Ledzo notation: Difference between revisions

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(And if you are using a prime higher than 101, either something's '''seriously gone wrong''' or this notation probably '''won't be useful''' for your use case anyway.)

Intervals with no '''a''' specified are assumed to be octave-reduced, you can specify '''0a''' if the distinction is significant. Due to this, octave complements can be specified by flipping the signs of the entries for each prime. For example, the major third is '''c''', while the minor sixth is '''-c'''.

'''LEGACY:''' <s>Intervals with no '''a''' specified are assumed to be octave-reduced, you can specify '''0a''' if the distinction is significant. Due to this, octave complements can be specified by flipping the signs of the entries for each prime. For example, the major third is '''c''', while the minor sixth is '''-c'''.</s>

Reduction of intervals is no longer assumed.

Octave complements can be specified by flipping the signs of the entries for each prime, and adding 1 to the '''a''' entry. For example, the major third is -2ac, while the minor sixth is 3a-c.

To provide a sense of the interval's size, ledzo notation can be paired with [[User:VectorGraphics/Walker brightness notation|'''WBN''']] or another system for specifying interval regions in order to give a sense of the size of the interval.

In order to disambiguate from other notation systems, letters are always written '''lowercase''' (so that, say, '''a''', when written, isn't confused for the note '''A''').

Additionally, if the ledzo for an interval is a single letter with no number or minus sign, it takes a 1 before it, so that when spoken, ledzos aren't confused for note names.

== Examples ==

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|[[2/1]]

|[1⟩

|a octave

|1a octave

|-

|[[3/2]]

|[-1 1⟩

|b fifth

| -ab fifth

|-

|[[4/3]]

|[2 -1⟩

| -b fourth

| 2a-b fourth

|-

|[[5/4]]

|[-2 0 1⟩

|c third

| -2ac third

|-

|[[8/5]]

|[3 0 -1⟩

| -c sixth

| 3a-c sixth

|-

|[[6/5]]

|[1 1 -1⟩

|b-c third

|ab-c third

|-

|[[5/3]]

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|[[7/6]]

|[-1 -1 0 1⟩

| -bd third

| -a-bd third

|-

|[[9/7]]

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|[[14/9]]

|[1 -2 0 1⟩

| -2bd sixth

| a-2bd sixth

|-

|[[7/5]]

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|[[10/7]]

|[1 0 1 -1⟩

|c-d tritone

|ac-d tritone

|-

|[[81/80]]

|[-4 4 -1⟩

|4b-c comma

| -4a4b-c comma

|-

|[[64/63]]

|[6 -2 0 -1⟩

| -2b-d comma

| 6a-2b-d comma

|-

|[[531441/524288]]

|[-19 12⟩

|~~12b~~ comma

| -19a12b comma

|-

|[[3/1]]

|[0 1⟩

|~~0ab~~ tritave

|1b tritave

|}

== [[Interval arithmetic]] ==

Ledzo notation supports a form of interval arithmetic natively, since intervals are stacked just by adding their monzos together. For example, a major third plus a minor third is a perfect fifth, and that's represented as follows: [-2 0 1⟩ * [1 1 -1⟩ = [-1 1 0⟩. This translates intuitively into the ledzo: c + b-c = b, analogously for the octave complements: -c + -bc = -b (note that concatenation here does not mean multiplication in the logarithmic scale this is using, positive entries have a hidden plus sign that is not written!)

Ledzo notation supports a form of interval arithmetic natively, since intervals are stacked just by adding their monzos together. For example, a major third plus a minor third is a perfect fifth, and that's represented as follows: [-2 0 1⟩ * [1 1 -1⟩ = [-1 1 0⟩. This translates intuitively into the ledzo: -2ac + ab-c = -ab, analogously for the octave complements: -c + -bc = -b (note that concatenation here does not mean multiplication in the logarithmic scale this is using, positive entries have a hidden plus sign that is not written!)

== Regular temperaments ==

[[Regular temperaments]] can be notated by taking a, b, c, etc to stand for the [[Majestazic system|tempered primes]] used in the temperament.

