User:Unque/Barbershop Tuning Theory: Difference between revisions

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While Barbershop music is typically said to use 7-limit Just Intonation, in practice it contains a number of idiosyncrasies and dialectal variation that differ from one performer to another, often dipping into tempered tunings or even into higher prime limits. On this page, I plan to document the chords of Barbershop harmony and the variations that occur among them; additionally, I will discuss the theoretical reasoning behind a performer's choice of one tuning over another, and theoretical extensions that could provide other potential tunings for a given chord. Any purely theoretical or otherwise unattested tunings will be explicitly disclaimed as such.

Note that the ratios given here are ''approximations'', not precise measurements. Most performers don't analyze the intervals as JI ratios, but rather slightly adjust the tunings to make the chords ring better. Even if they did analyze the ratios, the voice would not be precise enough to articulate certain minute differences, and as such these ratios would still not precisely represent the actual practical sounds being sung.

Note that the ratios given here are ''approximations'', not precise measurements. Most performers don't analyze the intervals as JI ratios (with the common exception of certain extremely simple chords, such as the famous 4:5:6:7 tetrad), but rather slightly adjust the tunings to make the chords ring better. Even if they did analyze the ratios, the voice would not be precise enough to articulate certain minute differences, and as such these ratios would still not precisely represent the actual practical sounds being sung.

Chords are given here as harmonic segments within the octave unless otherwise specified; true Barbershop performance often uses more open voicings, inversions, or other kinds of permutations with respect to the equivalent octaves.

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If one wishes to root the chord as a Half-Diminished Seventh, then inversions of 5:6:7:9 are the most common interpretation. If one instead wishes to root the chord as a Minor triad with an added Major Sixth, the chord is most commonly tuned as a standard 1/(6:5:4) Minor Triad with some type of Major Sixth (typically 5/3, or less commonly 12/7) added on top.

Additionally, the chord can theoretically be rooted as an inversion of a 1 - M2 - P4 - M6 tetrad, which could be interpreted as 9:10:12:13 or 9:10:12:14. While not technically invalid, however, this approach is purely theoretical, and does not seem to be used or even attested anywhere.

Finally, it should be noted that the ii<sup>ø7</sup> chord is used as a dominant due to being the negative harmony variant of V<sup>7</sup>. Because of this, some performers prefer to treat this chord as an undertonal variant of the standard Barbershop Seventh.

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|20/17, 6/5, 5/4, (17/15)

|1/(6:5:4) triad stacked on top of a 17:20 dyad. Most likely conceptualized as tempered tuning, not 17-limit JI.

|-

~~|10:12:13:18~~

~~|6/5, 13/10, 9/5~~

~~|6/5, 13/12, 18/13, (10/9)~~

~~|Purely theoretical. Seemingly unattested, despite ringing very well.~~

|}

'''See also:''' [[5afdo]] and [[7ifdo]], the low-complexity JI scales from which several of these tunings are derived.

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|7/6, 9/7, 7/6, (8/7)

|Standard 7-limit Minor Tetrad; much less common than 5-limit.

|-

|32:38:48:57

|19/16, 3/2, 57/32

|19/16, 24/19, 19/16, (64/57)

|Much closer to 12edo's minor third. Most likely conceptualized as tempered tuning, not 19-limit JI.

|}

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The Full Diminished tetrad is rarely used in Barbershop music, but is considered a part of the vocabulary by the Barbershop Harmony Society nonetheless. The simplest way to tune this tetrad is by stacking up intervals of 6/5, but this becomes complex and dissonant quite quickly.

Another option is to use the Half-Diminished tetrad, but flatten the seventh to yield a Full Diminished form; in order to do this, however, one must have a determined tuning for the Half-Diminished Seventh chord, which is already controversial. Additionally, the amount by which the seventh is flattened varies idiolectally, since there is no low-complexity tuning for the Diminished Seventh interval aside from the ~~enharmonically-equivalent~~ Major Sixth.

Another option is to use the Half-Diminished tetrad, but flatten the seventh to yield a Full Diminished form; in order to do this, however, one must have a determined tuning for the Half-Diminished Seventh chord, which is already controversial. Additionally, the amount by which the seventh is flattened varies idiolectally, since there is no low-complexity tuning for the Diminished Seventh interval aside from the enharmonic Major Sixth.

Due to the Diminished Chord's nature as a symmetrical chord in Western classical practices, rotations of the tetrad are also valid options.

