Rank-3 scale: Difference between revisions

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MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.

MOS scales are ''mirror-symmetric'', or ''achiral'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode''. The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.

MOS scales are ''mirror-symmetric'', or ''achiral'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern '''LsLLLsL'''. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode''. The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern '''sLssLssLss'''. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.

MOS scales and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.

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[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has ''exactly'' three sizes. SV3 scales are also called '''trivalent'''.

'''Conjecture:''' For all odd-cardinality SV3 scales apart from the scales ''abacaba'', and its repetitions ''abacabaabacaba'' etc., at least two of the three steps must occur the same number of times.

'''Conjecture:''' For all odd-cardinality SV3 scales apart from the scales '''''abacaba''''', and its repetitions '''''abacabaabacaba''''' etc., at least two of the three steps must occur the same number of times.

All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is ''abac''.

All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is '''''abac'''''.

'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form ''a…ba…c'' (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form '''''a…ba…c''''' (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

== Product words ==

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When associated with a mapping, product words are the rank-3 ''[[Fokker blocks]]''. Fokker blocks have ''unison vectors'', which generalize the concept of the chroma of MOS scales to higher ranks. If these intervals are plotted onto a plane representing rank-3 octave equivalent pitch space, they tile the space into Fokker blocks which differ by combinations of these unison vectors. Rank-2 Fokker blocks are the MOS scales, so Fokker blocks can be considered a generalization of MOS scales into higher ranks.

Product words have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as ''L'', ''l'', ''S'', ''s'', and they satisfy ''L'' - ''l'' = ''S'' - ''s''.

Product words have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''.

Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.

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When mappings are considered, PWF scales are rank-3 ''[[Gallery of wakalixes|wakalix]]es'' - Fokker blocks which are Fokker blocks in more than one way.

Not all SV3 scales are PWF. Only a single scale - ''abcba'' - is SV3 and not PWF.

Not all SV3 scales are PWF. Only a single scale - '''''abcba''''' - is SV3 and not PWF.

Only a single PWF scale is mirror-symmetric - ''abacaba''.

Only a single PWF scale is mirror-symmetric - '''''abacaba'''''.

Apart from ''abacaba'', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.

Apart from '''''abacaba''''', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.

PWF scales can only have odd numbers of notes.

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Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.

PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from ''abacaba''.

PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from '''''abacaba'''''.

The scale ''abacaba'' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric SV3 scale.

The scale '''''abacaba''''' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric SV3 scale.

The scales ''a…ba…c'', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.

The scales '''''a…ba…c''''', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.

There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales.

== 3-SN scales ==

The scales ''~~a…ba…c~~'' and ''abacaba'' are [[step-nested scale|step-nested]] (SN) scales, which are mirror-symmetric, and can be uniquely defined by a signature.

The scales '''''a'''…'''ba'''…'''c''''' and '''''abacaba''''' are [[step-nested scale|step-nested]] (SN) scales, which are mirror-symmetric, and can be uniquely defined by a signature.

SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.

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3-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.

'''Conjecture''': The only SN scale that is SV3 is ''abacaba''.

'''Conjecture 1''': The only SN scale that is SV3 is '''''abacaba'''''.

'''Conjecture''': The only SN scales that are MV3 are ''abacaba'', and scales of the form ''~~a…ba…c~~''.

'''Conjecture 2''': The only SN scales that are MV3 are '''''abacaba''''', and scales of the form '''''a'''…'''ba'''…'''c'''''.

'''Conjecture''': The only SN scales that are MV3, and have mean variety < 3 are those of the form ''~~a…ba…c~~''.

'''Conjecture 3''': The only SN scales that are MV3, and have mean variety < 3 are those of the form '''''a'''…'''ba'''…'''c'''''. This follows from '''Conjecture 1''' and '''2'''.

'''Conjecture''': ~~The~~ only SN scales that are [[Balanced word|balanced]] are the ''~~Power SNS~~''~~, which~~ are ~~equivalent to~~ the [[Fraenkel word|Fraenkel ~~words~~]].

