Kite's uniform solfege: Difference between revisions

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== 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"

|+[[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%.

=== 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 ==

@@ Line 562: / Line 562: @@
 == 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"
 |+[[Fifthspan|Fifthspans]] of various primes in various single-ring edos
@@ Line 675: / Line 677: @@
 |}
 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. 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 ==