Kite's uniform solfege - Revision history

TallKite: TallKite moved page Uniform solfege to Kite's uniform solfege: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

2025-08-20T00:46:11Z

TallKite moved page Uniform solfege to Kite's uniform solfege: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

← Older revision	Revision as of 00:46, 20 August 2025
(No difference)

TallKite: /* Solfeges for rank-2 temperaments */ updated EI to EU

2024-12-28T23:44:37Z

Solfeges for rank-2 temperaments: updated EI to EU

← Older revision		Revision as of 23:44, 28 December 2024
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	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.		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 EI (enharmonic ~~interval~~, "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 the ~~pergen, the EI~~ and the vowel sequence. For each pergen, there is one "official" solfege.		'''[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:		Sometimes -i and -e mean lift/drop not dup/dud. -i never means mid, so there are only two vowel sequences:

TallKite: /* Examples */

2024-05-10T23:20:41Z

Examples

← Older revision		Revision as of 23:20, 10 May 2024
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	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 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 ===

TallKite: /* Rank-2 temperaments */

2023-12-18T06:05:00Z

Rank-2 temperaments

← Older revision		Revision as of 06:05, 18 December 2023
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	\|}		\|}

	== ~~Application to fifthspan mappings~~ ==		== Applications ==

	=== EDOs ===		=== EDOs ===
	In any single-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.		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"		{\| class="wikitable"
	\|+[[Fifthspan\|Fifthspans]] of various primes in various single-ring edos		\|+[[Fifthspan\|Fifthspans]] of various primes in various single-ring edos
Line 629:		Line 629:
	Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.		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 [[archy]] edos have Tha as the 2nd syllable, ~~etc~~.		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.		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.
Line 671:		Line 671:
	\|vM3		\|vM3
	\|vvm7		\|vvm7
	\|^^^P4 or ~~vvvA4~~		\|^<sup>3</sup>P4 or v<sup>3</sup>A4
	\|^^^m6 or ~~vvvM6~~		\|^<sup>3</sup>m6 or v<sup>3</sup>M6
	\|MoThePeyoLeyo		\|MoThePeyoLeyo
	\|Peyo=Fiyu, Leyo=Fliyu		\|Peyo=Fiyu, Leyo=Fliyu
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	The second row of the temperament's mapping directly yields a solfege string. This string, plus the [[pergen]], can serve to concisely name the temperament.		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.		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 ~~the pergen/string naming format~~ covers ~~all <u>~~reasonably accurate~~</u>~~ temperaments. ~~For pergens that split the octave into 5 or more periods, [[Val#Shorthand notation\|wart notation]] can possibly be used. (to do: elaborate)~~		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.

	== ~~Application to~~ Bosanquet keyboards ==		=== 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.		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.

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	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.		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 ~~section~~). 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.		<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.		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]]		[[Category:Solfege]]

TallKite: /* Rank-2 temperaments */

2023-12-14T03:31:12Z

Rank-2 temperaments

← Older revision		Revision as of 03:31, 14 December 2023
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	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.		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 the pergen/string naming format covers all <u>~~reasonable~~</u> ~~varieties of the temperament~~. For pergens that split the octave into 5 or more periods, [[Val#Shorthand notation\|wart notation]] can possibly be used. (to do: elaborate)		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 the pergen/string naming format covers all <u>reasonably accurate</u> temperaments. For pergens that split the octave into 5 or more periods, [[Val#Shorthand notation\|wart notation]] can possibly be used. (to do: elaborate)

	== Application to Bosanquet keyboards ==		== Application to Bosanquet keyboards ==

TallKite: /* Application to fifthspan mappings */ more details

2023-12-14T03:29:26Z

Application to fifthspan mappings: more details

← Older revision		Revision as of 03:29, 14 December 2023
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	The second row of the temperament's mapping directly yields a solfege string. This string, plus the pergen, can serve to 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". The ~~temperament isn~~'t precisely ~~defined~~, 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 the pergen/string naming format ~~defines~~ all <u>reasonable</u> varieties of the temperament.		The second row of the temperament's mapping directly yields a solfege string. 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 the pergen/string naming format covers all <u>reasonable</u> varieties of the temperament. For pergens that split the octave into 5 or more periods, [[Val#Shorthand notation\|wart notation]] can possibly be used. (to do: elaborate)

	== Application to Bosanquet keyboards ==		== Application to Bosanquet keyboards ==

TallKite: /* Application to fifthspan mappings */ added temperament names using pergen and solfege strings

2023-12-13T22:50:51Z

Application to fifthspan mappings: added temperament names using pergen and solfege strings

← Older revision		Revision as of 22:50, 13 December 2023
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	== Application to fifthspan mappings ==		== Application to fifthspan mappings ==
	In any single-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 string 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.
			=== EDOs ===
			In any single-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"		{\| class="wikitable"
	\|+[[Fifthspan\|Fifthspans]] of various primes in various single-ring edos		\|+[[Fifthspan\|Fifthspans]] of various primes in various single-ring edos
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	\|}		\|}
	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%.		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. This string, plus the pergen, can serve to 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". The temperament isn't precisely defined, 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 the pergen/string naming format defines all <u>reasonable</u> varieties of the temperament.