'''Conjecture 4''': '''T'''he only 3-SN scales that are [[Balanced word|balanced]] are '''''abacaba''''', and scales of the form '''''a…ba…c'''''. Given that scales of the form '''''a…ba…c''''' are balanced (proof of this is left as an exercise for the reader), this follows from [[Fraenkel word|Fraenkel's conjecture]], and '''Conjecture 2.'''

See [[Gallery of 3-SN scales]] for examples of 3-SN scales.

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== Theorems, Proofs and Conjectures on 3-SN scales ==

'''Theorem:''' Scales of the form ''a...ba...c'' have mean variety (3''N''-4) / (''N''-1).

'''Theorem:''' Scales of the form '''''a...ba...c''''' have mean variety (3''N''-4) / (''N''-1).

'''Proof:'''

Since there are three step sizes, ''a'', ''b'', and ''c'', interval class ''N'' has variety 3.

Since there are three step sizes, '''''a''''', '''''b''''', and '''''c''''', interval class ''N'' has variety 3.

Scale segments of length 1 ≤ length ≤ ''N''/2-1 comprise either all ''a''’s, all ''a''’s but for a single ''b'', or all ''a''’s but for a single ''c'', and therefore interval classes of length 1 ≤ length ≤ ''N''/2-1 have variety 3. Interval classes of length ''N''/2+1 ≤ length ≤ ''N''-1 also have variety 3 by symmetry (given that scale segments of length ''N''/2+1 ≤ length ≤ ''N''-1 are the complement of scale segments of length 1 ≤ length ≤ ''N''/2-1.

Scale segments of length 1 ≤ length ≤ ''N''/2-1 comprise either all '''''a'''''’s, all '''''a'''''’s but for a single '''''b''''', or all '''''a'''''’s but for a single '''''c''''', and therefore interval classes of length 1 ≤ length ≤ ''N''/2-1 have variety 3. Interval classes of length ''N''/2+1 ≤ length ≤ ''N''-1 also have variety 3 by symmetry (given that scale segments of length ''N''/2+1 ≤ length ≤ ''N''-1 are the complement of scale segments of length 1 ≤ length ≤ ''N''/2-1.

Finally, scale segments of length ''N''/2 contain all a’s but for one ''b'', or all ''a''’s but for one ''c'', and so interval class ''N''/2 has variety 2.

Finally, scale segments of length ''N''/2 contain all a’s but for one '''''b''''', or all '''''a'''''’s but for one '''''c''''', and so interval class ''N''/2 has variety 2.

The total variety of the scale is then 2+(''N''-2)*3 = 3''N''-4, and the mean variety of the scale is (3''N''-4) / (''N''-1).

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Then the total number of specific intervals in X is (''N''/2-1)*2 + 2*3 + (''N''/2-2)*4 = 6+''N''-2+2''N''-8 = 3''N''-4, and the mean variety = (3''N''-4) / (''N''-1)

'''Conjecture:''' SN scales only of the form ''a…ba…c'', or generated by a single instance of a third ~~gen~~ at the top or bottom of each step of a WF scale, have mean variety < 3.

'''Conjecture:''' SN scales only of the form '''''a…ba…c''''', or generated by a single instance of a third generator at the top or bottom of each step of a WF scale (SN [[Flought scale|flought scales]]) have mean variety < 3.

'''Conjecture:''' Scales of the form ''a...ba...ba...c'' have mean variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).

'''Conjecture:''' Scales of the form '''''a...ba...ba...c''''' have mean variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).

'''Conjecture:''' Scales with 2 instances of a generator added to a WF scale have mean variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6) / (''N''-1)

'''Conjecture:''' Scales with 2 instances of a generator added to a WF scale have mean variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6) / (''N''-1).

'''Conjecture:''' ''abacaba'' and ''aabaabaac'' are the only SN scales with mean variety = 3.

'''Conjecture:''' '''''abacaba''''' and '''''aabaabaac''''' are the only SN scales with mean variety = 3.

'''Conjecture''': The only SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]], and SNS wherein two step sizes occur only once.

