Kite's uniform solfege: Difference between revisions

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==Overview==

Uniform solfeges are a type of solfege ~~developed~~ by [[Kite Giedraitis]]. They are closely related to his [[ups and downs notation]]. Like the notation, they work with both rank-1 and rank-2 temperaments. They use a uniform vowel sequence for each degree, hence the name. A uniform solfege lets one perform basic interval arithmetic directly within the solfege, without having to translate to note names or interval names and back.

Uniform solfeges are a type of solfege devised by [[Kite Giedraitis]]. They are closely related to his [[ups and downs notation]]. Like the notation, they work with both rank-1 and rank-2 temperaments. They use a uniform vowel sequence for each degree, hence the name. A uniform solfege lets one perform basic interval arithmetic directly within the solfege, without having to translate to note names or interval names and back.

Uniform solfeges use the conventional consonants D R M F S L T. But all consonants except D have an alternate form that indicates flattening or sharpening:

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=== Example Scales ===

{| class="wikitable center-all"

|+

! rowspan="2" |3-limit

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Once this is fully internalized, add in the other vowels.

== Interval Arithmetic ==

==Interval Arithmetic==

===Octave Complements===

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For example, Fru = minor-Re-up becomes major-Ti-down = To.

===~~The~~ Circle of Fifths===

===An Edo's Circle of Fifths===

The 13 -a notes form a chain of 5ths running from the dim 5th to the aug 4th:

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'''Da''' Sa Ra La Ma Ta Pa/Sho Fro Flo No Tho Fo '''Do''' So Ro Lo Mo To Po/Fu '''Du''' Su Ru Lu Mu Tu Pu/Sha Fra Fla Na Tha Fa '''Da'''

Thus as long as one spells the ~~tritone~~ correctly, all 5ths rhyme. This makes interval arithmetic very easy.

Thus as long as one spells the tritones correctly, all 5ths in an edo rhyme. This makes interval arithmetic very easy.

===Adding/subtracting 4ths and 5ths===

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However, consider the aug 4th, a P- note. The note a 5th above it would be an augmented 8ve, which doesn't exist in a uniform solfege. Therefore one must rename the tritone as a dim or mid 5th. Thus in 31edo Pa + 5th = Sho + 5th = Fro. Likewise, Sho and Sha need renaming when adding a 4th: Sha + 4th = Pu + 4th = Tu.

===Adding/subtracting ~~other intervals~~===

===Adding/subtracting Other Intervals===

The same rule for 4ths and 5ths mostly holds for plain major 2nds. Keep the vowel, and change the consonant as expected. Ra + M2 = Ma. But an aug 4th must be renamed as a 5th. Beware, this rule breaks down entirely for major and mid 7ths (the T- notes), due to the lack of aug and mid 8ves.

In general, one can add or subtract any conventional (i.e. plain) interval from any note, and the result will be as expected. But only if the expected answer exists in the solfege. It must exist on the 13-note chain of 5ths from dim5 to aug4. In other words, the expected answer must not be augmented or diminished, unless it's an aug4 or a dim5. (Otherwise, one must use an enharmonic equivalent.) For example, one can easily add a M3 to any note other than a L-, M-, T- or P- note. Thus Ro + M3 = Po and Na + M3 = Sa, but La + M3 is a Fr- note. Beware, sometimes a chain is not 13 notes long, and when adding to or subtracting, the expected answer must exist on the shorter chain. For example, in 31edo, the -u chain runs from P4 to A4.

In general, one can add or subtract any conventional (i.e. plain) interval from any note, and the result will be as expected. But only if the expected answer exists in the solfege. It must exist on the 13-note chain of 5ths from dim5 to aug4. In other words, the expected answer must not be augmented or diminished, unless it's an aug4 or a dim5. (Otherwise, one must use an enharmonic equivalent.) For example, one can easily add a M3 to any note other than a L-, M-, T- or P- note. Thus Ro + M3 = Po and Na + M3 = Sa, but La + M3 is a Fr- note. Beware, sometimes a chain is not 13 notes long, and when adding to or subtracting, the expected answer must exist on the shorter chain. For example, in 31edo, the -u chain only runs from P4 to A4.

One can often easily add/subtract an unconventional (upped or downed) interval as well. The ups and downs add up and cancel out as expected. Thus Ra + vM2 = Mo and Ru + vM2 = Ma. Obviously the vowel will change. Again, the expected answer must exist in the solfege. 3-vowel solfeges lack double-ups and double-downs. 4-vowel solfeges lack double-upmajor and double-downminor~~. 5-vowel solfeges lack triple-ups and triple-downs~~.

