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

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Blacksmith[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).
==Important features of Blacksmith[10] in 15edo==
<ul><li>As an 11-limit temperament, 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.</li><li>Blacksmith[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale.</li><li>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.</li><li>Blacksmith[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.</li><li>Another way to think about Blacksmith[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).</li></ul>
[[file:blacksmith_lattice_igs.png|frame|none|Blacksmith lattice]]
==Interval Classes in Blacksmith[10]==
{| class="wikitable"
|-
| | Step of 15edo
| | Cent Value
| | Interval Class
| | Guitar Notation
| | Decimal Notation
| | Approximated Ratios
| | Pseudo-Diatonic Category
|-
| | 0
| | 0
| | Unison
| | E
| | 1
| | 1/1
| | Unison
|-
| | 1
| | 80
| | Minor 2nd, Augmented Unison*
| | E#, Gbb
| | 2b, 1#
| | 16/15, 21/20, 22/21, 25/24
| | Minor 2nd
|-
| | 2
| | 160
| | Major 2nd, Diminished 3rd
| | Gb, Ex
| | 2, 3b
| | 10/9, 11/10, 12/11, 15/14
| | Flat Major 2nd
|-
| | 3
| | 240
| | Perfect 3rd, Augmented 2nd, Diminished 4th
| | G
| | 3, 2#, 4bb
| | 7/6, 8/7, 9/8
| | Major 2nd/Subminor 3rd
|-
| | 4
| | 320
| | Minor 4th, Augmented 3rd
| | G#, Abb
| | 4b, 3#
| | 6/5, 11/9
| | Minor 3rd
|-
| | 5
| | 400
| | Major 4th, Diminished 5th
| | Ab, Gx
| | 4, 5b
| | 5/4, 14/11,
| | Major 3rd
|-
| | 6
| | 480
| | Perfect 5th, Augmented 4th, Diminished 6th
| | A
| | 5, 4#, 6bb
| | 4/3, 21/16, 9/7
| | Perfect Fourth
|-
| | 7
| | 560
| | Minor 6th, Augmented 5th
| | A#, Bbb
| | 6b, 5#
| | 7/5, 11/8,
| | Augmented Fourth
|-
| | 8
| | 640
| | Major 6th, Diminished 7th
| | Bb, Ax
| | 6, 7b
| | 10/7, 16/11
| | Diminished 5th
|-
| | 9
| | 720
| | Perfect 7th, Augmented 6th, Diminished 8th
| | B
| | 7, 6#, 8bb
| | 3/2, 32/21, 14/9
| | Perfect Fifth
|-
| | 10
| | 800
| | Minor 8th, Augmented 7th
| | B#, Dbb
| | 8b, 7#
| | 8/5, 11/7,
| | Minor 6th
|-
| | 11
| | 880
| | Major 8th, Diminished 9th
| | Db, Bx
| | 8, 9b
| | 5/3, 18/11
| | Major 6th
|-
| | 12
| | 960
| | Perfect 9th, Augmented 8th, Diminished 10th
| | D
| | 9, 8#, 0bb
| | 12/7, 7/4, 16/9
| | Minor 7th/Supermajor 6th
|-
| | 13
| | 1040
| | Minor 10th, Augmented 9th
| | D#, Ebb
| | 0b, 9#
| | 9/5, 20/11, 11/6, 28/15
| | Sharp Minor 7th
|-
| | 14
| | 1120
| | Major 10th, Diminished Undecave
| | Eb, Dx
| | 0, 1b
| | 15/8, 40/21, 48/25
| | Major 7th
|-
| | 15
| | 1200
| | Undecave ("Octave")
| | E
| | 1
| | 2/1
| | Octave
|}
*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.
=Chords of Blacksmith[10]=
=Chords of Blacksmith[10]=


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