and offer more freedom of modulation, and thus for some listeners they might actually be less xenharmonic.<br /> | Titanium forms enneatonic scales which may be either of the form LLsLsLsLs or ssLsLsLsL; where both step sizes are equal this simply gives <a class="wiki_link" href="/9edo">9edo</a>. An enneatonic scale has six tetrads, and just like the diatonic scale it allows a variant of the familiar &quot;I-IV-V&quot; type of chord progression (which, using enneatonic notation, would be I-V-VI). Titanium's enneatonic scales provide an alternative to the Erlich decatonic; they are lower in accuracy, but are also simpler and offer more freedom of modulation, and thus for some listeners they might actually be less xenharmonic.<br /> | ||
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Ductile titanium does not extend well to the 9-odd-limit since the 3:2 is already extremely flat. However, beyond the 9th harmonic it performs better, with the 11th and 13th harmonics having passable mappings (tempering out 33:32 and 65:64, respectively). While pure octaves are possible, titanium (especially the ductile version) also benefits greatly from octave stretching, since the 3rd, 7th, 11th, and 13th harmonics are all flat, while the 5th is near-just in the above example. An octave of around 1205-1207 cents is worth trying.<br /> | Ductile titanium does not extend well to the 9-odd-limit since the 3:2 is already extremely flat. However, beyond the 9th harmonic it performs better, with the 11th and 13th harmonics having passable mappings (tempering out 33:32 and 65:64, respectively). While pure octaves are possible, titanium (especially the ductile version) also benefits greatly from octave stretching, since the 3rd, 7th, 11th, and 13th harmonics are all flat, while the 5th is near-just in the above example. An octave of around 1205-1207 cents is worth trying.<br /> | ||
<br /> | <br /> | ||
A good example of brittle titanium comes from using the <a class="wiki_link" href="/23edo">23cd</a> val, where the generator is about 261 cents and the fifths about 678 cents. The &quot;5:4&quot; is significantly sharp in this case and is actually very close to a just 14:11, making this situation akin to <a class="wiki_link" href="/fudging">fudging</a>. The 7:4 and 3:2 are both more accurate, but the 7:5 less so. The 11th harmonic is flatter than in ductile titanium, and the 13th harmonic is not matched well at all.</body></html></pre></div> | A good example of brittle titanium comes from using the <a class="wiki_link" href="/23edo">23cd</a> val, where the generator is about 261 cents and the fifths about 678 cents. The &quot;5:4&quot; is significantly sharp in this case and is actually very close to a just 14:11, making this situation akin to <a class="wiki_link" href="/fudging">fudging</a>. The 7:4 and 3:2 are both more accurate, but the 7:5 less so. The 11th harmonic is flatter than in ductile titanium, and the 13th harmonic is not matched well at all. <a class="wiki_link" href="/14edo">14edo</a>'s patent val gives a more expressive version of brittle titanium.</body></html></pre></div> | ||