Kite's thoughts on hi-lo notation: Difference between revisions

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=== Triads ===

For a triad in 3 voices, the basic, hiR and lo5 voicings (e.g. CEG, EGC and GCE) are all close voicings. The hi3, lo3 and hiRlo5 voicings (e.g. CGE, ECG and GEC) are all open voicings. These two categories of voicings, open and closed, are called '''generalized voicings'''. ~~Thinking~~ of the chord as notes in a '''tritonic''' scale (as opposed to a pentatonic or heptatonic scale)~~, all the notes in a close voicing are 1 scale step away from the adjacent notes. And in an open voicing, all notes are 2 scale steps away~~.

For a triad in 3 voices, the basic, hiR and lo5 voicings (e.g. CEG, EGC and GCE) are all close voicings. The hi3, lo3 and hiRlo5 voicings (e.g. CGE, ECG and GEC) are all open voicings. These two categories of voicings, open and closed, are called '''generalized voicings'''. We can think of the chord as notes in a '''tritonic''' scale (as opposed to a pentatonic or heptatonic scale).

{| class="wikitable"

|+

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|10th or 11th

|-

|etc.

| colspan="2" |etc.

|

|}

Thus the two generalized voicings can be named 1-1 and 2-2, pronounced ''one-one'' and ''two-two''. (Open and close are certainly preferable names, but this '''stepwise''' terminology is introduced here to pave the way for naming voicings of tetrads and pentads.)

In addition, 2nds, 7ths and 9ths might occur with a sus chord.

All the notes in a close voicing are 1 tritonic scale step away from adjacent notes. And in an open voicing, all notes are 2 scale steps away. Thus the two generalized voicings can be named 1-1 and 2-2, pronounced ''one-one'' and ''two-two''. (Open and close are certainly preferable names, but this '''stepwise''' terminology is introduced here to pave the way for naming voicings of tetrads and pentads.)

The following table assumes that the interval between any pair of adjacent notes is less than an octave, a reasonable assumption. Thus there are only 6 possible voicings of a triad, grouped into 2 generalized voicings (assuming only 3 voices). In any 1-1 voicing, the chord spans 2 steps of the tritonic scale. Likewise any 2-2 voicing spans 4 steps. Thus the '''stepspan''' of the voicing is simply the sum of the 2 numbers.

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|+Generalized voicings for a triad in 4 voices, empty octaves allowed

!stepspan

! colspan="2" |voicing

! colspan="3" |voicing

!root position

!1st inversion

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!1-1-3

!lift-2-3

!close over 8ve

|addloR

|hiRaddlo3

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!1-3-1

!lift-3 drop-2

!

|hi5add10

|hiRlo3addlo5

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!3-1-1

!drop-2-3

!close under 8ve

|addhi5

|lo35add8

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!2-2-3

!lift-3 drop-1-4

!open over 8ve

|hi3addloR

|hiR5addlo3

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!2-3-2

!lift-2 drop-3

!

|hi10addhi5

|hi5lo3add8

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!3-2-2

!lift-1-4 drop-2

!open under 8ve

|hi3addhi10

|lo3addhi5

|hiRlo5addhi8

|}

There are 42 five-part voicings of a triad, or 150 if allowing empty octaves.

=== Tetrads ===

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|hi38lo7

|}

The stepwise terminology has the advantage that it directly indicates the stepspan, and hence the "openness" of the voicing. Furthermore, a generalized voicing has only one stepwise name, but possibly more than one drop-N name. Drop-2 could possibly be called lift-3, and drop-2-4 could possibly be called lift-1-3.

The stepwise terminology has the advantage that it directly indicates the stepspan, and hence the "openness" of the voicing. Furthermore, a generalized voicing has only one stepwise name, but possibly more than one lift/drop name. Drop-2 could possibly be called lift-3, and drop-2-4 could possibly be called lift-1-3.

To find the stepwise name from the drop-N name: drop-3 changes 1234 to 1243, or more precisely 12(3)4(12)3. From 1 to 2 is 1 step, from 2 to 4 is 2 steps, and from 4 to 3 is 3 steps. Thus the stepwise name is 1-2-3.

To find the stepwise name from the lift/drop name: drop-3 changes 1234 to 1243, or more precisely 12(3)4(12)3. From 1 to 2 is 1 step, from 2 to 4 is 2 steps, and from 4 to 3 is 3 steps. Thus the stepwise name is 1-2-3.

Generalized voicings are particularly useful for dim7 chords.

There are 144 five-part voicings of a tetrad, or 240 if allowing empty octaves.

=== Pentads ===

For a pentad in 5 voices, there are 120 possible voicings grouped into 24 generalized voicings. Above span-9 the drop-N terminology becomes rather awkward. Because of the large number of generalized voicings, the stepspan becomes quite useful as a category of categories.

