Kite's uniform solfege: Difference between revisions

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~~==Overview==~~

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''' 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.

== Theory ==

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 13:

Line 14:

Sharpening and flattening refers to adding/subtracting the [[2187/2048|lawa unison (Lw1) aka apotome]]. Mnemonic for Pa: Sh- sharpens to S-, and Th- sharpens to T-, so if Fa were spelled Pha, it would sharpen to Pa.

The vowel sequence varies slightly depending on the context. ~~From high to low:~~

The vowel sequence runs from high to low -i -u -a -o -e. The meaning varies slightly depending on the context.

*-i = dup (~~double-up~~) ~~or double-augmented~~ (~~or for rank-2~~, ~~possibly lift)~~

(1) Whenever ups and downs are used (most edos, and all single-pair [[Pergen|pergens]]):

*-u = '''u'''p or augmented

*-a = pl'''a'''in i.e. 3-limit

*-o = d'''o'''wn or diminished

*-e = dud (double-down) or double-diminished (or for rank-~~2, possibly drop~~)

~~The augmented and diminished meanings only apply to tunings that don~~'~~t require ups and downs, such as sharp~~-~~1 edos and unsplit pergens.~~

*-i = dup (double-up)

*-u = '''u'''p

*-a = pl'''a'''in

*-o = d'''o'''wn

*-e = dud (double-down)

~~The 5 vowels are like those in Spanish or Italian. There are only 5 vowels because those are the most singable~~, and ~~also additional vowels would make the solfeges harder to learn. Shi is pronounced "she" and She is "shay". Fri~~ is ~~"free"~~.

However for certain edos such as 34 and 41, -i means m'''i'''d and -e is not used.

===Example ~~Scales~~===

(2) Whenever ups and downs are not used ([[Sharpness|sharp-1]] edos and the unsplit pergen):

*-i = double-augmented

*-u = a'''u'''gmented

*-a = n'''a'''tural

*-o = diminished

*-e = double-diminished

(3) Whenever both ups/downs and lifts/drops are used (double-pair pergens):

From high to low:

*-i = l'''i'''ft

*-u = '''u'''p

*-a = pl'''a'''in

*-o = d'''o'''wn

*-e = drop

The 5 vowels are pronounced like those in Spanish or Italian. There are only 5 vowels because those are the most singable, and also additional vowels would make the solfeges harder to learn. Shi is pronounced "she" and She is "shay". Fri is "free".

===Example scales===

{| class="wikitable center-all"

|+

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==Interval Arithmetic==

===Octave ~~Complements~~ ===

===Octave complements ===

To find the [[octave complement]] of any interval:

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

===An ~~Edo~~'s ~~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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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.

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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~~==

== Solfeges for edos==

In the perfect edos (7, 14, 21, 28 and 35), there is no need for the altered consonants, since ~~Lw1 is tempered out~~. ~~Some other~~ edos ~~(12~~, ~~??)~~ also omit ~~them~~.

In the perfect edos (7, 14, 21, 28 and 35), there is no need for the altered consonants, since major and minor are equated. Thus these edos only use 7 consonants. Edos 5, 11 and 13 also omit some of the consonants.

{| class="wikitable center-all"

|+the four vowel sequences, with example edos

!1 vowel

!5, 7, 12

!5, 7, 9, 12

|

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|-

!3 vowels

!10, 14-22~~, etc,~~

!10, 13b-19, 22

|

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

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|-

!4 vowels

!31, 41

!25, 27, 34, 41

|<nowiki>-i = mid</nowiki>

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

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|-

!5 vowels

!53, 60

!43, 46, 53, 60

| -e = dud

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

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Line 199:

| -i = dup

|}

There is only so much one can do with 5 vowels and 13 consonants. Not all edos are covered. The number of vowels an edo's solfege needs equals the edo's [[sharpness|sharpness or penta-sharpness]], whichever is larger. Thus an edo with a (penta)sharpness of 6 or higher needs 6 or more vowels and isn't covered. Every edo above 60 is such an edo. The excluded edos are the less efficient ones, with a fairly inaccurate 5th for their size. Thus they tend to be the less popular edos.

