Blackwood temperament modal harmony (in 15edo) - Revision history

VectorGraphics: VectorGraphics moved page Blacksmith temperament modal harmony (in 15edo) to Blackwood temperament modal harmony (in 15edo): Nobody uses the term "blacksmith"

2025-04-19T19:03:14Z

VectorGraphics moved page Blacksmith temperament modal harmony (in 15edo) to Blackwood temperament modal harmony (in 15edo): Nobody uses the term "blacksmith"

← Older revision	Revision as of 19:03, 19 April 2025
(No difference)

VectorGraphics: nobody uses the term "blacksmith"

2025-04-19T19:02:54Z

nobody uses the term "blacksmith"

← Older revision		Revision as of 19:02, 19 April 2025
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	== Introduction ==		== Introduction ==
	~~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).		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 ~~Blacksmith~~ tuning (and near to the POTE tuning), but much of it is also useful for ~~Blacksmith~~ temperament in general, even if played in other ~~Blacksmith~~-supported equal temperaments.		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 ~~Blacksmith~~[10] in 15edo ===		=== Important features of Blackwood[10] in 15edo ===
	* ~~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.		* 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.
	* ~~Blacksmith~~[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale.		* 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.		* 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.
	* ~~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.		* 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 ~~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).		* 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).

	[[file:blacksmith_lattice_igs.png\|frame\|none\|~~Blacksmith~~ lattice]]		[[file:blacksmith_lattice_igs.png\|frame\|none\|Blackwood lattice]]

	=== Interval Classes in ~~Blacksmith~~[10] ===		=== Interval Classes in Blackwood[10] ===

	{\| class="wikitable"		{\| class="wikitable"
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	*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.		*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 Blackwood[10] ==

	=== Basic Functional Chords ===		=== Basic Functional Chords ===
	All of the familiar triads and tetrads of the diatonic scale are found plentifully in ~~Blacksmith~~[10], which is pretty obvious when you just look at the notes available in the major and minor modes:		All of the familiar triads and tetrads of the diatonic scale are found plentifully in Blackwood[10], which is pretty obvious when you just look at the notes available in the major and minor modes:

	{\| class="wikitable"		{\| class="wikitable"
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	\|}		\|}

	== Diatonic Modal Harmony in ~~Blacksmith~~[10] ==		== 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).		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 ~~Blacksmith~~[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 ~~Blacksmith~~[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.		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 ~~Blacksmith~~[10] if we keep the same tonic, and 20 if we allow alterations of the tonic.		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.

	== MODMOSes of ~~Blacksmith~~[10] ==		== MODMOSes of Blackwood[10] ==

	=== Single-alteration MODMOSes ===		=== Single-alteration MODMOSes ===

Fredg999 category edits: Moving from Category:10-note scales to Category:10-tone scales using Cat-a-lot

2022-02-26T18:20:04Z

Moving from Category:10-note scales to Category:10-tone scales using Cat-a-lot

← Older revision		Revision as of 18:20, 26 February 2022
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	[[Category:Blackwood]]		[[Category:Blackwood]]
	[[Category:Harmony]]		[[Category:Harmony]]
	[[Category:10-~~note~~ scales]]		[[Category:10-tone scales]]
	[[Category:Triad]]		[[Category:Triad]]
	[[Category:Modes]]		[[Category:Modes]]

Fredg999: Fredg999 moved page Blacksmith Temperament Modal Harmony (in 15EDO) to Blacksmith temperament modal harmony (in 15edo): WP:NCCAPS

2022-01-18T17:55:41Z

Fredg999 moved page Blacksmith Temperament Modal Harmony (in 15EDO) to Blacksmith temperament modal harmony (in 15edo): WP:NCCAPS

← Older revision	Revision as of 17:55, 18 January 2022
(No difference)

Fredg999: Header levels, categories

2022-01-18T17:53:20Z

Header levels, categories

← Older revision		Revision as of 17:53, 18 January 2022
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	=Introduction=		== Introduction ==
	~~__FORCETOC__~~
	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).		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).

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

	==Important features of Blacksmith[10] in 15edo==		=== 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>~~		* 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.
			* Blacksmith[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.
			* 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.
			* 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).