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While the Dominant Diminished tetrad (a dominant chord with a diminished fifth) is officially recognized as part of the Barbershop vocabulary, it is rarely used in practice, and as such there are very few attestations of how to tune it. It is also called a Major Diminished chord, or simply a Flat-Five chord.

The Major Diminished chord is a barbershop tetrad with the fifth diminished; this interpretation yields the tuning 20:25:28:35. It can also be thought of as a Half-Diminished tetrad with the third raised, which may instead yield 20:25:28:36.

The Major Diminished chord can be seen as a barbershop tetrad with the fifth diminished; this interpretation yields the tuning 20:25:28:35. It can also be thought of as a Half-Diminished tetrad with the third raised, which may instead yield 20:25:28:36.

Additionally, rooting a Half-Diminished chord on the perfect fifth rather than the root yields a chord with intervals 1 - M3 - Aug4 - M6, which is very close to a Dominant Diminished chord, especially considering how sharp the M6 is tuned and how flat the dominant seventh is in Barbershop music. The major sixth can be raised to a typical minor seventh in tunings where it would be more beneficial to do so.

It can also be thought of as a Lydian Dominant chord with the fifth omitted; thus, 8:10:11:14 or 4:5:7:11. This tuning is seldom used in Barbershop practice, but is often used as the tuning of this chord in harmonic contexts.

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|5/4, 28/25, 9/7, (10/9)

|Half-diminished chord with raised third.

|-

|7:9:10:12

|9/7, 10/7, 12/7

|9/7, 10/9, 6/5, (7/6)

|Rotation of half-diminished chord.

|-

|12:15:17:21

|5/4, 17/12, 7/4

|5/4, 17/15, 21/12, (8/7)

|17-limit hald-diminished chord, but with the sixth raised.

|-

|10:13:14:18

|13/10, 7/5, 9/5

|13/10, 14/13, 9/7, (10/9)

|~~Most likely conceptualized as tempered tuning~~, ~~not~~ 13~~-limit JI.~~

|Purely theoretical; despite ringing very well, 13/10 is too sharp to be expected of typical Barbershop music

