Titanium
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**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), making it a [[Septisemi temperaments|septisemi]] temperament, and the slendro diesis (49:48), making it part of the [[slendro clan]]. As such, 6:5, 7:6, and 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). Finally, 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. 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 is also 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). 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 otonal and six utonal 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. An example of an enneatonic scale (using a generator of 271 cents) is given below. The step sizes are 116 and 155 cents. In this scale, the generator, while representing three different intervals, is quite close to a just 7:6 and thus has a rather stable sound, as does the 387-cent interval which is very close to a just 5:4. This helps compensate for the fifths and fourths being so out of tune. This is the reverse of what happens in Pythagorean scales (where the fifths are consonant and the thirds dissonant). Another thing to watch out for is that the so-called "wolf" fifth, at 697 cents, is actually quite close to just. It is a "wolf" interval only in the sense that it can't easily be used as a 10:7, unlike the other fifths, but it will certainly prove useful in other ways. This scale, in the standard major mode (i. e, the mode allowing for a I-IV-V chord progression), has the form sLsLsLssL. 0 116 271 387 542 658 813 929 1045 Enneatonic scales of this form can be extended to 13-note MOSes, while those where the generator is smaller than 2\9 extend to 14 notes. Either the 13 or 14 note scale could be considered the "chromatic" scale of titanium (in much the same way the enneatonic scale is analogous to the diatonic). When the generator is smaller than 2\9, the temperament and scales generated from it could be called "brittle", while if it is larger than 2\9 (as in the scale above), this variant of titanium temperament and its scales could be referred to as "ductile". (A reference to the fact that titanium metal undergoes a brittle-to-ductile transition at high temperatures). "Brittle" titanium gives a slightly closer approximation of 3:2, but "ductile" titanium gives a better 5:4 and 7:5. While pajara temperament and Paul Erlich's scales are closely related to Indian music theory (pajara temperament can be used to construct a 22-note MODMOS that easily represents the sruti system), titanium temperament and its enneatonic scales have a natural kinship with Indonesian music, being related to both slendro (via tempering out 49:48, and in fact the enneatonic scale can be considered a sort of extended slendro), and pelog scales (via the presence of the flat fifths; the complete pelog scale does not however occur as a subset of the enneatonic, but does occur as a subset of the 13- and 14-note "chromatic" titanium scales. Moreover, pelog scales are sometimes described as approximating a 7-note subset of 9edo, and 9edo falls within the titanium temperament's tuning range).
Original HTML content:
<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 (21:20), 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, 6:5, 7:6, and 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). Finally, 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 /> <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 /> In titanium, the 7-limit tetrad has very low <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> (only 3). The fact that 7:5 is also 4:3 allows a type of "tritone substitution" 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).<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 otonal and six utonal 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.<br /> <br /> An example of an enneatonic scale (using a generator of 271 cents) is given below. The step sizes are 116 and 155 cents. In this scale, the generator, while representing three different intervals, is quite close to a just 7:6 and thus has a rather stable sound, as does the 387-cent interval which is very close to a just 5:4. This helps compensate for the fifths and fourths being so out of tune. This is the reverse of what happens in Pythagorean scales (where the fifths are consonant and the thirds dissonant). Another thing to watch out for is that the so-called "wolf" fifth, at 697 cents, is actually quite close to just. It is a "wolf" interval only in the sense that it can't easily be used as a 10:7, unlike the other fifths, but it will certainly prove useful in other ways. This scale, in the standard major mode (i. e, the mode allowing for a I-IV-V chord progression), has the form sLsLsLssL.<br /> <br /> 0<br /> 116<br /> 271<br /> 387<br /> 542<br /> 658<br /> 813<br /> 929<br /> 1045<br /> <br /> Enneatonic scales of this form can be extended to 13-note MOSes, while those where the generator is smaller than 2\9 extend to 14 notes. Either the 13 or 14 note scale could be considered the "chromatic" scale of titanium (in much the same way the enneatonic scale is analogous to the diatonic). When the generator is smaller than 2\9, the temperament and scales generated from it could be called "brittle", while if it is larger than 2\9 (as in the scale above), this variant of titanium temperament and its scales could be referred to as "ductile". (A reference to the fact that titanium metal undergoes a brittle-to-ductile transition at high temperatures). "Brittle" titanium gives a slightly closer approximation of 3:2, but "ductile" titanium gives a better 5:4 and 7:5.<br /> <br /> While pajara temperament and Paul Erlich's scales are closely related to Indian music theory (pajara temperament can be used to construct a 22-note MODMOS that easily represents the sruti system), titanium temperament and its enneatonic scales have a natural kinship with Indonesian music, being related to both slendro (via tempering out 49:48, and in fact the enneatonic scale can be considered a sort of extended slendro), and pelog scales (via the presence of the flat fifths; the complete pelog scale does not however occur as a subset of the enneatonic, but does occur as a subset of the 13- and 14-note "chromatic" titanium scales. Moreover, pelog scales are sometimes described as approximating a 7-note subset of 9edo, and 9edo falls within the titanium temperament's tuning range).</body></html>