There is only so much one can do with 5 vowels and 13 consonants. Not all edos are covered. The number of vowels an edo's solfege needs equals the edo's [[sharpness|sharpness or penta-sharpness]], whichever is larger. Thus an edo with a (penta)sharpness of 6 or higher needs 6 or more vowels and isn't covered. Every edo above 60 is such an edo. The excluded edos are the less efficient ones, with a fairly large size, or a fairly inaccurate 5th for their size. Thus they tend to be the less popular edos.

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

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*21edo: Da Du Ro Ra Ru Mo Ma Mu Fo Fa Fu So Sa Su Lo La Lu To Ta Tu Do Da

Superflat edos have a very flat 5th. A uniform solfege can still be used, but the size of the interval won't match what its name implies very well.

Superflat edos (9, 11, 13b, 16, 18b and 23) have a very flat 5th. A uniform solfege can still be used, but the size of the interval won't match what its name implies very well.

In sharp-1 edos, to up an interval means to augment it. Thus Fu = Pa and So = Sha. Fru = Ra and Fra = Ro.

In flat-1 edos, to up an interval means to diminish it. Fo = Pa and Su = Sha. Fro = Ra and Fra = Ru.

In flat-1 edos (9, 16 and 23), to up an interval means to diminish it. Fo = Pa and Su = Sha. Fro = Ra and Fra = Ru.

In sharp-2 and sharp-4 edos, mid is spelled as downmajor, ~~or upperfect for~~ the 4th, ~~or downperfect for~~ the 5th.

In sharp-2 and sharp-4 edos, the mid 2nd/3rd/6th/7th is spelled as downmajor, the mid 4th is spelled as upperfect, and the mid 5th is downperfect.

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).

In edos with an even [[Sharpness|penta-sharpness]], there are interordinal 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:

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|}

==Solfeges for ~~Rank~~-2 ~~Temperaments~~==

==Solfeges for rank-2 temperaments==

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~~.

'''Genchains''' are distinct from fifthchains. Each pergen has one or more genchains, each of which contains one or more fifthchains. The number of genchains always equals the number of periods per octave. In general, the number of fifthchains per genchain equals the number of generators per '''multigen''', which is the 2nd interval in the pergen. Thus (P8, P4/3) has 1 genchain that contains 3 interwoven fifthchains. But an exception arises when the multigen is not perfect (i.e. major or minor). Then the fifthchains hop from one genchain to the next. The only such pergen that has a uniform solfege is (P8/2, M2/4). It has 2 genchains and a total of 4 fifthchains.

~~This pdf~~ '''[https://Tallkite.com/misc files/notation guide for rank-2 pergens.pdf ~~tallkite~~.com/misc_files/notation guide for rank-2 pergens.pdf]''' lists many pergens. The tuning of every interval and every accidental is defined in terms of c = P5 - 700¢. The EI (enharmonic ~~interval~~) can be added to or subtracted from any note or interval to get an equivalent note or interval. The entire solfege can be derived from the ~~pergen, the EI~~ and the vowel sequence.

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.

'''[https://Tallkite.com/misc files/notation guide for rank-2 pergens.pdf TallKite.com/misc_files/notation guide for rank-2 pergens.pdf]''' lists many pergens. The tuning of every interval and every accidental is defined in terms of c = P5 - 700¢. The EU (enharmonic unison, "E" in the pdf) can be added to or subtracted from any note or interval to get an equivalent note or interval. The entire solfege can be derived from just the EU and the vowel sequence. For each pergen, there is one "official" solfege.

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

~~{| class="wikitable center-all"~~

~~|+unsplit (no-pair) solfege~~

~~!-4~~

~~!-3~~

~~!-2~~

~~!-1~~

!0

~~! 1~~

!2

!3

!4

|-

~~!d4~~

~~!ddd~~

~~!dd~~

~~!dim~~

~~!plain~~

~~!aug~~

~~!AA~~

~~!AAA~~

~~!A4~~

|-

~~| -eye "ay-yay"~~

~~| -eyo~~

~~| -e~~

~~| -o~~

~~| -a~~

~~| -u~~

~~| -i~~

~~| -iyu~~

~~| -iyi "ee-yee"~~

|}

{| class="wikitable center-all"

|+single-pair solfeges

~~!-4~~

!-3

!-2

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!2

!3

!4

|-

~~!quud~~

!trud

!dud

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!dup

!trup

~~!quup~~

|-

| ~~-eye~~

| -eyo

| -e

| -o

| -a

| -u

| -i

| -iyu

~~| -iyi~~

|}

{| class="wikitable center-all"

|+double-pair solfeges

|

!