For a pentad in 5 voices, there are 120 possible voicings grouped into 24 generalized voicings. Above span-9 the lift/drop terminology becomes rather awkward. Because of the large number of generalized voicings, the stepspan becomes quite useful as a category of categories.

{| class="wikitable"

|+

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!span-4

|1-1-1-1 = close

|

! colspan="3" |

|

|-

!span-6

|2-1-1-2 = drop-2

|

! colspan="3" |

|

|-

!span-7

|1-2-1-3 = drop-3

|3-1-2-1 = lift-3

|

! colspan="2" |

|

|-

!span-8

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|2-4-3-4 = lift-1-3 drop-4

|4-3-4-2 = lift-2 drop-3-5

|

! colspan="2" |

|

|-

!span-14

|3-4-4-3 = lift-1-4 drop-2-5

|

! colspan="3" |

|

|-

!span-16

|4-4-4-4 = ???

|

! colspan="3" |

|

|}

The 9th of a pentad is treated as a 2nd in a pentatonic scale. Thus for G9 in root position, the 1-1-1-1 voicing is GABDF, the lo9 voicing. The ~~more consonant~~ GBDFA ~~voicing~~ is a 2-1-1-2 voicing.

The 9th of a pentad is treated as a 2nd in a pentatonic scale. Thus for G9 in root position, the 1-1-1-1 voicing is GABDF, the lo9 voicing. The basic voicing (GBDFA) is a 2-1-1-2 voicing.

{| class="wikitable"

|+Generalized voicings for a pentad in 5 voices, smallest 3 stepspans only

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=== Hexads, heptads, etc. ===

For an N-ad in N voices, there are N! possible voicings, which are grouped into (N-1)! generalized voicings, which are grouped into (N-2)<sup>2</sup> + 1 stepspans.{{Navbox notation}}

For an N-ad in N voices, there are N! possible voicings, which are grouped into (N-1)! generalized voicings, which are grouped into (N-2)<sup>2</sup> + 1 stepspans. For an N-ad in M voices (M > N), use Stirling numbers of the second kind.

