41edo solfege: Difference between revisions

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The aug 4th Pa is also the updim 5th Shu, which is the starting point for another 13-note chain of 5ths, all -u notes. Since Pu = Si, this leads to a 6-note chain of -i notes. This in turn leads to a 13-note -o chain, which leads back to the -a chain. 13 -a notes + 13 -u notes + 13 -o notes + 6 -i notes = 45 names = 41 notes with duplicate names for the 4 tritones.

To summarize, the 4 vowels create 4 separate chains of 5ths, and the 4 tritones with duplicate names connect those 4 chains into one 41-note circle. This is ~~the main~~ rationale for the 13th consonant, for it supplies most of the duplicate names.

To summarize, the 4 vowels create 4 separate chains of 5ths, and the 4 tritones with duplicate names connect those 4 chains into one 41-note circle. This is one rationale for the 13th consonant, for it supplies most of the duplicate names. (The other is to get symmetry.)

{| class="wikitable" style="text-align:center;"

|+The 45 note names in circle-of-5ths order <br>

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===Adding/subtracting 4ths, 5ths and major 2nds===

Because the aforementioned 4 chains connect up, it's very easy to find the note a 4th or 5th above any note. It always ~~has the same vowel~~, and the consonant is as would be expected from conventional interval arithmetic. Ra plus a 4th is Sa, Fro plus a 5th is Flo, etc. Thus in the example scales above, the 3rd, 6th and 7th always rhyme, as do the tonic, 2nd, 4th and 5th.

Because the aforementioned 4 chains connect up, it's very easy to find the note a 4th or 5th above any note. It always rhymes, and the consonant is as would be expected from conventional interval arithmetic. Ra plus a 4th is Sa, Fro plus a 5th is Flo, etc. Thus in the example scales above, the 3rd, 6th and 7th always rhyme, as do the tonic, 2nd, 4th and 5th.

However the note a 5th above an aug or mid 4th would be an aug or mid 8ve, which doesn't exist in this solfege. Therefore one must rename the aug/mid 4th to a dim/mid 5th. Thus Po + 5th = Sha + 5th = Fra. Dim/mid 5ths may also need renaming: Sha + 4th = Po + 4th = To.

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A few minor exceptions arise with the -i notes. Conventionally, M7 + 5th = A4, and indeed Ta + 5th = Pa. But Ti + 5th = Fi not Pi. Likewise Fa + 4th = Tha, a minor 7th as expected, but Fi + 4th = Ti not Thi. These exceptions are not an issue as long as you remember that there is no Pi or Thi in the solfege.

The same rule for 4ths and 5ths ~~often~~ holds for plain major 2nds. Keep the vowel, and change the consonant as expected. Ra + M2 = Ma. Again, aug/dim intervals must be renamed: Po + M2 = ~~Sha~~ + M2 = ~~Fla~~. ~~But beware,~~ Fi to Si is a minor 2nd~~, and Fi + M2 = Flo~~. ~~Furthermore~~, this rule breaks down for major and mid 7ths, due to the lack of aug and mid 8ves:

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. Again, aug/dim intervals must be renamed: Fi + M2 = Sho + M2 = Flo. Note that Fi to Si is a <u>minor</u> 2nd. Beware, this rule breaks down entirely for major and mid 7ths, due to the lack of aug and mid 8ves:

*Tu + M2 = Ri

*Tu + M2 = Ri (^M7 + M2 = ~2)

*Ta + M2 = Fru

*Ta + M2 = Fru (M7 + M2 = ^m2)

*To + M2 = Fra

*To + M2 = Fra (vM7 + M2 = m2)

*Ti + M2 = Fro

*Ti + M2 = Fro (~7 + M2 = vm2)

===Learning suggestion===

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|Ka

|}

The only single-vowel scales are the dorian scale, and subsets of it.

