Titanium: Difference between revisions

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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 one of the two standard major modes (i. e, one of the modes allowing for a I-IV-V chord progression), has the form sLsLsLssL. There is a second standard major mode of form sLsLssLsL, differing only in the position of the seventh scale degree. If this scale degree is considered to be movable, we can combine both modes and increase the total number of tetrads in the scale to seven.

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.

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 &amp;quot;wolf&amp;quot; fifth, at 697 cents, is actually quite close to just. It is a &amp;quot;wolf&amp;quot; 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 one of the two standard major modes (i. e, one of the modes allowing for a I-IV-V chord progression), has the form sLsLsLssL. There is a second standard major mode of form sLsLssLsL, differing only in the position of the seventh scale degree. If this scale degree is considered to be movable, we can combine both modes and increase the total number of tetrads in the scale to seven.&lt;br /&gt;

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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 &amp;quot;chromatic&amp;quot; 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 &amp;quot;brittle&amp;quot;, 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 &amp;quot;ductile&amp;quot;. (A reference to the fact that titanium metal undergoes a brittle-to-ductile transition at high temperatures). &amp;quot;Brittle&amp;quot; titanium gives a slightly closer approximation of 3:2, but &amp;quot;ductile&amp;quot; titanium gives a better 5:4 and 7:5.&lt;br /&gt;