!down

!plain

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|-

!lift

| -owi

| -i

| -uwi

|-

!plain

| -o

| -a

| -u

|-

! drop

| -owe

| -e

| -uwe

|}

Any compound vowel with a "w" is for double-pair only. Mnemonic: w = "double-U" = double-pair.

The whole (or unsplit) pergen (P8, P5) doesn't use ups and downs. It uses the single-pair vowel sequence, with -u and -o repurposed to mean augmented and diminished:

{| class="wikitable center-all"

|+(P8, P5) solfege

!-3

!-2

!-1

!0

! 1

!2

!3

|-

!ddd

!dd

!dim

!plain

!aug

!AA

!AAA

|-

| -eyo

| -e

| -o

| -a

| -u

| -i

| -iyu

|}

== Applications ==

=== EDOs ===

In any single-[[Ring number|ring]] edo, a prime can be mapped not only to a specific number of edosteps, but also to a specific number of fifths. This is called the [[fifthspan]]. The fifthspan of prime 2 is always zero and the fifthspan of prime 3 is always one. The fifthspans of all the primes is called the fifthspan mapping. The mapping can be expressed very concisely as a '''solfege string''', a list of uniform solfege syllables in which -u/-o means aug/dim. Note that this often differs from the EDO solfeges listed above, where -u/-o often refers to up/down. Primes 2 and 3 are always DaSa by definition, so these two primes are omitted from the string.

{| class="wikitable"

|+[[Fifthspan|Fifthspans]] of various primes in various single-ring edos

!

!prime 2

!prime 3

!prime 5

!prime 7

!prime 11

!prime 13

!solfege string

!alternates

|-

![[19-edo]]

|0

|1

|4

| -9

|6

| -4

|MaThoPaFla

|Tho=Lu

|-

![[22-edo]]

|0

|1

|9

| -2

| -6

| -9

|RuThaShaTho

|Ru=Sho, Tho=Pu

|-

![[31-edo]]

|0

|1

|4

|10

| -13

|15

|MaLuShoSi

|Sho=Mi, Si=The

|-

![[41-edo]]

|0

|1

| -8

| -14

| -18

|20

|FoDeFlePi

|Fle=Riyu, Pi=Deyo

|-

![[53edo|53-edo]]

|0

|1

| -8

| -14

|23

|20

|FoDeRiyuPi

|Riyu=Theye

|}

Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.

The solfege string for all [[meantone]] edos starts with Ma, all [[Schismatic family|schismatic]] edos start with Fo, all [[archy]] edos have Tha as the 2nd syllable, and so forth.

Each prime has a second, larger fifthspan which is found by adding/subtracting the edo itself. For example, 31edo's prime 13 fifthspan is 15 but also 15 - 31 = -16. Thus 31edo's alternate solfege string is MaLuShoThe. The alternate fifthspan is usually only of interest if the smaller fifthspan approaches half the edo, and the alternate fifthspan is only slightly more remote.

In a multi-ring edo such as 72, -u/-o must be repurposed to mean up/down. The alternates in the table below are exactly as remote as the primary names.

{| class="wikitable"

|+Solfege strings for various multi-ring edos

!

!prime 5

!prime 7

!prime 11

!prime 13

!solfege string

!alternates

|-

![[15-edo]]

|vM3

|m7

|^4

|(N/A)

|MoThaFu

|

|-

![[24-edo]]

|M3

|vm7

|^4 or vA4

|^m6 or vM6

|MaThoPoLo

|Po=Fu, Lo=Flu

|-

![[34-edo]]

|vM3

|(N/A)

|^^4

|^^m6

|MeAPoLo

|Po=Fu, Lo=Flu

|-

![[72-edo]]

|vM3

|vvm7

|^3P4 or v3A4

|^3m6 or v3M6

|MoThePeyoLeyo

|Peyo=Fiyu, Leyo=Fliyu

|}

34edo is an unusual case. Each ring has 17 notes, which is more than 13 consonants and 1 vowel can cover. So -u/-o means up/down within the ring, and -i/-e means lift/drop by an edostep from one ring to the next. Note the use of -A- to exclude prime 7, which in 34edo has a huge relative error of 45%.

