Xenharmonic Wiki

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

← Older edit
VisualWikitext
TallKite (talk | contribs)
autopatrolled, Sysops
4,223 edits
mNo edit summary
TallKite (talk | contribs)
autopatrolled, Sysops
4,223 edits
→Generalized Voicings: more small improvements
 
(5 intermediate revisions by the same user not shown)
Line 47: Line 47:
Use addhi8, not add15. No degrees higher than 13 are used, analogous to chord names.  
Use addhi8, not add15. No degrees higher than 13 are used, analogous to chord names.  


The terms ''hi'' and ''lo'' and degrees R-8 cover voicings up to a 3-octave range, because every note is in either the lo octave, the central octave or the hi octave. Degrees 9-13 go even higher. To extend the range an extra octave both above and below, use hihi and lolo. For example, lolo7 is a note a 9th below the root. Lolo is never used for degrees above 7. Lolo9 becomes lo2, lolo11 becomes lo4, and lolo13 becomes lo6.
The terms ''hi'' and ''lo'' and degrees R-8 cover voicings up to a 3-octave range, because every note is in either the lo octave, the central octave or the hi octave. Degrees 9-13 go even higher. To extend the range an extra octave both above and below, use dubhi and dublo, where dub- is short for double. For example, dublo7 is a note a 9th below the root. Dublo is never used for degrees above 7. Dublo9 becomes lo2, dublo11 becomes lo4, and dublo13 becomes lo6.


To find the number of voices in a chord, start with the obvious: a triad has 3, a tetrad has 4, etc. Then subtract the "no" notes and add the "add" notes. For example, A7no5hi3add8 has 4 (A7 is a tetrad) - 1 (no 5th) + 1 (add 8ve) = 4 notes.
To find the number of voices in a chord, start with the obvious: a triad has 3, a tetrad has 4, etc. Then subtract the "no" notes and add the "add" notes. For example, A7no5hi3add8 has 4 (A7 is a tetrad) - 1 (no 5th) + 1 (add 8ve) = 4 notes.
Line 101: Line 101:


=== Triads ===
=== 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 (as opposed to pentatonic, heptatonic, etc.) scale, all the notes in the close voicings are 1 scale step above the note below it. And in the open voicing, all notes are 2 scale steps above. 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.)
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"
|+
!tritonic stepspan
!heptatonic degree
|-
|1 step
|3rd or 4th
|-
|2 steps
|5th or 6th
|-
|3 steps
|octave
|-
|4 steps
|10th or 11th
|-
| colspan="2" |etc.
|}
In addition, 2nds, 7ths and 9ths might occur with a sus chord.


The following table assumes that the interval between any pair of adjacent notes is less than an octave, a very 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.
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.
{| class="wikitable"
{| class="wikitable"
|+Generalized voicings for a triad in 3 voices
|+Generalized voicings for a triad in 3 voices
Line 127: Line 149:
|}
|}
Generalized voicings are particularly useful when the chord has a [[Chord homonym|homonym]]. For example, a 12edo augmented chord is simultaneously in 3 different inversions and 3 different voicings, but in only 1 generalized voicing.
Generalized voicings are particularly useful when the chord has a [[Chord homonym|homonym]]. For example, a 12edo augmented chord is simultaneously in 3 different inversions and 3 different voicings, but in only 1 generalized voicing.
In 4 voices, there are 18 possible voicings of a triad, grouped into 6 generalized voicings. The [[wikipedia:Voicing_(music)#Drop_voicings|'''drop voicing''']] terminology that guitarists use can be extended via "lift" to name these generalized voicings.
{| class="wikitable"
|+Generalized voicings for a triad in 4 voices, no empty octaves
!stepspan
! colspan="2" |voicing
!root position
!1st inversion
!2nd inversion
|-
!span-3
!1-1-1
!close
|add8
|hiRadd10
|addlo5
|-
! rowspan="2" |span-4
!2-1-1
!drop-2
|add10
|hiRaddhi5
|lo5add8
|-
!1-1-2
!lift-3
|hi3add8
|addlo3
|hiRaddlo5
|-
! rowspan="2" |span-5
!1-2-2
!drop-3
|hi3addhi5
|lo3add8
|hiRlo5add10
|-
!2-2-1
!lift-2
|hi3addhi5
|hiRlo5addlo3
|hi3addlo5
|-
!span-6
!2-2-2
!drop-2-4
|hi3addhi8
|lo3add10
|hiR5addlo5
|}
In the 2-1-1 voicing, the 2nd highest note is 2 tritonic steps below the highest note, the 3rd highest note is 1 step below the 2nd, and the 4th is 1 step below the 3rd.
Allowing adjacent voices to be an octave apart adds 18 more voicings. Lift/drop names become awkward.
{| class="wikitable"
|+Generalized voicings for a triad in 4 voices, empty octaves allowed
!stepspan
! colspan="3" |voicing
!root position
!1st inversion
!2nd inversion
|-
! rowspan="3" |span-5
!1-1-3
!lift-2-3
!close over 8ve
|addloR
|hiRaddlo3
|hiR3addlo5
|-
!1-3-1
!lift-3 drop-2
!
|hi5add10
|hiRlo3addlo5
|hi3lo5add8
|-
!3-1-1
!drop-2-3
!close under 8ve
|addhi5
|lo35add8
|lo5add10
|-
! rowspan="3" |span-7
!2-2-3
!lift-3 drop-1-4
!open over 8ve
|hi3addloR
|hiR5addlo3
|hi38addlo5
|-
!2-3-2
!lift-2 drop-3
!
|hi10addhi5
|hi5lo3add8
|hi8lo5add10
|-
!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 ===
=== Tetrads ===
For a tetrad in 4 voices, there are 24 possible voicings grouped into 6 generalized voicings. The 2-1-2 voicings are known to guitarists as [[wikipedia:Voicing_(music)#Drop_voicings|drop-2 voicings]]. This "drop" terminology can be extended to the other generalized voicings, but only if there are no octave doublings.
For a tetrad in 4 voices, there are 24 possible voicings grouped into 6 generalized voicings.
{| class="wikitable"
{| class="wikitable"
|+Generalized voicings for a tetrad in 4 voices
|+Generalized voicings for a tetrad in 4 voices
Line 156: Line 284:
|-
|-
! rowspan="2" |span-6
! rowspan="2" |span-6
!3-2-1
!1-2-3
!drop-3
!drop-3
|hi35
|hi35
Line 163: Line 291:
|hiRlo37
|hiRlo37
|-
|-
!1-2-3
!3-2-1
!lift-2
!lift-2
|hi5
|hi5
Line 186: Line 314:
|hi38lo7
|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" 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 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.
Generalized voicings are particularly useful for dim7 chords.
There are 144 five-part voicings of a tetrad, or 240 if allowing empty octaves.