One can often easily add/subtract an unconventional (upped or downed) interval as well. The ups and downs add up and cancel out as expected. Thus Ra + vM2 = Mo and Ru + vM2 = Ma. Obviously the vowel will change. Again, the expected answer must exist in the solfege. 3-vowel solfeges lack double-ups and double-downs. 4-vowel solfeges lack double-upmajor and double-downminor.

==Solfeges for Edos==

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|-

!5 vowels

!53, 60

! 53, 60

| -e = dud

|<nowiki>-o = down</nowiki>

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Because 72edo is such a popular edo, an exception is made and it has 2 additional vowels.

===Examples===

=== Examples===

*12edo: Da Fra Ra Na Ma Fa Pa/Sha Sa Fla La Tha Ta Da

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In edos with an even [[Sharpness|penta-sharpness]], there are "in-between" notes with two names. For example, 4\19 is named as both a 2nd and a 3rd (Ru/No).

===Correlations with Color Notation===

=== Correlations with Color Notation===

-u/-o can mean not only up/down, but also under/over, meaning in the ratio's denominator or numerator. A [[color notation]] review:

*yo/gu = 5-over/5-under = subtract/add 81/80

* yo/gu = 5-over/5-under = subtract/add 81/80

*zo/ru = 7-over/7-under = subtract/add 64/63

*ilo/lu = 11-over/11-under = subtract/add [[729/704]]

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If 81/80 maps to 1 edostep, then yo/gu = down/up = -o/-u. Likewise with the other commas. The table below shows that almost every edo has at least one such correlation. Parentheses are used when the prime's relative error is high, e.g. 12edo's prime 11.

{| class="wikitable center-all"

|+

colorspeak correlations for all 25 diatonic edos that have uniform solfeges with 5 vowels or less

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!17

!19

!22

! 22

!24

!26

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<big>'''Main article: [[List of Uniform Solfeges For Pergens]]'''</big>

Rank-2 temperaments have an infinite number of notes, so a solfege can only cover a fraction of them. But often one only needs enough notes to make a MOS scale. [[Pergen|Pergens]] tell us how to use ups and downs to notate these temperaments, and the same consonants and vowels can be used. Instead of circles of 5ths, there are '''fifthchains'''. Each fifthchain requires its own vowel, so there is a maximum of 5 fifthchains. However this can be extended to 9 fifthchains by using compound vowels such as -iyu.

Rank-2 temperaments have an infinite number of notes, so a solfege can only cover a fraction of them. But often one only needs enough notes to make a MOS scale. [[Pergen|Pergens]] tell us how to use ups and downs to notate these temperaments, and the same consonants and vowels can be used. Instead of circles of 5ths, there are '''fifthchains'''. Each fifthchain requires its own vowel, so there is a maximum of 5 fifthchains. However this can be extended to 9 fifthchains by using compound vowels such as -iyu, see below.

'''Genchains''' are distinct from fifthchains. Each pergen has one or more genchains, each of which contains one or more fifthchains. The 13 consonants and 5 vowels without compound vowels cover 20 pergens. Of course, the genchains can only extend so far with only 13 consonants. But in general, it's enough to cover all the modes of any reasonably-sized MOS scale.

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Sometimes -i and -e mean lift/drop not dup/dud. -i never means mid, so there are only two vowel sequences:

{| class="wikitable"

{| class="wikitable center-all"

|+unsplit (no-pair) solfege

!-4

! -4

!-3

!-2

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!A<sup>4</sup>

|-

| -eye

| -eye<br>"ay-yay"

"ay-yay"

| -eyo

| -e

| -o

| -a

| -u

| -i

| -iyu

| -iyi

| -iyi<br>"ee-yee"

"ee-yee"

|}

{| class="wikitable"

{| class="wikitable center-all"

|+single-pair solfeges

!-4

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Line 525:

| -iyi

|}

{| class="wikitable"