=== Rank-2 temperaments ===

The second row of the temperament's mapping directly yields a solfege string (see the previous section). This string, plus the [[pergen]], can serve to concisely name the temperament.

For example, 11-limit [[Porcupine|Triyo/Porcupine]] has a mapping [⟨1 2 3 2 4], ⟨0 -3 -5 6 -4]]. The pergen is (P8, P4/3), and its solfege is given [[List of uniform solfeges for pergens#.237 .28P8.2C P4.2F3.29 third-4th|here]]. One simply uses syllables from columns 0, -3, -5, 6 and -4 to get DaSaMoThaFu. Since primes 2 and 3 are always DaSa by definition, they can be omitted. The temperament can be defined by the pergen plus the solfege string as "third-4th MoThaFu". Two 13-limit extensions are MoThaFuLo and MoThaFuSi. More examples: [[Pajara]] is "half-8ve MoTha" and [[Injera]] is "half-8ve MaThu". You can tell injera is in the meantone family because the first solfege is Ma. You can tell it's a weak extension of meantone because the pergen differs from meantone's.

The solfege string doesn't precisely define the temperament, since the first row of the mapping isn't used, and theoretically those numbers could change. But unless the period is a small fraction of an octave, such alternate mappings will be extremely inaccurate. So this nomenclature only covers reasonably accurate temperaments.

=== Bosanquet keyboards ===

Using fixed-solfege, each physical key on the Lumatone can be named. It's best to let -u/-o mean aug/dim not up/down, since the meaning of ups and downs changes in different edos. For example, in 31edo an up equals a step in the 5:00 direction, but in 41edo it's the opposite, a step in the 11:00 direction.

This picture shows the solfege names if Da corresponds to the note C.

[[File:Lumatone 41edo with solfege.jpg|none|thumb]]

The uppermost few keys use -iyi ("ee-yee"), meaning quadruple-aug. One could set Da to D not C, in order to get a more symmetrical layout, and thus change two of the three -iyi's to -eyo's.

Using movable-solfege, one can name the notes of a scale independently of the key. One can also name any physical interval on the lumatone. For example, one step in the 1:00 direction is always Du, two steps in the 2:30 direction is always Ma, etc.

Solfege strings: The placement of various primes on a Bosanquet keyboard is determined by the fifthspan mapping (see the previous sections). Thus an edo's solfege string tells a lumatone player the physical placement of various primes. Notes ending in -u/-i lie on the top half of the keyboard and those ending-o/-e lie on the bottom half. The alternate fifthspan is sometimes useful to bring one prime nearer the others. For example, 41edo's solfege string is FoDeFlePi, with prime 13 being an outlier. It's alternate string is FoDeFleDeyo, which makes for more compact chord shapes.

The solfege string can be used to compare edos. For example 41edo is FoDeFlePi (or FoDeFleDeyo) and 53edo is FoDeRiyuPi. This tells us that primes 5, 7 and perhaps 13 are placed similarly, but prime 11 differs. Thus any 7-limit chord's shape is the same in both 41edo and 53edo.

[[Category:Solfege]]