	== Application to Bosanquet keyboards ==		== Application to Bosanquet keyboards ==

Fredg999: Improve structure, add distinguish template

2023-12-12T06:28:04Z

Improve structure, add distinguish template

← Older revision		Revision as of 06:28, 12 December 2023
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:		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:

TallKite: Added a few paragraphs

2023-01-09T08:25:20Z

Added a few paragraphs

← Older revision		Revision as of 08:25, 9 January 2023
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	== Application to fifthspan mappings ==		== Application to fifthspan mappings ==
	In any single-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 ~~of interest above 3~~ is called the fifthspan mapping. The mapping can be expressed very concisely as a string 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.		In any single-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 string 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.
	{\| class="wikitable"		{\| class="wikitable"
	\|+[[Fifthspan\|Fifthspans]] of various primes in various single-ring edos		\|+[[Fifthspan\|Fifthspans]] of various primes in various single-ring edos
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	!prime 13		!prime 13
	!solfege string		!solfege string
			!alternates
	\|-		\|-
	![[19-edo]]		![[19-edo]]
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	\| -4		\| -4
	\|MaThoPaFla		\|MaThoPaFla
			\|Tho=Lu
	\|-		\|-
	![[22-edo]]		![[22-edo]]
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	\| -9		\| -9
	\|RuThaShaTho		\|RuThaShaTho
			\|Ru=Sho, Tho=Pu
	\|-		\|-
	![[31-edo]]		![[31-edo]]
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	\|15		\|15
	\|MaLuShoSi		\|MaLuShoSi
			\|Sho=Mi, Si=The
	\|-		\|-
	![[41-edo]]		![[41-edo]]
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	\|20		\|20
	\|FoDeFlePi		\|FoDeFlePi
			\|Fle=Riyu, Pi=Deyo
	\|-		\|-
	![[53edo\|53-edo]]		![[53edo\|53-edo]]
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	\|20		\|20
	\|FoDeRiyuPi		\|FoDeRiyuPi
			\|Riyu=Theye
	\|}		\|}
	Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.		Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.

	~~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 solfege string is also MaLuShoThe~~.		The solfege string for all meantone edos starts with Ma, all [[archy]] edos have Tha as the 2nd syllable, etc.

	In a multi-ring edo such as 72, ~~the vowels~~ must be repurposed to mean up/down. ~~Note that this system fails if a ring has more than 13 notes, since there~~ are ~~only 13 consonants and one vowel for each ring~~.		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"		{\| class="wikitable"
	\|+Solfege strings for various multi-ring edos		\|+Solfege strings for various multi-ring edos
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	!prime 13		!prime 13
	!solfege string		!solfege string
			!alternates
	\|-		\|-
	![[15-edo]]		![[15-edo]]
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	\|(N/A)		\|(N/A)
	\|MoThaFu		\|MoThaFu
			\|
	\|-		\|-
	![[24-edo]]		![[24-edo]]
Line 644:		Line 654:
	\|^4 or vA4		\|^4 or vA4
	\|^m6 or vM6		\|^m6 or vM6
	\|~~MaThoFuFlu or~~		\|MaThoPoLo
	~~MaThoPoLo~~		\|Po=Fu, Lo=Flu
	\|-		\|-
	![[34-edo]]		![[34-edo]]
	\|vM3		\|vM3
	\|m7		\|(N/A)
	\|~4		\|^^4
	\|~6		\|^^m6
	\|~~MoThaFiLi~~		\|MeAPoLo
			\|Po=Fu, Lo=Flu
	\|-		\|-
	![[72-edo]]		![[72-edo]]
Line 659:		Line 670:
	\|^^^P4 or vvvA4		\|^^^P4 or vvvA4
	\|^^^m6 or vvvM6		\|^^^m6 or vvvM6
	\|~~MoTheFiyuFliyu or~~		\|MoThePeyoLeyo
	~~MoThePeyoLeyo~~		\|Peyo=Fiyu, Leyo=Fliyu
	\|}		\|}
	~~The solfege string for all meantone edos starts with Ma~~, ~~all [[schismatic]] edos start with Fo~~, ~~all [[archy]] edos have Tha as~~ the ~~2nd syllable~~, ~~etc~~.		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%.

	== Application to Bosanquet keyboards ==		== Application to 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. This picture shows the names if Da corresponds to the note C.		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]]		[[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.		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.
Line 671:		Line 684:
	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.		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.

	The placement of various primes on a Bosanquet keyboard is determined by the fifthspan mapping (see the previous section). Thus ~~the~~ solfege string tells a lumatone player the physical placement of ~~all~~ the primes ~~of interest~~.		<u>Solfege strings</u>: The placement of various primes on a Bosanquet keyboard is determined by the fifthspan mapping (see the previous section). 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]]		[[Category:Solfege]]

TallKite: added a section on fifthspan mappings

2023-01-09T06:56:14Z

added a section on fifthspan mappings

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