=== Pentads ===
=== Pentads ===
For a pentad in 5 voices, there are 120 possible voicings grouped into 24 generalized voicings. Above span-9 the "drop" 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"
{| class="wikitable"
|+
|+
Line 200: Line 332:
!span-4
!span-4
|1-1-1-1 = close
|1-1-1-1 = close
|
! colspan="3" |
|
|
|-
|-
!span-6
!span-6
|2-1-1-2 = drop-2
|2-1-1-2 = drop-2
|
! colspan="3" |
|
|
|-
|-
!span-7
!span-7
|3-1-2-1 = drop-3
|1-2-1-3 = drop-3
|1-2-1-3 = lift-3
|3-1-2-1 = lift-3
|
! colspan="2" |
|
|-
|-
!span-8
!span-8
|2-2-2-2 = drop-2-4
|2-2-2-2 = drop-2-4
|1-3-3-1 = drop-3-4
|1-3-3-1 = drop-3-4
|4-2-1-1 = drop-4
|1-1-2-4 = drop-4
|1-1-2-4 = lift-2
|4-2-1-1 = lift-2
|-
|-
!span-9
!span-9
|3-3-1-2 = drop-2-5
|2-1-3-3 = drop-2-5
|2-1-3-3 = lift-1-4
|3-3-1-2 = lift-1-4
|2-4-2-1 = drop-3-5
|1-2-4-2 = drop-3-5
|1-2-4-2 = lift-1-3
|2-4-2-1 = lift-1-3
|-
|-
!span-11
!span-11
|4-3-1-3 = lift-3 drop-4
|3-1-3-4 = lift-3 drop-4
|3-1-3-4 = lift-2 drop-3
|4-3-1-3 = lift-2 drop-3
|3-4-2-2 = lift-2-4 drop-3
|2-2-4-3 = lift-2-4 drop-3
|2-2-4-3 = lift-3 drop-2-4
|3-4-2-2 = lift-3 drop-2-4
|-
|-
!span-12
!span-12
|4-2-2-4 = lift-2 drop-4
|4-2-2-4 = lift-2 drop-4
|3-3-3-3 = lift-3 drop-2-5
|3-3-3-3 = lift-3 drop-2-5
|4-4-3-1 = lift-2-3 drop-4
|1-3-4-4 = lift-2-3 drop-4
|1-3-4-4 = lift-2 drop-3-4
|4-4-3-1 = lift-2 drop-3-4
|-
|-
!span-13
!span-13
|4-3-4-2 = lift-1-3 drop-4
|2-4-3-4 = lift-1-3 drop-4
|2-4-3-4 = lift-2 drop-3-5
|4-3-4-2 = lift-2 drop-3-5
|
! colspan="2" |
|
|-
|-
!span-14
!span-14
|3-4-4-3 = lift-1-4 drop-2-5
|3-4-4-3 = lift-1-4 drop-2-5
|
! colspan="3" |
|
|
|-
|-
!span-16
!span-16
|4-4-4-4 = ???
|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"
{| class="wikitable"
|+Generalized voicings for a pentad in 5 voices, smallest 3 stepspans only
|+Generalized voicings for a pentad in 5 voices, smallest 3 stepspans only
Line 288: Line 410:
|-
|-
! rowspan="2" |span-7
! rowspan="2" |span-7
!3-1-2-1
!1-2-1-3
!drop-3
!drop-3
|hi3
|hi3
Line 296: Line 418:
|lo27
|lo27
|-
|-
!1-2-1-3
!3-1-2-1
!lift-3
!lift-3
|hi3lo9
|hi3lo9
Line 306: Line 428:


=== Hexads, heptads, etc. ===
=== 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}}
Retrieved from "https://en.xen.wiki/w/Kite%27s_thoughts_on_hi-lo_notation"