41edo solfege

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Uniform Solfege

Overview

See Uniform solfege for a full explanation.

41edo edosteps solfege ups and downs names
unisons 0-1 Da Du P1 ^1 C ^C
2nds 2-8 Fro Fra Fru Ri Ro Ra Ru vm2 m2 ^m2 ~2 vM2 M2 ^M2 vDb Db ^Db vvD vD D ^D
3rds 9-15 No Na Nu Mi Mo Ma Mu vm3 m3 ^m3 ~3 vM3 M3 ^M3 vEb Eb ^Eb vvE vE E ^E
4ths 16-18 Fo Fa Fu v4 P4 ^4 vF F ^F
tritones 19-22 Fi/Sho Po/Sha Pa/Shu Pu/Si ~4/vd5 vA4/d5 A4/^d5 ^A4/~5 ^^F/vGb vF#/Gb F#/^Gb ^F#/vvG
5ths 23-25 So Sa Su v5 P5 ^5 vG G ^G
6ths 26-32 Flo Fla Flu Li Lo La Lu vm6 m6 ^m6 ~6 vM6 M6 ^M6 vAb Ab ^Ab vvA vA A ^A
7ths 33-39 Tho Tha Thu Ti To Ta Tu vm7 m7 ^m7 ~7 vM7 M7 ^M7 vBb Bb ^Bb vvB vB B ^B
8ves 40-41 Do Da v8 P8 vC C

The seven 2nds illustrate the solfege's logic:

  • Fro = flat-Re-down = vm2
  • Fra = flat-Re-plain = m2
  • Fru = flat-Re-up = ^m2
  • Ri = Re-mid = ~2
  • Ro = Re-down = vM2
  • Ra = Re-plain = M2
  • Ru = Re-up = ^M2

The vowels relate to color notation: -a = wa, -o = yo or zo = over/otonal, -u = gu or ru = under/utonal, and -i = ila. The zogu 5th is Sha because the -o and -u in zogu cancel to make -a.

Example scales & tags

3-limit Plain major scale Da Ra Ma Fa Sa La Ta Da
Plain minor scale Da Ra Na Fa Sa Fla Tha Da
5-limit Downmajor scale Da Ra Mo Fa Sa Lo To Da
Upminor scale Da Ra Nu Fa Sa Flu Thu Da
7-limit Upmajor scale Da Ra Mu Fa Sa Lu Tu Da
Downminor scale Da Ra No Fa Sa Flo Tho Da
11-limit Mid scale Da Ra Mi Fa Sa Li Ti Da

See also these barbershop tags: Sweet Sweet Harmony (original tag) and Kite's translations of barbershop tags.

Kite Guitar fretboard

The various rainbows run either -o -u -o -u or else -a -i -a.

<-- nut bridge -->
Pa So Su Fla Li La Tho Thu To Tu Da
Ru Na Mi Ma Fo Fu Sha Si Sa Flo Flu Lo Lu Tha Ti Ta Do Du Fra Ri
Ti Ta Do Du Fra Ri Ra No Nu Mo Mu Fa Fi Pa So Su Fla Li La Tho
Da Fro Fru Ro Ru Na Mi Ma Fo Fu Sha

Suggestion for learning

Even with many familiar consonants and a consistent vowel sequence, it can take a while to master 45 syllables. One might want to take a divide-and-conquer approach. Start with replacing Do Re Mi etc. with this solfege:

Da - Ra - Ma - Fa - Sa - La - Ta - Da

This helps with unlearning the syllables Do, Mi, So and Ti, which are still present but have a changed meaning. (For those familiar with the full 17-name solfege, note that Ra, Ri, Fi, Si and Li are also present but changed.)

Once this solfege feels natural, add in the 6 altered consonants, making a 12-edo-like solfege:

Da - Fra - Ra - Na - Ma - Fa - Pa/Sha - Sa - Fla - La - Tha - Ta - Da

Once this is fully memorized, add in the other 3 vowels.

Octave Complements

To find the octave complement of any interval:

  • change the degree as usual: 2nd <--> 7th, 3rd <--> 6th, and 4th <--> 5th
  • change the quality as usual: major <--> minor, aug <--> dim, but perfect and mid are unchanged
  • get the new consonant from the degree and quality
  • change the vowel as expected: -o <--> -u, but -a and -i are unchanged

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

The Circle of Fifths

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

Sha - Fra - Fla - Na - Tha - Fa - Da - Sa - Ra - La - Ma - Ta - Pa

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 the ending point Pu is also 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 + 6 -i notes + 13 -o 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 one rationale for the 13th consonant P-, for it supplies most of the duplicate names.