@@ Line 1: / Line 1: @@
-==Overview==
+{{Distinguish|Universal solfege}}
-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''' 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.
+== Theory ==
 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 13: / Line 14: @@
 Sharpening and flattening refers to adding/subtracting the [[2187/2048|lawa unison (Lw1) aka apotome]]. Mnemonic for Pa: Sh- sharpens to S-, and Th- sharpens to T-, so if Fa were spelled Pha, it would sharpen to Pa.
-The vowel sequence varies slightly depending on the context. From high to low:
+The vowel sequence runs from high to low -i -u -a -o -e. The meaning varies slightly depending on the context.
-*-i = dup (double-up) or double-augmented (or for rank-2, possibly lift)
+(1) Whenever ups and downs are used (most edos, and all single-pair [[Pergen|pergens]]):
-*-u = '''u'''p or augmented
-*-a = pl'''a'''in i.e. 3-limit
-*-o = d'''o'''wn or diminished
-*-e = dud (double-down) or double-diminished (or for rank-2, possibly drop)
-The augmented and diminished meanings only apply to tunings that don't require ups and downs, such as sharp-1 edos and unsplit pergens.
+*-i = dup (double-up)
+*-u = '''u'''p
+*-a = pl'''a'''in
+*-o = d'''o'''wn
+*-e = dud (double-down)
-The 5 vowels are like those in Spanish or Italian. There are only 5 vowels because those are the most singable, and also additional vowels would make the solfeges harder to learn. Shi is pronounced "she" and She is "shay". Fri is "free".
+However for certain edos such as 34 and 41, -i means m'''<u>i</u>'''d and -e is not used.
-===Example Scales===
+(2) Whenever ups and downs are not used ([[Sharpness|sharp-1]] edos and the unsplit pergen):
+*-i = double-augmented
+*-u = a'''u'''gmented
+*-a = n'''a'''tural
+*-o = diminished
+*-e = double-diminished
+(3) Whenever both ups/downs and lifts/drops are used (double-pair pergens):
+From high to low:
+*-i = l'''<u>i</u>'''ft
+*-u = '''u'''p
+*-a = pl'''a'''in
+*-o = d'''o'''wn
+*-e = drop
+The 5 vowels are pronounced like those in Spanish or Italian. There are only 5 vowels because those are the most singable, and also additional vowels would make the solfeges harder to learn. Shi is pronounced "she" and She is "shay". Fri is "free".
+===Example scales===
 {| class="wikitable center-all"
 |+
@@ Line 105: / Line 126: @@
 ==Interval Arithmetic==
-===Octave Complements ===
+===Octave complements ===
 To find the [[octave complement]] of any interval:
@@ Line 115: / Line 136: @@
 For example, Fru = minor-Re-up becomes major-Ti-down = To.
-===An Edo's 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 133: / Line 154: @@
 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.
@@ Line 140: / Line 161: @@
 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==
+== Solfeges for edos==
 {{Main|List of uniform solfeges for EDOs}}
-In the perfect edos (7, 14, 21, 28 and 35), there is no need for the altered consonants, since Lw1 is tempered out. Some other edos (12, ??) also omit them.
+In the perfect edos (7, 14, 21, 28 and 35), there is no need for the altered consonants, since major and minor are equated. Thus these edos only use 7 consonants. Edos 5, 11 and 13 also omit some of the consonants.
 {| class="wikitable center-all"
 |+the four vowel sequences, with example edos
 !1 vowel
-!5, 7, 12
+!5, 7, 9, 12
 |
 |
@@ Line 155: / Line 176: @@
 |-
 !3 vowels
-!10, 14-22, etc,
+!10, 13b-19, 22
 |
 |<nowiki>-o = down</nowiki>
@@ Line 163: / Line 184: @@
 |-
 !4 vowels
-!31, 41
+!25, 27, 34, 41
 |<nowiki>-i = mid</nowiki>
 |<nowiki>-o = down</nowiki>
@@ Line 171: / Line 192: @@
 |-
 !5 vowels
-!53, 60
+!43, 46, 53, 60
 | -e = dud
 |<nowiki>-o = down</nowiki>
@@ Line 178: / Line 199: @@
 |  -i = dup
 |}