{| class="wikitable center-all"

|+double-pair solfeges

|

@@ Line 1: / Line 1: @@
 ==Overview==
-Uniform solfeges are a type of solfege developed by [[Kite Giedraitis]]. They are closely related to his [[ups and downs notation]]. Like the notation, they work with both rank-1 and rank-2 temperaments. They use a uniform vowel sequence for each degree, hence the name. A uniform solfege lets one perform basic interval arithmetic directly within the solfege, without having to translate to note names or interval names and back.
+Uniform solfeges are a type of solfege devised by [[Kite Giedraitis]]. They are closely related to his [[ups and downs notation]]. Like the notation, they work with both rank-1 and rank-2 temperaments. They use a uniform vowel sequence for each degree, hence the name. A uniform solfege lets one perform basic interval arithmetic directly within the solfege, without having to translate to note names or interval names and back.
 Uniform solfeges use the conventional consonants D R M F S L T. But all consonants except D have an alternate form that indicates flattening or sharpening:
@@ Line 26: / Line 26: @@
 === Example Scales ===
-{| class="wikitable  center-all"
+{| class="wikitable center-all"
 |+
 ! rowspan="2" |3-limit
@@ Line 103: / Line 103: @@
 Once this is fully internalized, add in the other vowels.
-== Interval Arithmetic ==
+==Interval Arithmetic==
 ===Octave Complements===
@@ Line 115: / Line 115: @@
 For example, Fru = minor-Re-up becomes major-Ti-down = To.
-===The Circle of Fifths===
+===An Edo's Circle of Fifths===
 The 13 -a notes form a chain of 5ths running from the dim 5th to the aug 4th:
@@ Line 126: / Line 126: @@
 '''Da''' Sa Ra La Ma Ta Pa/Sho Fro Flo No Tho Fo '''Do''' So Ro Lo Mo To Po/Fu '''Du''' Su Ru Lu Mu Tu Pu/Sha Fra Fla Na Tha Fa '''Da'''
-Thus as long as one spells the tritone correctly, all 5ths rhyme. This makes interval arithmetic very easy.
+Thus as long as one spells the tritones correctly, all 5ths in an edo rhyme. This makes interval arithmetic very easy.
 ===Adding/subtracting 4ths and 5ths===
@@ Line 133: / Line 133: @@
 However, consider the aug 4th, a P- note. The note a 5th above it would be an augmented 8ve, which doesn't exist in a uniform solfege. Therefore one must rename the tritone as a dim or mid 5th. Thus in 31edo Pa + 5th = Sho + 5th = Fro. Likewise, Sho and Sha need renaming when adding a 4th: Sha + 4th = Pu + 4th = Tu.
-===Adding/subtracting other intervals===
+===Adding/subtracting Other Intervals===
 The same rule for 4ths and 5ths mostly holds for plain major 2nds. Keep the vowel, and change the consonant as expected. Ra + M2 = Ma. But an aug 4th must be renamed as a 5th. Beware, this rule breaks down entirely for major and mid 7ths (the T- notes), due to the lack of aug and mid 8ves.
-In general, one can add or subtract any conventional (i.e. plain) interval from any note, and the result will be as expected. But only if the expected answer exists in the solfege. It must exist on the 13-note chain of 5ths from dim5 to aug4. In other words, the expected answer must not be augmented or diminished, unless it's an aug4 or a dim5. (Otherwise, one must use an enharmonic equivalent.) For example, one can easily add a M3 to any note other than a L-, M-, T- or P- note. Thus Ro + M3 = Po and Na + M3 = Sa, but La + M3 is a Fr- note. Beware, sometimes a chain is not 13 notes long, and when adding to or subtracting, the expected answer must exist on the shorter chain. For example, in 31edo, the -u chain runs from P4 to A4.
+In general, one can add or subtract any conventional (i.e. plain) interval from any note, and the result will be as expected. But only if the expected answer exists in the solfege. It must exist on the 13-note chain of 5ths from dim5 to aug4. In other words, the expected answer must not be augmented or diminished, unless it's an aug4 or a dim5. (Otherwise, one must use an enharmonic equivalent.) For example, one can easily add a M3 to any note other than a L-, M-, T- or P- note. Thus Ro + M3 = Po and Na + M3 = Sa, but La + M3 is a Fr- note. Beware, sometimes a chain is not 13 notes long, and when adding to or subtracting, the expected answer must exist on the shorter chain. For example, in 31edo, the -u chain only runs from P4 to A4.
