Tablet - Revision history

Overthink: + inacc template

2025-10-24T23:11:32Z

+ inacc template

← Older revision		Revision as of 23:11, 24 October 2025
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			{{inacc}}
	A '''tablet''' (the name by analogy with {{w\|tablature}}) is a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].		A '''tablet''' (the name by analogy with {{w\|tablature}}) is a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].

VectorGraphics at 22:42, 17 May 2025

2025-05-17T22:42:24Z

← Older revision		Revision as of 22:42, 17 May 2025
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	~~By a~~ '''tablet''' (the name by analogy with {{w\|tablature}}) is ~~meant~~ a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].		A '''tablet''' (the name by analogy with {{w\|tablature}}) is a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].

	There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.		There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.

Sintel: -legacy

2025-03-29T18:11:19Z

-legacy

← Older revision		Revision as of 18:11, 29 March 2025
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	~~{{Legacy}}~~
	By a '''tablet''' (the name by analogy with {{w\|tablature}}) is meant a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].		By a '''tablet''' (the name by analogy with {{w\|tablature}}) is meant a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].

Lériendil: deploying new cat

2025-02-17T16:18:31Z

deploying new cat

← Older revision		Revision as of 16:18, 17 February 2025
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			{{Legacy}}
	By a '''tablet''' (the name by analogy with {{w\|tablature}}) is meant a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].		By a '''tablet''' (the name by analogy with {{w\|tablature}}) is meant a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].

Lériendil: Changed protection level for "Tablet" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

2025-02-17T16:03:47Z

Changed protection level for "Tablet" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:03, 17 February 2025
(No difference)

FloraC: Fix header level. Promote the first section to intro. Mark with todo.

2024-10-16T10:34:16Z

Fix header level. Promote the first section to intro. Mark with todo.

← Older revision		Revision as of 10:34, 16 October 2024
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	~~__FORCETOC__~~		By a '''tablet''' (the name by analogy with {{w\|tablature}}) is meant a pair [''n'', ''c''] consisting of an approximate note-number ''n'' (an integer) and a chord-denoting element ''c'', typically a {{w\|tuple}} of integers, which defines a type of [[chord]] up to [[octave equivalence]]. Together they define a note in a [[just intonation subgroup\|just intonation group]] or [[regular temperament]]. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a [[val]] or val-like mapping, the number ''n'' in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].
	~~=What is a tablet?=~~
	By a ''tablet'' (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a ~~[http://en.wikipedia.org/wiki/Tuple~~ tuple] of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].

	There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.		There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.

	=The 5-limit 3et tablet=		== The 5-limit 3et tablet ==
	If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted \|* e3 e5>, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that [r e3 e5] defines a major triad with root given by \|* e3 e5> when r is even, and a minor triad when r is odd.		If we want a way to uniquely denote the octave-equivalent class of a 5-limit major triad, we can simply give the root of the chord, ignoring octaves. In terms of monzos that means a major triad would be denoted \|* e3 e5>, where the asterisk can be anything. Since the asterisk is not being used to impart information, we can use it to distinguish major from minor triads. So we might decide that [r e3 e5] defines a major triad with root given by \|* e3 e5> when r is even, and a minor triad when r is odd.

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	Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7\|note(n, c) = n.		Then if c is a tablet, note(n, c) gives the note belonging to the triad denoted by c, satisfying <3 5 7\|note(n, c) = n.

	=4et tablets=		== 4et tablets ==
			=== The 7-limit 4et tablet ===
	==The 7-limit 4et tablet==
	Suppose m0, m1, m2 and m3 are four [[monzo]]s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being \|* e3 e5 e7>, where the asterisk can be any integer value, then m = \|* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord \|* e3 e5 e7> then m = \|* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the [[The Seven Limit Symmetrical Lattices\|cubic lattice of 7-limit tetrads]].		Suppose m0, m1, m2 and m3 are four [[monzo]]s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being \|* e3 e5 e7>, where the asterisk can be any integer value, then m = \|* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord \|* e3 e5 e7> then m = \|* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the [[The Seven Limit Symmetrical Lattices\|cubic lattice of 7-limit tetrads]].