-There is only so much one can do with 5 vowels and 13 consonants. Not all edos are covered. The number of vowels an edo's solfege needs equals the edo's [[sharpness|sharpness or penta-sharpness]], whichever is larger. Thus an edo with a (penta)sharpness of 6 or higher needs 6 or more vowels and isn't covered. Every edo above 60 is such an edo. The excluded edos are the less efficient ones, with a fairly inaccurate 5th for their size. Thus they tend to be the less popular edos.
+There is only so much one can do with 5 vowels and 13 consonants. Not all edos are covered. The number of vowels an edo's solfege needs equals the edo's [[sharpness|sharpness or penta-sharpness]], whichever is larger. Thus an edo with a (penta)sharpness of 6 or higher needs 6 or more vowels and isn't covered. Every edo above 60 is such an edo. The excluded edos are the less efficient ones, with a fairly large size, or a fairly inaccurate 5th for their size. Thus they tend to be the less popular edos.
 Because 72edo is such a popular edo, an exception is made and it has 2 additional vowels.
@@ Line 187: / Line 208: @@
 *21edo: Da Du Ro Ra Ru Mo Ma Mu Fo Fa Fu So Sa Su Lo La Lu To Ta Tu Do Da
-Superflat edos have a very flat 5th. A uniform solfege can still be used, but the size of the interval won't match what its name implies very well.
+Superflat edos (9, 11, 13b, 16, 18b and 23) have a very flat 5th. A uniform solfege can still be used, but the size of the interval won't match what its name implies very well.
 In sharp-1 edos, to up an interval means to augment it. Thus Fu = Pa and So = Sha. Fru = Ra and Fra = Ro.
-In flat-1 edos, to up an interval means to diminish it. Fo = Pa and Su = Sha. Fro = Ra and Fra = Ru.
+In flat-1 edos (9, 16 and 23), to up an interval means to diminish it. Fo = Pa and Su = Sha. Fro = Ra and Fra = Ru.
-In sharp-2 and sharp-4 edos, mid is spelled as downmajor, or upperfect for the 4th, or downperfect for the 5th.
+In sharp-2 and sharp-4 edos, the mid 2nd/3rd/6th/7th is spelled as downmajor, the mid 4th is spelled as upperfect, and the mid 5th is downperfect.
-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).
+In edos with an even [[Sharpness|penta-sharpness]], there are interordinal 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:
@@ Line 451: / Line 472: @@
 |}
-==Solfeges for Rank-2 Temperaments==
+==Solfeges for rank-2 temperaments==
-{{Main|List of Uniform Solfeges For Pergens}}
+{{Main|List of uniform solfeges for pergens}}
 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.
+'''Genchains''' are distinct from fifthchains. Each pergen has one or more genchains, each of which contains one or more fifthchains. The number of genchains always equals the number of periods per octave. In general, the number of fifthchains per genchain equals the number of generators per '''multigen''', which is the 2nd interval in the pergen. Thus (P8, P4/3) has 1 genchain that contains 3 interwoven fifthchains. But an exception arises when the multigen is not perfect (i.e. major or minor). Then the fifthchains hop from one genchain to the next. The only such pergen that has a uniform solfege is (P8/2, M2/4). It has 2 genchains and a total of 4 fifthchains.
-This pdf '''[https://Tallkite.com/misc&#x20;files/notation&#x20;guide&#x20;for&#x20;rank-2&#x20;pergens.pdf tallkite.com/misc_files/notation guide for rank-2 pergens.pdf]''' lists many pergens. The tuning of every interval and every accidental is defined in terms of c = P5 - 700¢. The EI (enharmonic interval) can be added to or subtracted from any note or interval to get an equivalent note or interval. The entire solfege can be derived from the pergen, the EI and the vowel sequence.
+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.