@@ Line 101: / Line 101: @@
 === Triads ===
-For a triad in 3 voices, the basic, hiR and lo5 voicings (e.g. CEG, EGC and GCE) are all close voicings. The hi3, lo3 and hiRlo5 voicings (e.g. CGE, ECG and GEC) are all open voicings. These two categories of voicings, open and closed, are called '''generalized voicings'''. Thinking of the chord as notes in a '''tritonic''' scale (as opposed to a pentatonic or heptatonic scale), all the notes in a close voicing are 1 scale step away from the adjacent notes. And in an open voicing, all notes are 2 scale steps away.
+For a triad in 3 voices, the basic, hiR and lo5 voicings (e.g. CEG, EGC and GCE) are all close voicings. The hi3, lo3 and hiRlo5 voicings (e.g. CGE, ECG and GEC) are all open voicings. These two categories of voicings, open and closed, are called '''generalized voicings'''. We can think of the chord as notes in a '''tritonic''' scale (as opposed to a pentatonic or heptatonic scale).
 {| class="wikitable"
 |+
@@ Line 119: / Line 119: @@
 |10th or 11th
 |-
-|etc.
+| colspan="2" |etc.
-|
 |}
-Thus the two generalized voicings can be named 1-1 and 2-2, pronounced ''one-one'' and ''two-two''. (Open and close are certainly preferable names, but this '''stepwise''' terminology is introduced here to pave the way for naming voicings of tetrads and pentads.)
+In addition, 2nds, 7ths and 9ths might occur with a sus chord.
+All the notes in a close voicing are 1 tritonic scale step away from adjacent notes. And in an open voicing, all notes are 2 scale steps away. Thus the two generalized voicings can be named 1-1 and 2-2, pronounced ''one-one'' and ''two-two''. (Open and close are certainly preferable names, but this '''stepwise''' terminology is introduced here to pave the way for naming voicings of tetrads and pentads.)
 The following table assumes that the interval between any pair of adjacent notes is less than an octave, a reasonable assumption. Thus there are only 6 possible voicings of a triad, grouped into 2 generalized voicings (assuming only 3 voices). In any 1-1 voicing, the chord spans 2 steps of the tritonic scale. Likewise any 2-2 voicing spans 4 steps. Thus the '''stepspan''' of the voicing is simply the sum of the 2 numbers.
@@ Line 204: / Line 205: @@
 |+Generalized voicings for a triad in 4 voices, empty octaves allowed
 !stepspan
-! colspan="2" |voicing
+! colspan="3" |voicing
 !root position
 !1st inversion
@@ Line 212: / Line 213: @@
 !1-1-3
 !lift-2-3
+!close over 8ve
 |addloR
 |hiRaddlo3
@@ Line 218: / Line 220: @@
 !1-3-1
 !lift-3 drop-2
+!
 |hi5add10
 |hiRlo3addlo5
@@ Line 224: / Line 227: @@
 !3-1-1
 !drop-2-3
+!close under 8ve
 |addhi5
 |lo35add8
@@ Line 231: / Line 235: @@
 !2-2-3
 !lift-3 drop-1-4
+!open over 8ve
 |hi3addloR
 |hiR5addlo3
@@ Line 237: / Line 242: @@
 !2-3-2
 !lift-2 drop-3
+!
 |hi10addhi5
 |hi5lo3add8
@@ Line 243: / Line 249: @@
 !3-2-2
 !lift-1-4 drop-2
+!open under 8ve
 |hi3addhi10
 |lo3addhi5
 |hiRlo5addhi8
 |}
+There are 42 five-part voicings of a triad, or 150 if allowing empty octaves.
 === Tetrads ===
@@ Line 306: / Line 314: @@
 |hi38lo7
 |}
-The stepwise terminology has the advantage that it directly indicates the stepspan, and hence the "openness" of the voicing. Furthermore, a generalized voicing has only one stepwise name, but possibly more than one drop-N name. Drop-2 could possibly be called lift-3, and drop-2-4 could possibly be called lift-1-3.
+The stepwise terminology has the advantage that it directly indicates the stepspan, and hence the "openness" of the voicing. Furthermore, a generalized voicing has only one stepwise name, but possibly more than one lift/drop name. Drop-2 could possibly be called lift-3, and drop-2-4 could possibly be called lift-1-3.
-To find the stepwise name from the drop-N name: drop-3 changes 1234 to 1243, or more precisely 12(3)4(12)3. From 1 to 2 is 1 step, from 2 to 4 is 2 steps, and from 4 to 3 is 3 steps. Thus the stepwise name is 1-2-3.
+To find the stepwise name from the lift/drop name: drop-3 changes 1234 to 1243, or more precisely 12(3)4(12)3. From 1 to 2 is 1 step, from 2 to 4 is 2 steps, and from 4 to 3 is 3 steps. Thus the stepwise name is 1-2-3.
 Generalized voicings are particularly useful for dim7 chords.
+There are 144 five-part voicings of a tetrad, or 240 if allowing empty octaves.
 === Pentads ===
-For a pentad in 5 voices, there are 120 possible voicings grouped into 24 generalized voicings. Above span-9 the drop-N terminology becomes rather awkward. Because of the large number of generalized voicings, the stepspan becomes quite useful as a category of categories.
+For a pentad in 5 voices, there are 120 possible voicings grouped into 24 generalized voicings. Above span-9 the lift/drop terminology becomes rather awkward. Because of the large number of generalized voicings, the stepspan becomes quite useful as a category of categories.
 {| class="wikitable"
 |+
@@ Line 322: / Line 332: @@
 !span-4
 |1-1-1-1 = close
-|
+! colspan="3" |
-|
-|
 |-
 !span-6
 |2-1-1-2 = drop-2
-|
+! colspan="3" |
-|
-|
 |-
 !span-7
 |1-2-1-3 = drop-3
 |3-1-2-1 = lift-3
-|
+! colspan="2" |
-|
 |-
 !span-8
@@ Line 365: / Line 370: @@
 |2-4-3-4 = lift-1-3 drop-4
 |4-3-4-2 = lift-2 drop-3-5
-|
+! colspan="2" |
-|
 |-
 !span-14
 |3-4-4-3 = lift-1-4 drop-2-5
-|
+! colspan="3" |
-|
-|
 |-
 !span-16
 |4-4-4-4 = ???
-|
+! colspan="3" |
-|
-|
 |}
-The 9th of a pentad is treated as a 2nd in a pentatonic scale. Thus for G9 in root position, the 1-1-1-1 voicing is GABDF, the lo9 voicing. The more consonant GBDFA voicing is a 2-1-1-2 voicing.
+The 9th of a pentad is treated as a 2nd in a pentatonic scale. Thus for G9 in root position, the 1-1-1-1 voicing is GABDF, the lo9 voicing. The basic voicing (GBDFA) is a 2-1-1-2 voicing.
 {| class="wikitable"
 |+Generalized voicings for a pentad in 5 voices, smallest 3 stepspans only
@@ Line 428: / Line 428: @@
 === Hexads, heptads, etc. ===
-For an N-ad in N voices, there are N! possible voicings, which are grouped into (N-1)! generalized voicings, which are grouped into (N-2)<sup>2</sup> + 1 stepspans.{{Navbox notation}}
+For an N-ad in N voices, there are N! possible voicings, which are grouped into (N-1)! generalized voicings, which are grouped into (N-2)<sup>2</sup> + 1 stepspans. For an N-ad in M voices (M > N), use Stirling numbers of the second kind.
+{{Navbox notation}}