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	If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and <4 6 9 11\|note(n, t) = n.		If a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). Then note(t) is the note defined by the tablet t. If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w)] is a 7-limit tetrad, and <4 6 9 11\|note(n, t) = n.

	==The keenanismic tablet==		=== The keenanismic tablet ===
	This is based on the five [[keenanismic tetrads]], which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords [[Dyadic chord\|dyadic]]. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.		This is based on the five [[keenanismic tetrads]], which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords [[Dyadic chord\|dyadic]]. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.

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	If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a [http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html positive definite quaratic form] on 3-tuples [x y z] by Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root √Q(a-b) is a Eulidean measure of distance, and √8 is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.		If a and b are two allowable triples of integers representing tetrads (so that their reduction modulo four is one of the seven allowed types) then we may derive a [http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html positive definite quaratic form] on 3-tuples [x y z] by Q(x, y, z) = 4x^2 + 3y^2 + 3z^2 + 4xy + 4xz + 2xy, such that Q(a-b) = 8 if a and b are 3-tuples representing tetrads with a common triad. The square root √Q(a-b) is a Eulidean measure of distance, and √8 is the minimum distance between tetrad 3-tuples. Such 3-tuple pairs a minimum distance apart share at least an interval in common, and usually a triad, and are therefore closely related.

	==The pele tablet==		=== The pele tablet ===
	This is a tablet for the rank 3 13-limit temperament [[Hemifamity family#Pele\|pele]]. It is based on the following 71 chords, in the 5-limit transversal:		This is a tablet for the rank 3 13-limit temperament [[Hemifamity family#Pele\|pele]]. It is based on the following 71 chords, in the 5-limit transversal:

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	14/11-13/11-13/11-9/8]]:		14/11-13/11-13/11-9/8]]:

	=5et tablets=		== 5et tablets ==
			=== The 7-limit 5et tablet ===
	==The 7-limit 5et tablet==

	If we define u = n - 9a - 5b - 3c then supposing a+b+c is even,		If we define u = n - 9a - 5b - 3c then supposing a+b+c is even,
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	Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit pentad, where <5 8 12 14\|note(n, t) = n.		Once again, if a+b+c is odd, then define note(t) as -note(-n, [-1-a -1-b -1-c]). If t = [n, w], where w is a 3-tuple, then [note([n, w)), note(n+1, w), note(n+2), w), note(n+3, w), note(n+4), w)] is a complete 9-odd-limit pentad, where <5 8 12 14\|note(n, t) = n.

	==The meantone add6/9 tablet==		=== The meantone add6/9 tablet ===
	The meantone add6/9 tablet is based on the [[Meantone add6-9 pentad\|meantone add6/9 pentad]], which can also be called the add2/9 pentad, the meantone pentatonic scale or Meantone[5]. The tablet is extremely simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where		The meantone add6/9 tablet is based on the [[Meantone add6-9 pentad\|meantone add6/9 pentad]], which can also be called the add2/9 pentad, the meantone pentatonic scale or Meantone[5]. The tablet is extremely simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where

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	In all cases <5 8\|note(n, c) = n. Tempering the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.		In all cases <5 8\|note(n, c) = n. Tempering the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.

	==The 5et portent tablet==		=== The 5et portent tablet ===
	This is based on the following twelve chords, which are expressed in terms of the 2.5.7 transversal of the 11-limit rank three temperament portent, which tempers out 385/384, 441/440 and hence also 1029/1024 and 3025/3024.		This is based on the following twelve chords, which are expressed in terms of the 2.5.7 transversal of the 11-limit rank three temperament portent, which tempers out 385/384, 441/440 and hence also 1029/1024 and 3025/3024.

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	Each of the other eleven chords shares a triad with the first, otonal, pentad, and all of the chords can be related by an infinite but locally finite graph by drawing an edge between chords with a common triad.		Each of the other eleven chords shares a triad with the first, otonal, pentad, and all of the chords can be related by an infinite but locally finite graph by drawing an edge between chords with a common triad.