[[Category:Rank-3 scales| ]]

[[Category:Rank 3]]

[[Category:Pages with open problems]]

@@ Line 12: / Line 12: @@
 MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.
-MOS scales are ''mirror-symmetric'', or ''achiral'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode''. The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.
+MOS scales are ''mirror-symmetric'', or ''achiral'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern '''LsLLLsL'''. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode''. The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern '''sLssLssLss'''. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.
 MOS scales and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.
@@ Line 59: / Line 59: @@
 [[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has ''exactly'' three sizes. SV3 scales are also called '''trivalent'''.
-'''Conjecture:''' For all odd-cardinality SV3 scales apart from the scales ''abacaba'', and its repetitions ''abacabaabacaba'' etc., at least two of the three steps must occur the same number of times.
+'''Conjecture:''' For all odd-cardinality SV3 scales apart from the scales '''''abacaba''''', and its repetitions '''''abacabaabacaba''''' etc., at least two of the three steps must occur the same number of times.
-All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is ''abac''.
+All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is '''''abac'''''.
-'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form ''a…ba…c'' (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
+'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form '''''a…ba…c''''' (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
 == Product words ==
@@ Line 70: / Line 70: @@
 When associated with a mapping, product words are the rank-3 ''[[Fokker blocks]]''. Fokker blocks have ''unison vectors'', which generalize the concept of the chroma of MOS scales to higher ranks. If these intervals are plotted onto a plane representing rank-3 octave equivalent pitch space, they tile the space into Fokker blocks which differ by combinations of these unison vectors. Rank-2 Fokker blocks are the MOS scales, so Fokker blocks can be considered a generalization of MOS scales into higher ranks.
-Product words have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as ''L'', ''l'', ''S'', ''s'', and they satisfy ''L'' - ''l'' = ''S'' - ''s''.
+Product words have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''.
 Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.
@@ Line 83: / Line 83: @@
 When mappings are considered, PWF scales are rank-3 ''[[Gallery of wakalixes|wakalix]]es'' - Fokker blocks which are Fokker blocks in more than one way.
-Not all SV3 scales are PWF. Only a single scale - ''abcba'' - is SV3 and not PWF.
+Not all SV3 scales are PWF. Only a single scale - '''''abcba''''' - is SV3 and not PWF.
-Only a single PWF scale is mirror-symmetric - ''abacaba''.
+Only a single PWF scale is mirror-symmetric - '''''abacaba'''''.
-Apart from ''abacaba'', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.
+Apart from '''''abacaba''''', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.
 PWF scales can only have odd numbers of notes.
@@ Line 96: / Line 96: @@
 Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.
-PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from ''abacaba''.
+PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from '''''abacaba'''''.
-The scale ''abacaba'' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric SV3 scale.
+The scale '''''abacaba''''' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric SV3 scale.
-The scales ''a…ba…c'', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.
+The scales '''''a…ba…c''''', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.
 There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales.
 == 3-SN scales ==
-The scales ''a…ba…c'' and ''abacaba'' are [[step-nested scale|step-nested]] (SN) scales, which are mirror-symmetric, and can be uniquely defined by a signature.
+The scales '''''a'''…'''ba'''…'''c''''' and '''''abacaba''''' are [[step-nested scale|step-nested]] (SN) scales, which are mirror-symmetric, and can be uniquely defined by a signature.
 SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.
@@ Line 113: / Line 113: @@
 -SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.
-'''Conjecture''': The only SN scale that is SV3 is ''abacaba''.
+'''Conjecture 1''': The only SN scale that is SV3 is '''''abacaba'''''.
