User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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=== Neutrals ===

N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds and ~~whitewood~~ temperaments, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.

N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds and Whitewood temperaments, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.

Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.

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=== Intermediates ===

To provide native support for Semaphore, Pajara and Injera, intermediates are also added to the system.

To provide native support for Semaphore, Pajara and Injera, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. In 'lists of all edos and MOS Scales', all edos and scales described that include intermediates are given alternative primary interval names leaving them out.

‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixths-seventh’ or ‘sinth’.

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‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixths’.

5edo can be ~~spelt~~ with the list of only these intermediates:

5edo can be spelled with the list of only these intermediates:

1-2 2-3 3-4 5-6 6-7 7-8.

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== How to ==

Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide:.

Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.

For ''n-''edo:

# Label P1 = 1, P8 = ''n''

# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).

#* P5 = round(ln(3/2)/ln(''n'')) steps

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#* M7 = (5*P5) mod ''n''

#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)

# ~~Check for~~ the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).

# Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).

#* If A is even, then intervals half way between M and m may be labelled N

# The initial chromatic intervals may be labelled:

# ~~Check for~~ the size of meantone and septimal comma alterations

#* A1 = A

#* ~~small/supra~~ = M3 - sM3. If ~~small/super =~~ 0 then 81/80 is tempered out, the edo ~~supports meantone~~ temperament and alterations of 81/80 are not needed.

#* d8 = P8 - A

#* ~~sub/super~~ = m7 - sm7. If ~~sub/super =~~ 0 then 64/63 is tempered out, the edo ~~supports superpyth~~ temperament, and alterations of 64/63 are not needed.

#* A4 = P4 + A

# ~~Name any unnamed~~ intervals by

#* d5 = P5 - A

# If A is even, then intervals half way between M and m within a degree may be labelled N.

# Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)

#* meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.

#* septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.

# If sub/super < 0, alterations of 64/63 should not be applied.

# If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.

# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:

#* sM (small major) = M - meantone comma

#* Sm (supra minor) = m + meantone comma

#* SM (super major) = M + septimal comma

#* sm (sub minor) = m - septimal comma

# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:

#* S4 (supra 4th) = P4 + meantone comma

#* s5 (small 5th) = P5 - meantone comma

#* s4 (sub 4th) = P4 - septimal comma

#* S5 (super 5th) = P5 + septimal comma

# If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.

# Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.

# If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.

# 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.

# If desired and if ''n'' is even, the interval subtended by ''n''/2 steps may be labelled 4-5.

# If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.

#

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=== Primary interval names for edos ===

2edo: P1 P4/4-5/P5 P8

2edo: P1 P4/4-5/P5 P8 (P1 P4/P5 P8)

3edo: P1 3-4/P4 P5/5-6 P8

3edo: P1 3-4/P4 P5/5-6 P8 (P1 P4 P5 P8)

4edo: P1 2-3 P4/4-5/P5 6-7 P8

4edo: P1 2-3 P4/4-5/P5 6-7 P8 (P1 SM2/sm3 P4/P5 SM6/sm7 P8)

5edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8

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 === Neutrals ===
-N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds and whitewood temperaments, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.
+N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds and Whitewood temperaments, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.
 Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.
@@ Line 98: / Line 98: @@
 === Intermediates ===
-To provide native support for Semaphore, Pajara and Injera, intermediates are also added to the system.
+To provide native support for Semaphore, Pajara and Injera, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. In 'lists of all edos and MOS Scales', all edos and scales described that include intermediates are given alternative primary interval names leaving them out.
 ‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixths-seventh’ or ‘sinth’.
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 ‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixths’.
-edo can be spelt with the list of only these intermediates:
+edo can be spelled with the list of only these intermediates:
 -2 2-3 3-4 5-6 6-7 7-8.
@@ Line 468: / Line 468: @@
 == How to ==
-Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide:.
+Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.
 For ''n-''edo:
+# Label P1 = 1, P8 = ''n''
 # Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).
 #* P5 = round(ln(3/2)/ln(''n'')) steps
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 #* M7 = (5*P5) mod ''n''
 #* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)
-# Check for the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
+# Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
-#* If A is even, then intervals half way between M and m may be labelled N
+# The initial chromatic intervals may be labelled:
-# Check for the size of meantone and septimal comma alterations
+#* A1 = A
-#* small/supra = M3 - sM3. If small/super = 0 then 81/80 is tempered out, the edo supports meantone temperament and alterations of 81/80 are not needed.
+#* d8 = P8 - A
-#* sub/super = m7 - sm7. If sub/super = 0 then 64/63 is tempered out, the edo supports superpyth temperament, and alterations of 64/63 are not needed.
+#* A4 = P4 + A
-# Name any unnamed intervals by
+#* d5 = P5 - A
+# If A is even, then intervals half way between M and m within a degree may be labelled N.
+# Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)
+#* meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.
+#* septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.
+# If sub/super < 0, alterations of 64/63 should not be applied.
+# If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
+# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:
+#* sM (small major) = M - meantone comma
+#* Sm (supra minor) = m + meantone comma
+#* SM (super major) = M + septimal comma
+#* sm (sub minor)  = m - septimal comma
+# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:
+#* S4 (supra 4th) = P4 + meantone comma
+#* s5 (small 5th) = P5 - meantone comma
+#* s4 (sub 4th) = P4 - septimal comma
+#* S5 (super 5th) = P5 + septimal comma
+# If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.
+# Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
+# If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
+# 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
+# If desired and if ''n'' is even, the interval subtended by ''n''/2 steps may be labelled 4-5.
+# If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.
 #
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 === Primary interval names for edos ===
-edo: P1 P4/4-5/P5 P8
+edo: P1 P4/4-5/P5 P8 (P1 P4/P5 P8)
-edo: P1 3-4/P4 P5/5-6 P8
+edo: P1 3-4/P4 P5/5-6 P8 (P1 P4 P5 P8)
-edo: P1 2-3 P4/4-5/P5 6-7 P8
+edo: P1 2-3 P4/4-5/P5 6-7 P8 (P1 SM2/sm3 P4/P5 SM6/sm7 P8)
 edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8