	=The 6et tutone tutonic tablet=		== The 6et tutone tutonic tablet ==
	This tablet is based on the [[tutonic sextad]], which in terms of the 99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in [[Chromatic pairs#Tutone\|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.		This tablet is based on the [[tutonic sextad]], which in terms of the 99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in [[Chromatic pairs#Tutone\|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.

	If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = \|(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19\|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19\|note(n, c) = n, where note(n, c) in subgroup monzo terms is \|(u-i)/6-3i c+i>.		If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = \|(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19\|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19\|note(n, c) = n, where note(n, c) in subgroup monzo terms is \|(u-i)/6-3i c+i>.

	=The 13-limit 7et tablet=		== The 13-limit 7et tablet ==
	Let <r e3 e5 e7 e11 e13\| denote an otonal 13-limit septad with root given by \|* e3 e5 e7 e11 e13> when r is even, which in close position is 9/8-5/4-11/8-3/2-13/8-7/4-2. If r is odd, let it denote a utonal pentad which in close position is 12/11-6/5-4/3-3/2-12/7-24/13-2		Let <r e3 e5 e7 e11 e13\| denote an otonal 13-limit septad with root given by \|* e3 e5 e7 e11 e13> when r is even, which in close position is 9/8-5/4-11/8-3/2-13/8-7/4-2. If r is odd, let it denote a utonal pentad which in close position is 12/11-6/5-4/3-3/2-12/7-24/13-2

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	<7 11 16 20 24 26\|note(n, [r e3 e5 e7 e11 e13]) = n.		<7 11 16 20 24 26\|note(n, [r e3 e5 e7 e11 e13]) = n.

	=The orwell nonad tablet=		== The orwell nonad tablet ==
	The [[orwell tetrad\|orwell nonad]] is the Orwell[9] MOS, considered as an essentially tempered dyadic chord (which it is, in the 15-limit.) While nine notes is a lot of notes for a chord, obviously in practice subchords could be used.		The [[orwell tetrad\|orwell nonad]] is the Orwell[9] MOS, considered as an essentially tempered dyadic chord (which it is, in the 15-limit.) While nine notes is a lot of notes for a chord, obviously in practice subchords could be used.

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	if i is odd. We then have <9 14 21 25\|note(n, c) = n.		if i is odd. We then have <9 14 21 25\|note(n, c) = n.

	[[Category:Chords]]		[[Category:Chords]]
	[[Category:Composition]]		[[Category:Composition]]
	[[Category:Notation]]		[[Category:Notation]]
			{{Todo\| improve readability }}

FloraC: FloraC moved page Tablets to Tablet: WP:PLURAL

2024-10-16T09:46:49Z

FloraC moved page Tablets to Tablet: WP:PLURAL

← Older revision	Revision as of 09:46, 16 October 2024
(No difference)

Mike Battaglia: Protected "Tablets" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:22:37Z

Protected "Tablets" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:22, 15 November 2021
(No difference)

ResonantFrequencies: re-categorisation (AWB), typos fixed: On the other hand → On the other hand,, extremly → extremely, the the → the

2019-06-30T23:59:08Z

re-categorisation (AWB), typos fixed: On the other hand → On the other hand,, extremly → extremely, the the → the

← Older revision		Revision as of 23:59, 30 June 2019
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	__FORCETOC__		__FORCETOC__
	=What is a tablet?=		=What is a tablet?=
	By a ''tablet'' (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a [http://en.wikipedia.org/wiki/Tuple tuple] of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[~~Composing_with_tablets\|~~Composing with tablets]].		By a ''tablet'' (the name by analogy with tablature) is meant a pair [n, c] consisting of an approximate note-number n (an integer) and a chord denoting element c, typically a [http://en.wikipedia.org/wiki/Tuple tuple] of integers, which defines a type of chord up to octave equivalence. Together they define a note in a just intonation group or regular temperament. The representation of the note is non-unique, as for any note there will be a variety of tablets for it depending on the specified type of chord and chord element, but by means of a val or val-like mapping, the number n in the tablet is definable from the note. For a discussion of how tablets can be used as a compositional tool, see [[Composing with tablets]].

