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User:Ganaram inukshuk/Notes: Difference between revisions

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→Proposal: strict and weak definitions for a chromatic pair: Added stipulations, per talk page for chromatic pairs; definition (or at least my attempt at such) is still wip
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→Nk note name notation: N(k) is denoted like a function
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=== Nk note name notation ===
=== N(k) note name notation (work-in-progress) ===
Rather than using alphabetical names, notes of the form Nk are used. These are used to indicate position on a staff, where N0 is middle C, unless specified otherwise. These names serve as an alternative to using different notations for different scales, but may be interpreted as blanks for one to fill in with different, more specific notation. If k is unbounded, then this notation denotes position on a staff. However, k may be bounded within the range [0, n), where n is the note count, to indicate pitch classes.
Rather than using alphabetical names, notes of the form N(k) are used. These are used to indicate position on a staff, where N(0) is the root. These names serve as an alternative to using different notations for different scales, but may be interpreted as blanks for one to fill in with different, more specific notation. If k is unbounded, then this notation denotes position on a staff. However, k may be bounded within the range [0, n), where n is the note count, to indicate pitch classes.


For a given mos xL ys, note names are based on a mode u|p; the choice of mode is up to the user. Starting at N0, or 0-mosstep, successive pitch classes are named N1, N2, and so on. If two pitches, reached by going up or down some quantity of mossteps, have the same remainder when divided by xL+ys (which is the same as octave-reducing), then they are in the same pitch class.
For a given mos xL ys, note names are based on a mode u|p; the choice of mode is up to the user. Starting at the root of N(0), successive pitch classes are named N(1), N(2), and so on. If note names are given and assuming N(0) is the root, then N(k) can be thought of as a function that returns an unaltered note name corresponding to the k-mosdegree of a mos xL ys in the mode u|p. In standard notation, N(0) is C, N(1), is D, and so on. Since this is cyclical, N(7) and N(0) are both the same value of C.
 
If two pitches, reached by going up or down some quantity of mossteps, have the same remainder when divided by xL+ys (which is the same as octave-reducing), then they are in the same pitch class.
{| class="wikitable"
{| class="wikitable"
|+Example for 5L 2s (LLsLLLs, mode 5|Example with standard notation (5L 2s, mode 5|1)
|+ Example for 5L 2s (LLsLLLs, mode 5 |Example with standard notation (5L 2s, mode 5|1)
!Mossteps from root
!Mossteps from root
!Substring
!Substring
Line 1,493: Line 1,495:
|0
|0
|C
|C
|N0
|N(0)
|-
|-
|1
|1
Line 1,499: Line 1,501:
|L
|L
|D
|D
|N1
|N(1)
|-
|-
|2
|2
Line 1,505: Line 1,507:
|2L
|2L
|E
|E
|N2
|N(2)
|-
|-
|3
|3
Line 1,511: Line 1,513:
|2L+s
|2L+s
|F
|F
|N3
|N(3)
|-
|-
|4
|4
Line 1,517: Line 1,519:
|3L+s
|3L+s
|G
|G
|N4
|N(4)
|-
|-
|5
|5
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|4L+s
|4L+s
|A
|A
|N5
|N(5)
|-
|-
|6
|6
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|5L+s
|5L+s
|B
|B
|N6
|N(6)
|-
|-
|7
|7
Line 1,535: Line 1,537:
|5L+2s
|5L+2s
|C
|C
|N7 (same as N0)
|N(7) (same as N(0))
|}
|}
Chromas are denoted using the letter c, and are expressed as a multiple of c being added (or subtracted) from a note Nk. Half-accidentals are denoted as fractions (such as c/2) or decimals (such as 0.5c). Dieses, if present, are expressed similarly using the letter d. If this notation denotes position on a staff, then chromas and dieses don't change position on a staff, but modify the pitch at that position. If this notation is treated as placeholders for more specific notation, then adding or subtracting c represents the use of sharp or flat (or equivalent) accidentals.
Chromas are denoted using the letter c, and are expressed as a multiple of c being added (or subtracted) from a note N(k). Half-accidentals are denoted as fractions (such as c/2) or decimals (such as 0.5c). Dieses, if present, are expressed similarly using the letter d. If this notation denotes position on a staff, then chromas and dieses don't change position on a staff, but modify the pitch at that position. If this notation is treated as placeholders for more specific notation, then adding or subtracting c represents the use of sharp or flat (or equivalent) accidentals.


Since chromas and dieses can be expressed in terms of L and s – where a chroma is L - s and a diesis is the absolute value of L - 2s – modifying a note by a chroma or diesis can equivalently expressed as going up (or down) some interval iL+js. If, for a given step ratio L:s, two pitch classes Np and Nq are modified by different amounts of chromas uc and vc to produce pitch classes Np+uc and Nq+vc, if dividing both by xL+ys produces the same remainder, then the two pitches are enharmonic equivalents.
Since chromas and dieses can be expressed in terms of L and s – where a chroma is L - s and a diesis is the absolute value of L - 2s – modifying a note by a chroma or diesis can equivalently expressed as going up (or down) some interval iL+js. If, for a given step ratio L:s, two pitch classes Np and Nq are modified by different amounts of chromas uc and vc to produce pitch classes N(p)+uc and N(q)+vc, if dividing both by xL+ys produces the same remainder, then the two pitches are enharmonic equivalents.


As an example, the table below denotes diatonic (5L 2s) pitch classes as sums of L's and s's, and shows how different step ratios produce different enharmonic equivalences; namely, in 12edo, C# and Db are equivalent, but in 19edo, C# and Db are not equivalent but B# and Cb are equivalent.
As an example, the table below denotes diatonic (5L 2s) pitch classes as sums of L's and s's, and shows how different step ratios produce different enharmonic equivalences; namely, in 12edo, C# and Db are equivalent, but in 19edo, C# and Db are not equivalent but B# and Cb are equivalent.
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|+Examples with standard diatonic notation
|+Examples with standard diatonic notation
!Note name
!Note name
!Nk note name with chroma
!N(k) note name with chroma
!Mosstep sum
!Mosstep sum
!Like terms combined
!Like terms combined
Line 1,552: Line 1,554:
|-
|-
|C
|C
|N0
|N(0)
|0
|0
|0
|0
Line 1,559: Line 1,561:
|-
|-
|C#
|C#
|N0+c
|N(0)+c
|L-s
|L-s
|L-s
|L-s
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|-
|-
|Db
|Db
|N1-c
|N(1)-c
|L-(L-s)
|L-(L-s)
|s
|s
Line 1,573: Line 1,575:
|-
|-
|D
|D
|N1
|N(1)
|L
|L
|L
|L
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|-
|-
|B
|B
|N6
|N(6)
|5L+s
|5L+s
|5L+s
|5L+s
Line 1,587: Line 1,589:
|-
|-
|B#
|B#
|N6+c
|N(6)+c
|5L+s+(L-s)
|5L+s+(L-s)
|6L
|6L
Line 1,594: Line 1,596:
|-
|-
|Cb
|Cb
|N7-c
|N(7)-c
|5L+2s-(L-s)
|5L+2s-(L-s)
|4L+3s
|4L+3s
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|-
|-
|C (one octave up)
|C (one octave up)
|N7 (same as N0, as a pitch class)
|N(7) (same as N(0), as a pitch class)
|5L+2s (reduced to 0 due to modular arithmetic)
|5L+2s (reduced to 0 due to modular arithmetic)
|5L+2s (reduced to 0)
|5L+2s (reduced to 0)
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