Blackwood temperament modal harmony (in 15edo): Difference between revisions

== Introduction ==

Blackwood[10] in [[15edo|15edo]] refers to the 10-note symmetric 5L5s [[MOSScales|MOS]] scale in 15edo, which has two modes: 2 1 2 1 2 1 2 1 2 1 and 1 2 1 2 1 2 1 2 1 2. It can be thought of as a 5-limit temperament tempering out 256/243 (the Pythagorean diatonic semitone), a 7-limit temperament tempering out 28/27 and 49/48, and an 11-limit temperament tempering out 28/27, 49/48, and 55/54 (though in 15edo 121/120 and 100/99 are both tempered out as well, making the tuning identical to Ferrier and the unnamed 5c&15 temperament). In 15edo it has a period of 240 cents (5 periods per octave) and a generator of 80 or 160 cents (though it is more commonly described as having a generator of 400 cents).

Most of this page focuses on [[15-EDO]], perhaps the canonical Blackwood tuning (and near to the POTE tuning), but much of it is also useful for Blackwood temperament in general, even if played in other Blackwood-supported equal temperaments.

=== Important features of Blackwood[10] in 15edo ===

* 11-limit blackwood ("blacksmith") is extremely simple and efficient, and while it does fairly high damage to many ratios of 3 and 9, it does a very acceptable job of approximating most ratios of 5, 7, and 11. 9/8, 7/6, 11/9, 4/3, and their octave inversions are the most heavily-damaged, but 6/5, 12/11, and their octave inversions are tuned with good to tolerable accuracy.

* Blackwood[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale.

* Because it is a 10-note scale with a period of 1/5 of an octave, any arbitrary harmony will occur either 5 or 10 times within the 10-note scale, and for otonal harmonies consisting of three or more notes, the utonal counterpart of the harmony will also occur either 5 or 10 times within the scale; this is a property that is only held by other scales with 5 periods per octave.

* Blackwood[10] is also a "mode of limited transposition" like the Diminished and Augmented scales in 12edo: since the scale is built by applying the generator only a single time within each period, the scale has only two modes.

* Another way to think about Blackwood[10] is as a superposition of seven 7-note 5-limit Fokker blocks, representing untempered variations of the diatonic modes, built on a single tonic (more on this below).

| | Degrees of Minor:

| | 1/(6:8:11)

| | 1/(5:8:9)

| | 1/(5:7:9)

| | 7:8:9, 6:7:8

| | 1/(7:8:9), 1/(6:7:8)

| | 1/(9:11:16)

| | 1/(5:6:7)

== Diatonic Modal Harmony in Blackwood[10] ==

Because 15edo is not a meantone temperament, and thus does not temper out the syntonic comma of 81/80, the usual 5L2s diatonic scale is not available. In fact, in 15edo the syntonic comma, which is normally only 21.51 cents, is tuned quite wide: it is mapped to one step of 15edo, and is thus 80 cents! However, one can approximate the diatonic scale (or rather, approximate the various untempered 5-limit JI versions of it) using 3 step-sizes—a large whole tone of 240 cents representing 9/8, a small whole tone of 160 cents representing 10/9, and a semitone of 80 cents representing 16/15. Since these non-MOS diatonic scales do not temper out the syntonic comma, they will only have at most five consonant 5-limit triads (unless an 8th note is included in the right place). They may have even fewer, depending on how the steps are permuted (for instance, if the step-pattern 240-240-80-240-160-160-80 is used, only two consonant 5-limit triads are available).

However, if one insists on using only the versions of the diatonic scale that have the maximum number of consonant triads available, then it turns out all of these scales will be 7-note subsets of Blackwood[10]. They will also be the most compact arrangement of those five consonant triads possible on the 5-limit triangular lattice, which is just a fancy way of saying those five chords will be maximally connected to each other by common tones. This suggests that one can approach melody in Blackwood[10] by treating it not as one 10-note scale, but as several related 7-note scales, each of which functions like a 5-limit untempered version of the diatonic scale.

This approach allows one to apply the usual principles of diatonic tonality and modality, with the caveat that each familiar mode of the diatonic scale will come in two flavors, depending which of the 7 notes one wants to build consonant triads on. The two flavors will share six notes in common, but one of the seven will differ by 81/80 (i.e. one step of 15edo). Interesting relationships do arise if one maintains the tonic but switches through its different modes (i.e. 1 mixolydian to 1 ionian to 1 lydian), and an "extra" mode appears, because of the fact that 256/243 is tempered out (a way to understand 256/243 vanishing in musical terms is that V/V/V/V = IV, unlike in meantone where V/V/V/V = iii). All together, there are 14 modes of the 7-note diatonic to be found in Blackwood[10] if we keep the same tonic, and 20 if we allow alterations of the tonic.

<span style="line-height: 1.5;">0-160-240-400-480-</span><span style="color: #0000ff; line-height: 1.5;">560</span><span style="line-height: 1.5;">-720-880-960-1120-1200</span>

0-80-240-320-480-560-720-800-960-<span style="color: #ff0000;">1120</span>-1200

0-<span style="color: #ff0000;">160</span>-240-320-480-560-720-800-960-<span style="color: #ff0000; line-height: 1.5;">1120</span><span style="line-height: 1.5;">-1200</span>

0-160-240-<span style="color: #0000ff;">320</span>-480-640-720-<span style="color: #0000ff;">800</span>-960-1120-1200

0-<span style="color: #ff0000;">160</span>-240-320-480-<span style="color: #ff0000;">640</span>-720-800-960-1040-1200

[[Category:15edo]]

[[Category:Blacksmith]]

[[Category:Blackwood]]

[[Category:Harmony]]

[[Category:10-tone scales]]

[[Category:Triad]]

[[Category:Modes]]