	There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.		There is really no better way of defining more precisely what a tablet is than by giving examples, which are considered below.
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	==The 7-limit 4et tablet==		==The 7-limit 4et tablet==
	Suppose m0, m1, m2 and m3 are four [[~~monzo\|~~monzo]]s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being \|* e3 e5 e7>, where the asterisk can be any integer value, then m = \|* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord \|* e3 e5 e7> then m = \|* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the [[~~The_Seven_Limit_Symmetrical_Lattices~~\|cubic lattice of 7-limit tetrads]].		Suppose m0, m1, m2 and m3 are four [[monzo]]s denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being \|* e3 e5 e7>, where the asterisk can be any integer value, then m = \|* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord \|* e3 e5 e7> then m = \|* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal. Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the [[The Seven Limit Symmetrical Lattices\|cubic lattice of 7-limit tetrads]].

	If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then		If t = [n, [a b c]] is a tablet, define u = n - 7a - 4b - 2c. If a+b+c is even, then
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	==The keenanismic tablet==		==The keenanismic tablet==
	This is based on the five [[~~keenanismic_tetrads\|~~keenanismic tetrads]], which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords [[~~Dyadic_chord~~\|dyadic]]. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.		This is based on the five [[keenanismic tetrads]], which are essentially tempered chords tempering out 385/384, plus the otonal and utonal tetrads. If we take the intervals for an otonal tetrad in close position, 5/4-6/5-7/6-8/7, we can permute them in 24 ways. The number of circular permutations, leading to chords which are not simply transpositions, is six. Keenanismic tempering makes all six chords [[Dyadic chord\|dyadic]]. in addition the steps 8/7-5/4-8/7-14/11 lead to a keenanismic tempered 35/32-5/4-3/2-12/7 chord.

	Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos \|* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to \|* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions:		Taking the product of all four notes of the tetrad and ignoring octaves. we can uniquely associate these chords to no-twos monzos \|* w x y>. If the chord is transposed by some quantity with monzo M, the chord product is changed to \|* x y z> + 4M, so that the no-twos monzos modulo four represent the types of chords. It's convenient to modify this slightly, and represent the chords by [2-x 2-y 2-z]. We then have the following for the mod four reductions:
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	==The pele tablet==		==The pele tablet==
	This is a tablet for the rank 3 13-limit temperament [[~~Hemifamity_family~~#Pele\|pele]]. It is based on the following 71 chords, in the 5-limit transversal:		This is a tablet for the rank 3 13-limit temperament [[Hemifamity family#Pele\|pele]]. It is based on the following 71 chords, in the 5-limit transversal:

	chords = [		chords = [
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	==The meantone add6/9 tablet==		==The meantone add6/9 tablet==
	The meantone add6/9 tablet is based on the [[~~Meantone_add6~~-~~9_pentad~~\|meantone add6/9 pentad]], which can also be called the add2/9 pentad, the meantone pentatonic scale or Meantone[5]. The tablet is ~~extremly~~ simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where		The meantone add6/9 tablet is based on the [[Meantone add6-9 pentad\|meantone add6/9 pentad]], which can also be called the add2/9 pentad, the meantone pentatonic scale or Meantone[5]. The tablet is extremely simple, consiting of an ordered pair [n, c], where we have a meantone transversal for the notes defined by u = n-8c, where

	<ul><li>If u mod 5 = 0 then</li></ul>note(n, c) = \|u/5 c>		<ul><li>If u mod 5 = 0 then</li></ul>note(n, c) = \|u/5 c>
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	<ul><li>If u mod 5 = 4 then</li></ul>note(n, c) = \|(u-4)/5-4 c+3>		<ul><li>If u mod 5 = 4 then</li></ul>note(n, c) = \|(u-4)/5-4 c+3>

	In all cases <5 8\|note(n, c) = n. Tempering ~~the~~ the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.		In all cases <5 8\|note(n, c) = n. Tempering the Pythgorean transversal by flattening 3 gives, as usual, a meantone tuning.

	==The 5et portent tablet==		==The 5et portent tablet==
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	Once again, <5 8 12 14 17\|note(n, c) = c.		Once again, <5 8 12 14 17\|note(n, c) = c.