@@ Line 33: / Line 33: @@
 (And if you are using a prime higher than 101, either something's '''seriously gone wrong''' or this notation probably '''won't be useful''' for your use case anyway.)
-Intervals with no '''a''' specified are assumed to be octave-reduced, you can specify '''0a''' if the distinction is significant. Due to this, octave complements can be specified by flipping the signs of the entries for each prime. For example, the major third is '''c''', while the minor sixth is '''-c'''.
+'''LEGACY:''' <s>Intervals with no '''a''' specified are assumed to be octave-reduced, you can specify '''0a''' if the distinction is significant. Due to this, octave complements can be specified by flipping the signs of the entries for each prime. For example, the major third is '''c''', while the minor sixth is '''-c'''.</s>
+Reduction of intervals is no longer assumed.
+Octave complements can be specified by flipping the signs of the entries for each prime, and adding 1 to the '''a''' entry. For example, the major third is -2ac, while the minor sixth is 3a-c.
 To provide a sense of the interval's size, ledzo notation can be paired with [[User:VectorGraphics/Walker brightness notation|'''WBN''']] or another system for specifying interval regions in order to give a sense of the size of the interval.
 In order to disambiguate from other notation systems, letters are always written '''lowercase''' (so that, say, '''a''', when written, isn't confused for the note '''A''').
+Additionally, if the ledzo for an interval is a single letter with no number or minus sign, it takes a 1 before it, so that when spoken, ledzos aren't confused for note names.
 == Examples ==
@@ Line 48: / Line 54: @@
 |[[2/1]]
 |[1⟩
-|a octave
+|1a octave
 |-
 |[[3/2]]
 |[-1 1⟩
-|b fifth
+| -ab fifth
 |-
 |[[4/3]]
 |[2 -1⟩
-| -b fourth
+| 2a-b fourth
 |-
 |[[5/4]]
 |[-2 0 1⟩
-|c third
+| -2ac third
 |-
 |[[8/5]]
 |[3 0 -1⟩
-| -c sixth
+| 3a-c sixth
 |-
 |[[6/5]]
 |[1 1 -1⟩
-|b-c third
+|ab-c third
 |-
 |[[5/3]]
@@ Line 76: / Line 82: @@
 |[[7/6]]
 |[-1 -1 0 1⟩
-| -bd third
+|  -a-bd third
 |-
 |[[9/7]]
@@ Line 84: / Line 90: @@
 |[[14/9]]
 |[1 -2 0 1⟩
-| -2bd sixth
+| a-2bd sixth
 |-
 |[[7/5]]
@@ Line 92: / Line 98: @@
 |[[10/7]]
 |[1 0 1 -1⟩
-|c-d tritone
+|ac-d tritone
 |-
 |[[81/80]]
 |[-4 4 -1⟩
-|4b-c comma
+| -4a4b-c comma
 |-
 |[[64/63]]
 |[6 -2 0 -1⟩
-| -2b-d comma
+| 6a-2b-d comma
 |-
 |[[531441/524288]]
 |[-19 12⟩
-|12b comma
+| -19a12b comma
 |-
 |[[3/1]]
 |[0 1⟩
-|0ab tritave
+|1b tritave
 |}
 == [[Interval arithmetic]] ==
-Ledzo notation supports a form of interval arithmetic natively, since intervals are stacked just by adding their monzos together. For example, a major third plus a minor third is a perfect fifth, and that's represented as follows: [-2 0 1⟩ * [1 1 -1⟩ = [-1 1 0⟩. This translates intuitively into the ledzo: c + b-c = b, analogously for the octave complements: -c + -bc = -b (note that concatenation here does not mean multiplication in the logarithmic scale this is using, positive entries have a hidden plus sign that is not written!)
+Ledzo notation supports a form of interval arithmetic natively, since intervals are stacked just by adding their monzos together. For example, a major third plus a minor third is a perfect fifth, and that's represented as follows: [-2 0 1⟩ * [1 1 -1⟩ = [-1 1 0⟩. This translates intuitively into the ledzo: -2ac + ab-c = -ab, analogously for the octave complements: -c + -bc = -b (note that concatenation here does not mean multiplication in the logarithmic scale this is using, positive entries have a hidden plus sign that is not written!)
 == Regular temperaments ==
 [[Regular temperaments]] can be notated by taking a, b, c, etc to stand for the [[Majestazic system|tempered primes]] used in the temperament.