-'''Conjecture''': The only SN scales that are MV3 are ''abacaba'', and scales of the form ''a…ba…c''.
+'''Conjecture 2''': The only SN scales that are MV3 are '''''abacaba''''', and scales of the form '''''a'''…'''ba'''…'''c'''''.
-'''Conjecture''': The only SN scales that are MV3, and have mean variety < 3 are those of the form ''a…ba…c''.
+'''Conjecture 3''': The only SN scales that are MV3, and have mean variety < 3 are those of the form '''''a'''…'''ba'''…'''c'''''. This follows from '''Conjecture 1''' and '''2'''.
-'''Conjecture''': The only SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]].
+'''Conjecture 4''': '''T'''he only 3-SN scales that are [[Balanced word|balanced]] are '''''abacaba''''', and scales of the form '''''a…ba…c'''''. Given that scales of the form '''''a…ba…c''''' are balanced (proof of this is left as an exercise for the reader), this follows from [[Fraenkel word|Fraenkel's conjecture]], and '''Conjecture 2.'''
 See [[Gallery of 3-SN scales]] for examples of 3-SN scales.
@@ Line 129: / Line 129: @@
 == Theorems, Proofs and Conjectures on 3-SN scales ==
-'''Theorem:''' Scales of the form ''a...ba...c'' have mean variety (3''N''-4) / (''N''-1).
+'''Theorem:''' Scales of the form '''''a...ba...c''''' have mean variety (3''N''-4) / (''N''-1).
 '''Proof:'''
-Since there are three step sizes, ''a'', ''b'', and ''c'', interval class ''N'' has variety 3.
+Since there are three step sizes, '''''a''''', '''''b''''', and '''''c''''', interval class ''N'' has variety 3.
-Scale segments of length 1 ≤ length ≤ ''N''/2-1 comprise either all ''a''’s, all ''a''’s but for a single ''b'', or all ''a''’s but for a single ''c'', and therefore interval classes of length 1 ≤ length ≤ ''N''/2-1 have variety 3. Interval classes of length ''N''/2+1 ≤ length ≤ ''N''-1 also have variety 3 by symmetry (given that scale segments of length ''N''/2+1 ≤ length ≤ ''N''-1 are the complement of scale segments of length 1 ≤ length ≤ ''N''/2-1.
+Scale segments of length 1 ≤ length ≤ ''N''/2-1 comprise either all '''''a'''''’s, all '''''a'''''’s but for a single '''''b''''', or all '''''a'''''’s but for a single '''''c''''', and therefore interval classes of length 1 ≤ length ≤ ''N''/2-1 have variety 3. Interval classes of length ''N''/2+1 ≤ length ≤ ''N''-1 also have variety 3 by symmetry (given that scale segments of length ''N''/2+1 ≤ length ≤ ''N''-1 are the complement of scale segments of length 1 ≤ length ≤ ''N''/2-1.
-Finally, scale segments of length ''N''/2 contain all a’s but for one ''b'', or all ''a''’s but for one ''c'', and so interval class ''N''/2 has variety 2.
+Finally, scale segments of length ''N''/2 contain all a’s but for one '''''b''''', or all '''''a'''''’s but for one '''''c''''', and so interval class ''N''/2 has variety 2.
 The total variety of the scale is then 2+(''N''-2)*3 = 3''N''-4, and the mean variety of the scale is (3''N''-4) / (''N''-1).
@@ Line 161: / Line 161: @@
 Then the total number of specific intervals in X is (''N''/2-1)*2 + 2*3 + (''N''/2-2)*4 = 6+''N''-2+2''N''-8 = 3''N''-4, and the mean variety = (3''N''-4) / (''N''-1)
-'''Conjecture:''' SN scales only of the form ''a…ba…c'', or generated by a single instance of a third gen at the top or bottom of each step of a WF scale, have mean variety < 3.
+'''Conjecture:''' SN scales only of the form '''''a…ba…c''''', or generated by a single instance of a third generator at the top or bottom of each step of a WF scale (SN [[Flought scale|flought scales]]) have mean variety < 3.
-'''Conjecture:''' Scales of the form ''a...ba...ba...c'' have mean variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).
+'''Conjecture:''' Scales of the form '''''a...ba...ba...c''''' have mean variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).
-'''Conjecture:''' Scales with 2 instances of a generator added to a WF scale have mean variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6) / (''N''-1)
+'''Conjecture:''' Scales with 2 instances of a generator added to a WF scale have mean variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6) / (''N''-1).
-'''Conjecture:''' ''abacaba'' and ''aabaabaac'' are the only SN scales with mean variety = 3.
+'''Conjecture:''' '''''abacaba''''' and '''''aabaabaac''''' are the only SN scales with mean variety = 3.
+'''Conjecture''': The only SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]], and SNS wherein two step sizes occur only once.
 [[Category:Rank-3 scales| ]] <!--main article-->
 [[Category:Rank 3]]
 [[Category:Pages with open problems]]