+'''[https://Tallkite.com/misc&#x20;files/notation&#x20;guide&#x20;for&#x20;rank-2&#x20;pergens.pdf TallKite.com/misc_files/notation guide for rank-2 pergens.pdf]''' lists many pergens. The tuning of every interval and every accidental is defined in terms of c = P5 - 700¢. The EU (enharmonic unison, "E" in the pdf) can be added to or subtracted from any note or interval to get an equivalent note or interval. The entire solfege can be derived from just the EU and the vowel sequence. For each pergen, there is one "official" solfege.
 Sometimes -i and -e mean lift/drop not dup/dud. -i never means mid, so there are only two vowel sequences:
-{| class="wikitable center-all"
-|+unsplit (no-pair) solfege
-!-4
-!-3
-!-2
-!-1
-!0
-! 1
-!2
-!3
-!4
-|-
-!d<sup>4</sup>
-!ddd
-!dd
-!dim
-!plain
-!aug
-!AA
-!AAA
-!A<sup>4</sup>
-|-
-| -eye<br>"ay-yay"
-| -eyo
-| -e
-| -o
-| -a
-| -u
-| -i
-| -iyu
-|  -iyi<br>"ee-yee"
-|}
 {| class="wikitable center-all"
 |+single-pair solfeges
-!-4
 !-3
 !-2
@@ Line 503: / Line 492: @@
 !2
 !3
-!4
 |-
-!quud
 !trud
 !dud
@@ Line 513: / Line 500: @@
 !dup
 !trup
-!quup
 |-
-| -eye
+|  -eyo
-| -eyo
+|  -e
-| -e
+|  -o
-| -o
+|  -a
-| -a
+|  -u
-| -u
+|  -i
-| -i
+|  -iyu
-| -iyu
-| -iyi
 |}
 {| class="wikitable center-all"
 |+double-pair solfeges
-|
+!
 !down
 !plain
@@ Line 533: / Line 517: @@
 |-
 !lift
-| -owi
+|  -owi
-| -i
+|  -i
-| -uwi
+|  -uwi
 |-
 !plain
-| -o
+|  -o
-| -a
+|  -a
-| -u
+|  -u
 |-
 ! drop
-| -owe
+|  -owe
-| -e
+|  -e
-| -uwe
+|  -uwe
 |}
 Any compound vowel with a "w" is for double-pair only. Mnemonic: w = "double-U" = double-pair.
+The whole (or unsplit) pergen (P8, P5) doesn't use ups and downs. It uses the single-pair vowel sequence, with -u and -o repurposed to mean augmented and diminished:
+{| class="wikitable center-all"
+|+(P8, P5) solfege
+!-3
+!-2
+!-1
+!0
+! 1
+!2
+!3
+|-
+!ddd
+!dd
+!dim
+!plain
+!aug
+!AA
+!AAA
+|-
+|  -eyo
+|  -e
+|  -o
+|  -a
+|  -u
+|  -i
+|  -iyu
+|}
+== Applications ==
+=== EDOs ===
+In any single-[[Ring number|ring]] edo, a prime can be mapped not only to a specific number of edosteps, but also to a specific number of fifths. This is called the [[fifthspan]]. The fifthspan of prime 2 is always zero and the fifthspan of prime 3 is always one. The fifthspans of all the primes is called the fifthspan mapping. The mapping can be expressed very concisely as a '''solfege string''', a list of uniform solfege syllables in which -u/-o means aug/dim. Note that this often differs from the EDO solfeges listed above, where -u/-o often refers to up/down. Primes 2 and 3 are always DaSa by definition, so these two primes are omitted from the string.
+{| class="wikitable"
+|+[[Fifthspan|Fifthspans]] of various primes in various single-ring edos
+!
+!prime 2
+!prime 3
+!prime 5
+!prime 7
+!prime 11
+!prime 13
+!solfege string
+!alternates
+|-
+![[19-edo]]
+|0
+|1
+|4
+| -9
+|6
+| -4
+|MaThoPaFla
+|Tho=Lu
+|-
+![[22-edo]]
+|0
+|1
+|9
+| -2
+| -6
+| -9
+|RuThaShaTho
+|Ru=Sho, Tho=Pu
+|-
+![[31-edo]]
+|0
+|1
+|4
+|10
+| -13
+|15
+|MaLuShoSi
+|Sho=Mi, Si=The
+|-
+![[41-edo]]
+|0
+|1
+| -8
+| -14
+| -18
+|20
+|FoDeFlePi
+|Fle=Riyu, Pi=Deyo
+|-
+![[53edo|53-edo]]
+|0
+|1
+| -8
+| -14
+|23
+|20
+|FoDeRiyuPi
+|Riyu=Theye
+|}
+Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.
+The solfege string for all [[meantone]] edos starts with Ma, all [[Schismatic family|schismatic]] edos start with Fo, all [[archy]] edos have Tha as the 2nd syllable, and so forth.
+Each prime has a second, larger fifthspan which is found by adding/subtracting the edo itself. For example, 31edo's prime 13 fifthspan is 15 but also 15 - 31 = -16. Thus 31edo's alternate solfege string is MaLuShoThe. The alternate fifthspan is usually only of interest if the smaller fifthspan approaches half the edo, and the alternate fifthspan is only slightly more remote.
+In a multi-ring edo such as 72, -u/-o must be repurposed to mean up/down. The alternates in the table below are exactly as remote as the primary names.
+{| class="wikitable"
+|+Solfege strings for various multi-ring edos