-One can often easily add/subtract an unconventional (upped or downed) interval as well. The ups and downs add up and cancel out as expected. Thus Ra + vM2 = Mo and Ru + vM2 = Ma. Obviously the vowel will change. Again, the expected answer must exist in the solfege. 3-vowel solfeges lack double-ups and double-downs. 4-vowel solfeges lack double-upmajor and double-downminor. 5-vowel solfeges lack triple-ups and triple-downs.
+One can often easily add/subtract an unconventional (upped or downed) interval as well. The ups and downs add up and cancel out as expected. Thus Ra + vM2 = Mo and Ru + vM2 = Ma. Obviously the vowel will change. Again, the expected answer must exist in the solfege. 3-vowel solfeges lack double-ups and double-downs. 4-vowel solfeges lack double-upmajor and double-downminor.
 ==Solfeges for Edos==
@@ Line 171: / Line 171: @@
 |-
 !5 vowels
-!53, 60
+! 53, 60
 | -e = dud
 |<nowiki>-o = down</nowiki>
@@ Line 182: / Line 182: @@
 Because 72edo is such a popular edo, an exception is made and it has 2 additional vowels.
-===Examples===
+=== Examples===
 *12edo: Da Fra Ra Na Ma Fa Pa/Sha Sa Fla La Tha Ta Da
@@ Line 197: / Line 197: @@
 In edos with an even [[Sharpness|penta-sharpness]], there are "in-between" notes with two names. For example, 4\19 is named as both a 2nd and a 3rd (Ru/No).
-===Correlations with Color Notation===
+=== Correlations with Color Notation===
 -u/-o can mean not only up/down, but also under/over, meaning in the ratio's denominator or numerator. A [[color notation]] review:
-*yo/gu = 5-over/5-under = subtract/add 81/80
+* yo/gu = 5-over/5-under = subtract/add 81/80
 *zo/ru = 7-over/7-under = subtract/add 64/63
 *ilo/lu = 11-over/11-under = subtract/add [[729/704]]
@@ Line 208: / Line 208: @@
 If 81/80 maps to 1 edostep, then yo/gu = down/up = -o/-u. Likewise with the other commas. The table below shows that almost every edo has at least one such correlation. Parentheses are used when the prime's relative error is high, e.g. 12edo's prime 11.
-{| class="wikitable  center-all"
+{| class="wikitable center-all"
 |+
 colorspeak correlations for all 25 diatonic edos that have uniform solfeges with 5 vowels or less
@@ Line 215: / Line 215: @@
 !17
 !19
-!22
+! 22
 !24
 !26
@@ Line 454: / Line 454: @@
 <big>'''Main article: [[List of Uniform Solfeges For Pergens]]'''</big>
-Rank-2 temperaments have an infinite number of notes, so a solfege can only cover a fraction of them. But often one only needs enough notes to make a MOS scale. [[Pergen|Pergens]] tell us how to use ups and downs to notate these temperaments, and the same consonants and vowels can be used. Instead of circles of 5ths, there are '''fifthchains'''. Each fifthchain requires its own vowel, so there is a maximum of 5 fifthchains. However this can be extended to 9 fifthchains by using compound vowels such as -iyu.
+Rank-2 temperaments have an infinite number of notes, so a solfege can only cover a fraction of them. But often one only needs enough notes to make a MOS scale. [[Pergen|Pergens]] tell us how to use ups and downs to notate these temperaments, and the same consonants and vowels can be used. Instead of circles of 5ths, there are '''fifthchains'''. Each fifthchain requires its own vowel, so there is a maximum of 5 fifthchains. However this can be extended to 9 fifthchains by using compound vowels such as -iyu, see below.
 '''Genchains''' are distinct from fifthchains. Each pergen has one or more genchains, each of which contains one or more fifthchains. The 13 consonants and 5 vowels without compound vowels cover 20 pergens. Of course, the genchains can only extend so far with only 13 consonants. But in general, it's enough to cover all the modes of any reasonably-sized MOS scale.
@@ Line 461: / Line 461: @@
 Sometimes -i and -e mean lift/drop not dup/dud. -i never means mid, so there are only two vowel sequences:
-{| class="wikitable"
+{| class="wikitable center-all"
 |+unsplit (no-pair) solfege
-!-4
+! -4
 !-3
 !-2
@@ Line 483: / Line 483: @@
 !A<sup>4</sup>
 |-
-| -eye
+| -eye<br>"ay-yay"
-"ay-yay"
 | -eyo
 | -e
 | -o
-| -a
+|  -a
 | -u
 | -i
 | -iyu
-| -iyi
+|  -iyi<br>"ee-yee"
-"ee-yee"
 |}
-{| class="wikitable"
+{| class="wikitable center-all"
 |+single-pair solfeges
 !-4
@@ Line 527: / Line 525: @@
 | -iyi
 |}
-{| class="wikitable"
+{| class="wikitable center-all"
 |+double-pair solfeges
 |