|-

|15:19:21:27

@@ Line 1: / Line 1: @@
 While Barbershop music is typically said to use 7-limit Just Intonation, in practice it contains a number of idiosyncrasies and dialectal variation that differ from one performer to another, often dipping into tempered tunings or even into higher prime limits.  On this page, I plan to document the chords of Barbershop harmony and the variations that occur among them; additionally, I will discuss the theoretical reasoning behind a performer's choice of one tuning over another, and theoretical extensions that could provide other potential tunings for a given chord.  Any purely theoretical or otherwise unattested tunings will be explicitly disclaimed as such.
-Note that the ratios given here are ''approximations'', not precise measurements.  Most performers don't analyze the intervals as JI ratios, but rather slightly adjust the tunings to make the chords ring better.  Even if they did analyze the ratios, the voice would not be precise enough to articulate certain minute differences, and as such these ratios would still not precisely represent the actual practical sounds being sung.
+Note that the ratios given here are ''approximations'', not precise measurements.  Most performers don't analyze the intervals as JI ratios (with the common exception of certain extremely simple chords, such as the famous 4:5:6:7 tetrad), but rather slightly adjust the tunings to make the chords ring better.  Even if they did analyze the ratios, the voice would not be precise enough to articulate certain minute differences, and as such these ratios would still not precisely represent the actual practical sounds being sung.
 Chords are given here as harmonic segments within the octave unless otherwise specified; true Barbershop performance often uses more open voicings, inversions, or other kinds of permutations with respect to the equivalent octaves.
@@ Line 34: / Line 34: @@
 If one wishes to root the chord as a Half-Diminished Seventh, then inversions of 5:6:7:9 are the most common interpretation.  If one instead wishes to root the chord as a Minor triad with an added Major Sixth, the chord is most commonly tuned as a standard 1/(6:5:4) Minor Triad with some type of Major Sixth (typically 5/3, or less commonly 12/7) added on top.
-Additionally, the chord can theoretically be rooted as an inversion of a 1 - M2 - P4 - M6 tetrad, which could be interpreted as 9:10:12:13 or 9:10:12:14.  While not technically invalid, however, this approach is purely theoretical, and does not seem to be used or even attested anywhere.
 Finally, it should be noted that the ii<sup>ø7</sup> chord is used as a dominant due to being the negative harmony variant of V<sup>7</sup>.  Because of this, some performers prefer to treat this chord as an undertonal variant of the standard Barbershop Seventh.
@@ Line 64: / Line 62: @@
 |20/17, 6/5, 5/4, (17/15)
 |1/(6:5:4) triad stacked on top of a 17:20 dyad.  Most likely conceptualized as tempered tuning, not 17-limit JI.
-|-
-|10:12:13:18
-|6/5, 13/10, 9/5
-|6/5, 13/12, 18/13, (10/9)
-|Purely theoretical.  Seemingly unattested, despite ringing very well.
 |}
 '''See also:''' [[5afdo]] and [[7ifdo]], the low-complexity JI scales from which several of these tunings are derived.
@@ Line 112: / Line 105: @@
 |7/6, 9/7, 7/6, (8/7)
 |Standard 7-limit Minor Tetrad; much less common than 5-limit.
+|-
+|32:38:48:57
+|19/16, 3/2, 57/32
+|19/16, 24/19, 19/16, (64/57)
+|Much closer to 12edo's minor third.  Most likely conceptualized as tempered tuning, not 19-limit JI.
 |}
@@ Line 134: / Line 132: @@
 The Full Diminished tetrad is rarely used in Barbershop music, but is considered a part of the vocabulary by the Barbershop Harmony Society nonetheless.  The simplest way to tune this tetrad is by stacking up intervals of 6/5, but this becomes complex and dissonant quite quickly.
-Another option is to use the Half-Diminished tetrad, but flatten the seventh to yield a Full Diminished form; in order to do this, however, one must have a determined tuning for the Half-Diminished Seventh chord, which is already controversial.  Additionally, the amount by which the seventh is flattened varies idiolectally, since there is no low-complexity tuning for the Diminished Seventh interval aside from the enharmonically-equivalent Major Sixth.
+Another option is to use the Half-Diminished tetrad, but flatten the seventh to yield a Full Diminished form; in order to do this, however, one must have a determined tuning for the Half-Diminished Seventh chord, which is already controversial.  Additionally, the amount by which the seventh is flattened varies idiolectally, since there is no low-complexity tuning for the Diminished Seventh interval aside from the enharmonic Major Sixth.
 Due to the Diminished Chord's nature as a symmetrical chord in Western classical practices, rotations of the tetrad are also valid options.
@@ Line 169: / Line 167: @@
 While the Dominant Diminished tetrad (a dominant chord with a diminished fifth) is officially recognized as part of the Barbershop vocabulary, it is rarely used in practice, and as such there are very few attestations of how to tune it.  It is also called a Major Diminished chord, or simply a Flat-Five chord.
-The Major Diminished chord is a barbershop tetrad with the fifth diminished; this interpretation yields the tuning 20:25:28:35.  It can also be thought of as a Half-Diminished tetrad with the third raised, which may instead yield 20:25:28:36.
+The Major Diminished chord can be seen as a barbershop tetrad with the fifth diminished; this interpretation yields the tuning 20:25:28:35.  It can also be thought of as a Half-Diminished tetrad with the third raised, which may instead yield 20:25:28:36.
+Additionally, rooting a Half-Diminished chord on the perfect fifth rather than the root yields a chord with intervals 1 - M3 - Aug4 - M6, which is very close to a Dominant Diminished chord, especially considering how sharp the M6 is tuned and how flat the dominant seventh is in Barbershop music.  The major sixth can be raised to a typical minor seventh in tunings where it would be more beneficial to do so.
 It can also be thought of as a Lydian Dominant chord with the fifth omitted; thus, 8:10:11:14 or 4:5:7:11.  This tuning is seldom used in Barbershop practice, but is often used as the tuning of this chord in harmonic contexts.
@@ Line 188: / Line 188: @@
 |5/4, 28/25, 9/7, (10/9)
 |Half-diminished chord with raised third.
+|-
+|7:9:10:12
+|9/7, 10/7, 12/7
+|9/7, 10/9, 6/5, (7/6)
+|Rotation of half-diminished chord.
+|-
+|12:15:17:21
+|5/4, 17/12, 7/4
+|5/4, 17/15, 21/12, (8/7)
+|17-limit hald-diminished chord, but with the sixth raised.
 |-
 |10:13:14:18
 |13/10, 7/5, 9/5
 |13/10, 14/13, 9/7, (10/9)
-|Most likely conceptualized as tempered tuning, not 13-limit JI.
+|Purely theoretical; despite ringing very well, 13/10 is too sharp to be expected of typical Barbershop music
 |-
 |15:19:21:27