*

[[Category:41edo]]

[[Category:Solfege]]

@@ Line 156: / Line 156: @@
 The aug 4th Pa is also the updim 5th Shu, which is the starting point for another 13-note chain of 5ths, all -u notes. Since Pu = Si, this leads to a 6-note chain of -i notes. This in turn leads to a 13-note -o chain, which leads back to the -a chain. 13 -a notes + 13 -u notes + 13 -o notes + 6 -i notes = 45 names = 41 notes with duplicate names for the 4 tritones.
-To summarize, the 4 vowels create 4 separate chains of 5ths, and the 4 tritones with duplicate names connect those 4 chains into one 41-note circle. This is the main rationale for the 13th consonant, for it supplies most of the duplicate names.
+To summarize, the 4 vowels create 4 separate chains of 5ths, and the 4 tritones with duplicate names connect those 4 chains into one 41-note circle. This is one rationale for the 13th consonant, for it supplies most of the duplicate names. (The other is to get symmetry.)
 {| class="wikitable" style="text-align:center;"
 |+The 45 note names in circle-of-5ths order <br>
@@ Line 230: / Line 230: @@
 ===Adding/subtracting 4ths, 5ths and major 2nds===
-Because the aforementioned 4 chains connect up, it's very easy to find the note a 4th or 5th above any note. It always has the same vowel, and the consonant is as would be expected from conventional interval arithmetic. Ra plus a 4th is Sa, Fro plus a 5th is Flo, etc. Thus in the example scales above, the 3rd, 6th and 7th always rhyme, as do the tonic, 2nd, 4th and 5th.
+Because the aforementioned 4 chains connect up, it's very easy to find the note a 4th or 5th above any note. It always rhymes, and the consonant is as would be expected from conventional interval arithmetic. Ra plus a 4th is Sa, Fro plus a 5th is Flo, etc. Thus in the example scales above, the 3rd, 6th and 7th always rhyme, as do the tonic, 2nd, 4th and 5th.
 However the note a 5th above an aug or mid 4th would be an aug or mid 8ve, which doesn't exist in this solfege. Therefore one must rename the aug/mid 4th to a dim/mid 5th. Thus Po + 5th = Sha + 5th = Fra. Dim/mid 5ths may also need renaming: Sha + 4th = Po + 4th = To.
@@ Line 236: / Line 236: @@
 A few minor exceptions arise with the -i notes. Conventionally, M7 + 5th = A4, and indeed Ta + 5th = Pa. But Ti + 5th = Fi not Pi. Likewise Fa + 4th = Tha, a minor 7th as expected, but Fi + 4th = Ti not Thi. These exceptions are not an issue as long as you remember that there is no Pi or Thi in the solfege.
-The same rule for 4ths and 5ths often holds for plain major 2nds. Keep the vowel, and change the consonant as expected. Ra + M2 = Ma. Again, aug/dim intervals must be renamed: Po + M2 = Sha + M2 = Fla. But beware, Fi to Si is a minor 2nd, and Fi + M2 = Flo. Furthermore, this rule breaks down for major and mid 7ths, due to the lack of aug and mid 8ves:
+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. Again, aug/dim intervals must be renamed: Fi + M2 = Sho + M2 = Flo. Note that Fi to Si is a <u>minor</u> 2nd. Beware, this rule breaks down entirely for major and mid 7ths, due to the lack of aug and mid 8ves:
-*Tu + M2 = Ri
+*Tu + M2 = Ri (^M7 + M2 = ~2)
-*Ta + M2 = Fru
+*Ta + M2 = Fru (M7 + M2 = ^m2)
-*To + M2 = Fra
+*To + M2 = Fra (vM7 + M2 = m2)
-*Ti + M2 = Fro
+*Ti + M2 = Fro (~7 + M2 = vm2)
 ===Learning suggestion===
@@ Line 509: / Line 509: @@
 |Ka
 |}
+The only single-vowel scales are the dorian scale, and subsets of it.
 *
 [[Category:41edo]]
 [[Category:Solfege]]