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