+!
+!prime 5
+!prime 7
+!prime 11
+!prime 13
+!solfege string
+!alternates
+|-
+![[15-edo]]
+|vM3
+|m7
+|^4
+|(N/A)
+|MoThaFu
+|
+|-
+![[24-edo]]
+|M3
+|vm7
+|^4 or vA4
+|^m6 or vM6
+|MaThoPoLo
+|Po=Fu, Lo=Flu
+|-
+![[34-edo]]
+|vM3
+|(N/A)
+|^^4
+|^^m6
+|MeAPoLo
+|Po=Fu, Lo=Flu
+|-
+![[72-edo]]
+|vM3
+|vvm7
+|^<sup>3</sup>P4 or v<sup>3</sup>A4
+|^<sup>3</sup>m6 or v<sup>3</sup>M6
+|MoThePeyoLeyo
+|Peyo=Fiyu, Leyo=Fliyu
+|}
+edo is an unusual case. Each ring has 17 notes, which is more than 13 consonants and 1 vowel can cover. So -u/-o means up/down within the ring, and -i/-e means lift/drop by an edostep from one ring to the next. Note the use of -A- to exclude prime 7, which in 34edo has a huge relative error of 45%.
+=== Rank-2 temperaments ===
+The second row of the temperament's mapping directly yields a solfege string (see the previous section). This string, plus the [[pergen]], can serve to concisely name the temperament.
+For example, 11-limit [[Porcupine|Triyo/Porcupine]] has a mapping [⟨1 2 3 2 4], ⟨0 -3 -5 6 -4]]. The pergen is (P8, P4/3), and its solfege is given [[List of uniform solfeges for pergens#.237 .28P8.2C P4.2F3.29 third-4th|here]]. One simply uses syllables from columns 0, -3, -5, 6 and -4 to get DaSaMoThaFu. Since primes 2 and 3 are always DaSa by definition, they can be omitted. The temperament can be defined by the pergen plus the solfege string as "third-4th MoThaFu". Two 13-limit extensions are MoThaFuLo and MoThaFuSi. More examples: [[Pajara]] is "half-8ve MoTha" and [[Injera]] is "half-8ve MaThu". You can tell injera is in the meantone family because the first solfege is Ma. You can tell it's a weak extension of meantone because the pergen differs from meantone's.
+The solfege string doesn't precisely define the temperament, since the first row of the mapping isn't used, and theoretically those numbers could change. But unless the period is a small fraction of an octave, such alternate mappings will be extremely inaccurate. So this nomenclature only covers reasonably accurate temperaments.
+=== Bosanquet keyboards ===
+Using fixed-solfege, each physical key on the Lumatone can be named. It's best to let -u/-o mean aug/dim not up/down, since the meaning of ups and downs changes in different edos. For example, in 31edo an up equals a step in the 5:00 direction, but in 41edo it's the opposite, a step in the 11:00 direction.
+This picture shows the solfege names if Da corresponds to the note C.
+[[File:Lumatone 41edo with solfege.jpg|none|thumb]]
+The uppermost few keys use -iyi ("ee-yee"), meaning quadruple-aug. One could set Da to D not C, in order to get a more symmetrical layout, and thus change two of the three -iyi's to -eyo's.
+Using movable-solfege, one can name the notes of a scale independently of the key. One can also name any physical interval on the lumatone. For example, one step in the 1:00 direction is always Du, two steps in the 2:30 direction is always Ma, etc.
+<u>Solfege strings</u>: The placement of various primes on a Bosanquet keyboard is determined by the fifthspan mapping (see the previous sections). Thus an edo's solfege string tells a lumatone player the physical placement of various primes. Notes ending in -u/-i lie on the top half of the keyboard and those ending-o/-e lie on the bottom half. The alternate fifthspan is sometimes useful to bring one prime nearer the others. For example, 41edo's solfege string is FoDeFlePi, with prime 13 being an outlier. It's alternate string is FoDeFleDeyo, which makes for more compact chord shapes.
+The solfege string can be used to compare edos. For example 41edo is FoDeFlePi (or FoDeFleDeyo) and 53edo is FoDeRiyuPi. This tells us that primes 5, 7 and perhaps 13 are placed similarly, but prime 11 differs. Thus any 7-limit chord's shape is the same in both 41edo and 53edo.
 [[Category:Solfege]]