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 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). 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, 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 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.

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). 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.

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.

cents generator steps
0 0
116 -4
271 1
387 -3
542 2
658 -2
813 (or 774) 3 (or -6)
929 -1
1045 -5

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. Also, 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).

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 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.

Suggested timbre

If using brittle titanium (23cd, 14edo, etc.), one might want to consider using this as a guideline. With this spectrum, no partials are more than 25 cents above or below their perfectly harmonic values, and when using 14edo, no intervals will be more than 26 cents out of tune. This is only a guideline, and only with synthesized tones would it be possible to achieve this perfectly. With physical idiophones (celestas, gamelans, etc.) it should still be possible to get a great approximation using CAD (or trial and error).

Fundamental (1st): just

Octave (2nd): just

3rd: -12.5

4th: just

5th: +25

6th: -12.5 cents

7th: -25 cents

8th: just

9th: -25 cents

10th: +25 cents

11th: -25 cents

12th: -12.5 cents

13th: +15 cents

14th: -25 cents

15th: +12.5 cents

16th: just