	The selection of these particular representatives for each of the twelve types of chords is based on each of them having a common ~~triad--three~~ common ~~notes--in~~ common with the utonal pentad, the first chord in the chords list. The chord representatives do not need to be given in terms of a 2.5.7 subgroup transversal, and a more perspicuous way of expressing the same portent temperament pentads is		The selection of these particular representatives for each of the twelve types of chords is based on each of them having a common triad—three common notes—in common with the utonal pentad, the first chord in the chords list. The chord representatives do not need to be given in terms of a 2.5.7 subgroup transversal, and a more perspicuous way of expressing the same portent temperament pentads is

	<ul><li>JI pentads</li></ul>1-9/8-5/4-3/2-7/4		<ul><li>JI pentads</li></ul>1-9/8-5/4-3/2-7/4
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	=The 6et tutone tutonic tablet=		=The 6et tutone tutonic tablet=
	This tablet is based on the [[~~tutonic_sextad\|~~tutonic sextad]], which in terms of the 99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in [[~~Chromatic_pairs~~#Tutone\|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.		This tablet is based on the [[tutonic sextad]], which in terms of the 99/98 (Huygens) version of 11-limit meantone consists of a chain of five tones, followed by an augmented second; in other words a {81/80, 126/125, 99/98}-tempered version of 9/8-9/8-9/8-9/8-9/8-8/7, which in terms of notes rather than steps is a tempered 1-9/8-5/4-7/5-11/7-7/4. Using this chord as the basis for harmony puts one in [[Chromatic pairs#Tutone\|tutone temperament]], a 2.9.7.11 subgroup temperament, and the sextad can be called Tutone[6], the tutone haplotonic scale.

	If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = \|(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19\|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19\|note(n, c) = n, where note(n, c) in subgroup monzo terms is \|(u-i)/6-3i c+i>.		If the tablet is the ordered pair [n, c] and if u = n-19c, then if i = u mod 6, define note(n, c) = \|(u-i)/6-3i 2c+2i>. This gives a 3-limit interval which tempers to a note of tutone satisfying the identity <12 19\|note(n, c) = 2n. We can also express this in terms of a subgroup monzo as <6 19\|note(n, c) = n, where note(n, c) in subgroup monzo terms is \|(u-i)/6-3i c+i>.
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	<ul><li>If u mod 7 = 6, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = \|(u-20)/7 e3 e5 e7+1 e11 e13>		<ul><li>If u mod 7 = 6, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = \|(u-20)/7 e3 e5 e7+1 e11 e13>

	On the other hand if r is odd:		On the other hand, if r is odd:

	<ul><li>If u mod 7 = 0, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = \|u/7 e3 e5 e7 e11 e13>		<ul><li>If u mod 7 = 0, then</li></ul>note(n, [r e3 e5 e7 e11 e13]) = \|u/7 e3 e5 e7 e11 e13>
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	=The orwell nonad tablet=		=The orwell nonad tablet=
	The [[~~orwell_tetrad~~\|orwell nonad]] is the Orwell[9] MOS, considered as an essentially tempered dyadic chord (which it is, in the 15-limit.) While nine notes is a lot of notes for a chord, obviously in practice subchords could be used.		The [[orwell tetrad\|orwell nonad]] is the Orwell[9] MOS, considered as an essentially tempered dyadic chord (which it is, in the 15-limit.) While nine notes is a lot of notes for a chord, obviously in practice subchords could be used.

	If the tablet is the ordered pair [n, c] and if u = n-11c, then setting i = u mod 9, define a transversal for note(n, c) by determining if i is even or odd, and setting		If the tablet is the ordered pair [n, c] and if u = n-11c, then setting i = u mod 9, define a transversal for note(n, c) by determining if i is even or odd, and setting
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	if i is odd. We then have <9 14 21 25\|note(n, c) = n.		if i is odd. We then have <9 14 21 25\|note(n, c) = n.
	[[Category:~~chord~~]]		[[Category:Chords]]
	[[Category:~~composition~~]]		[[Category:Composition]]
	[[Category:~~notation~~]]		[[Category:Notation]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

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