The 45 note names in circle-of-5ths order
(read left-to-right, top-row-to-bottom-row)
d5 m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7 A4 (d5)
-a Da Sa Ra La Ma Ta Pa (Shu)
-u Shu Fru Flu Nu Thu Fu Du Su Ru Lu Mu Tu Pu (Si)
-i Si Ri Li Mi Ti Fi (Sho)
-o Sho Fro Flo No Tho Fo Do So Ro Lo Mo To Po (Sha)
-a Sha Fra Fla Na Tha Fa Da

Adding/subtracting 4ths and 5ths

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, consider the four tritones Fi, Po, Pa and Pu. The note a 5th above any of these would be some sort of augmented or mid 8ve, which doesn't exist in this solfege. Therefore one must rename the tritone as a dim or mid 5th. Thus Po + 5th = Sha + 5th = Fra. Likewise, Sho, Sha, Shu and Si need renaming when adding a 4th: Shu + 4th = Pa + 4th = Ta.

One minor exception arises with Ti and Fi. Conventionally, M7 + 5th = A4, and indeed Tu/Ta/To + 5th = Pu/Pa/Po. But Ti + 5th = Fi not Pi. Likewise Fu/Fa/Fo + 4th = Thu/Tha/Tho, minor 7ths 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. (What if we fix this by renaming Fi as Pi? Another issue arises: one would expect that Pi's octave complement would be Shi, but instead it's Si. What if Si were renamed Shi? Then Shi plus a 5th would make not Fri but rather Ri. So some sort of minor exception is inevitable.)

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 again the 4 tritones must be named as 5ths not 4ths: Fi + M2 = Sho + M2 = Flo. Note that Fi to Si is a minor 2nd. Beware, this rule breaks down entirely for major and mid 7ths (the four T- notes), due to the lack of aug and mid 8ves:

  • Tu + M2 = Ri (^M7 + M2 = ~9)
  • Ta + M2 = Fru (M7 + M2 = ^m9)
  • To + M2 = Fra (vM7 + M2 = m9)
  • Ti + M2 = Fro (~7 + M2 = vm9)

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 = Fru. Beware, because the -i chain is only 6 notes long, when adding to or subtracting from an -i note, the expected answer must exist on the P5-A4 chain.

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. No double-upmajor, double-upminor or double-downminor intervals! (Double-downmajor is mid, thus Ro + vM2 = Mi.)

Andrew Heathwaite's Solfege

Andrew's solfege expands on the conventional Do - Di/Ra - Re - Ri/Me - Mi - Fa - Fi/Se - Sol - Si/Le - La - Li/Te - Ti - Do. There are 8 vowels, with -ih, -eh, -aa and -u added. There are 6 different vowel sequences.

41edo solfege names ups and downs names edosteps
unisons Do Di P1 ^1 0-1
2nds Ro Rih Ra Ru Reh Re Ri vm2 m2 ^m2 ~2 vM2 M2 ^M2 2-8
3rds Ma Meh Me Mu Mi Maa Mo vm3 m3 ^m3 ~3 vM3 M3 ^M3 9-15
4ths Fe Fa Fih Fu Fi v4 P4 ^4 ~4 vA4 16-20
5ths Se Su Sih So (or Sol) Si ^d5 ~5 v5 P5 ^5 21-25
6ths Lo Leh Le Lu La Laa Li vm6 m6 ^m6 ~6 vM6 M6 ^M6 26-32
7ths Ta Teh Te Tu Ti Taa To vm7 m7 ^m7 ~7 vM7 M7 ^M7 33-39
8ves Da Do (Di) v8 P8 (^8) 40-41 (42)

See also Andrew's 31edo solfege, which is a subset of this solfege, and Phylingual‎'s 53edo solfege, which is very nearly a superset. (It names the M7 as Tih.)

Example scales

3- limit Plain major scale Do Re Maa Fa Sol Laa Taa Do
Plain minor scale Do Re Meh Fa Sol Leh Teh Do
5-limit Downmajor scale Do Re Mi Fa Sol La Ti Do
Upminor scale Do Re Me Fa Sol Le Te Do
7-limit Upmajor scale Do Re Mo Fa Sol Li To Do
Downminor scale Do Re Ma Fa Sol Lo Ta Do
11-limit Mid scale Do Re Mu Fa Sol Lu Tu Do

The two 5-limit scales are the same as conventional solfege.