	[[file:blacksmith_lattice_igs.png\|frame\|none\|Blacksmith lattice]]		[[file:blacksmith_lattice_igs.png\|frame\|none\|Blacksmith lattice]]

	==Interval Classes in Blacksmith[10]==		=== Interval Classes in Blacksmith[10] ===

	{\| class="wikitable"		{\| class="wikitable"
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	*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.		*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] ==

	==Basic Functional Chords==		=== Basic Functional Chords ===
	All of the familiar triads and tetrads of the diatonic scale are found plentifully in Blacksmith[10], which is pretty obvious when you just look at the notes available in the major and minor modes:		All of the familiar triads and tetrads of the diatonic scale are found plentifully in Blacksmith[10], which is pretty obvious when you just look at the notes available in the major and minor modes:

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	\|}		\|}

	==Additional Functional Chords==		=== Additional Functional Chords ===

	{\| class="wikitable"		{\| class="wikitable"
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	\|}		\|}

	==Chords of Extended JI & Essentially-Tempered Chords==		=== Chords of Extended JI & Essentially-Tempered Chords ===
	The dominant 7th and minor-major 6th are both 7-limit chords (4:5:6:7 and 1/(4:5:6:7), respectively). The diminished triad also approximates 5:6:7. Because of the extreme tempering, it's also the case that 9:11:14 = 10:12:15 (i.e., the minor triad). There are no full otonal 11-limit hexads in the 10-note scale, but there are lots of smaller 11-limit chords (otonal, utonal, and essentially-tempered) approximated:		The dominant 7th and minor-major 6th are both 7-limit chords (4:5:6:7 and 1/(4:5:6:7), respectively). The diminished triad also approximates 5:6:7. Because of the extreme tempering, it's also the case that 9:11:14 = 10:12:15 (i.e., the minor triad). There are no full otonal 11-limit hexads in the 10-note scale, but there are lots of smaller 11-limit chords (otonal, utonal, and essentially-tempered) approximated:

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	\|}		\|}

	=Diatonic Modal Harmony in Blacksmith[10]=		== Diatonic Modal Harmony in Blacksmith[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).		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).

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	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 Blacksmith[10] if we keep the same tonic, and 20 if we allow alterations of the tonic.		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 Blacksmith[10] if we keep the same tonic, and 20 if we allow alterations of the tonic.

	=MODMOSes of Blacksmith[10]=		== MODMOSes of Blacksmith[10] ==

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

	==Double-alteration MODMOSes==		=== Double-alteration MODMOSes ===
	<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-</span><span style="color: #0000ff; line-height: 1.5;">800</span><span style="line-height: 1.5;">-960-1120-1200</span>		<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-</span><span style="color: #0000ff; line-height: 1.5;">800</span><span style="line-height: 1.5;">-960-1120-1200</span>

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

	[[Category:15edo]]		[[Category:15edo]]
	[[Category:~~blacksmith~~]]		[[Category:Blacksmith]]
	[[Category:~~blackwood~~]]		[[Category:Blackwood]]
	[[Category:~~harmony~~]]		[[Category:Harmony]]
	[[Category:~~interval~~]]		[[Category:10-note scales]]
	[[Category:~~triad~~]]		[[Category:Triad]]
	[[Category:~~modes~~]]		[[Category:Modes]]

Keenan Pepper: rm weird empty section

2021-05-08T21:31:12Z

rm weird empty section

← Older revision		Revision as of 21:31, 8 May 2021
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	\|}		\|}

	~~== ==~~

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

Mike Battaglia: moved to category:modes

2021-05-08T20:26:22Z

moved to category:modes

← Older revision		Revision as of 20:26, 8 May 2021
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	[[Category:interval]]		[[Category:interval]]
	[[Category:triad]]		[[Category:triad]]
	[[Category:~~modal_harmony~~]]		[[Category:modes]]

Mike Battaglia: Added modal harmony category

2021-05-08T20:21:19Z

Added modal harmony category

← Older revision		Revision as of 20:21, 8 May 2021
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	[[Category:interval]]		[[Category:interval]]
	[[Category:triad]]		[[Category:triad]]
			[[Category:modal_harmony]]

Mike Battaglia: Added small note about much of this being also useful for Blacksmith in general, not just 15

2021-05-08T20:20:52Z

Added small note about much of this being also useful for Blacksmith in general, not just 15

← Older revision		Revision as of 20:20, 8 May 2021
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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).		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).

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

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

Mike Battaglia: Mike Battaglia moved page Harmony in 15edo Blacksmith to Blacksmith Temperament Modal Harmony (in 15EDO): Changed name to match Porcupine Temperament Modal Harmony and Mavila Temperament Modal Harmony, so that if one searches for "Modal Harmony" this page is also there

2021-05-08T20:18:26Z

Mike Battaglia moved page Harmony in 15edo Blacksmith to Blacksmith Temperament Modal Harmony (in 15EDO): Changed name to match Porcupine Temperament Modal Harmony and Mavila Temperament Modal Harmony, so that if one searches for "Modal Harmony" this page is also there

← Older revision	Revision as of 20:18, 8 May 2021
(No difference)