The "wrap-around" mnemonic for native nominals

When music students first learn standard meantone-based notation, how do they learn that A-C is a minor third but C-E is a major third, A-G is a perfect fifth but B-F is a diminished fifth, and so on? You basically just have to memorize them, right? It would make about as much sense if A-A were any mode other than aeolian.

With MOSes of the form 1Lxs or xL1s (where the generator is one step), on the other hand, there is one assignment of nominals that clearly makes more sense than the others, which is to make consecutive letters of the alphabet always be generators. The step that "wraps around" back to A is the odd step. This mnemonic doesn't just apply to steps, though, but to all intervals! I'll explain using porcupine as an example.

This dictates that porcupine[7] should be notated such that A-B, B-C... F-G are all porcupine generators, and G-A is the large step because it wraps back around.

Now, what about thirds? It's simple: if the letters are in the usual order in the alphabet (A-C, B-D, C-E, D-F, E-G), it's a minor third; whereas if the letters "wrap around" back to the beginning of the alphabet (F-A, G-B), it's a major third.

Similarly, if the letters of a fourth are in alphabet order (A-D, B-E, C-F, D-G), it's a porcupine minor fourth (~4/3), whereas if the letters wrap around (E-A, F-B, G-C), it's a porcupine major fourth (~11/8).

If you are have standard notation in mind, all of this will be just as confusing as any other assignment of nominals. But if you forget standard notation and just think of the alphabet, I think this is a great mnemonic with the potential to be intuitive and fast.

Other examples:

For slendric[5], A B C D E A where E-A is the large step (~7/6). Three-step intervals that don't wrap are 3/2 whereas ones that wrap are 32/21, a comma wide of 3/2.

For machine[6], A B C D E F A where F-A is the small step. Thirds that don't wrap are major (~9/7) whereas thirds that wrap are minor (~11/9). Three-step intervals that don't wrap are 16/11 whereas ones that do wrap are 11/8.

For tetracot[7], A B C D E F G A where G-A is the small step. Thirds that don't wrap are 11/9 whereas thirds that wrap are 6/5. Fifths that don't wrap are 3/2 whereas fifths that wrap are narrow.

For bleu[9], A B C D E F G H I A where I-A is the small step. Two-step intervals that don't wrap are 7/6 whereas ones that wrap are 8/7; three-step intervals that don't wrap are 9/7 whereas ones that wrap are 11/9; four-step intervals that don't wrap are 11/8 whereas ones that wrap are 4/3.

Keenan

This is a great mnemonic, I never even thought about it this way. I guess that adds a bonus to Porcupine as a notational basis for 15-ET...though, a pentatonic notation based on Blackwood[10] has NO wrap-around steps, consecutive letters are always the same interval, unless altered by accidentals....

-Igs

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> When music students first learn standard meantone-based notation, how do they learn that A-C is a minor third but C-E is a major third, A-G is a perfect fifth but B-F is a diminished fifth, and so on? You basically just have to memorize them, right? It would make about as much sense if A-A were any mode other than aeolian.
>
> With MOSes of the form 1Lxs or xL1s (where the generator is one step), on the other hand, there is one assignment of nominals that clearly makes more sense than the others, which is to make consecutive letters of the alphabet always be generators. The step that "wraps around" back to A is the odd step. This mnemonic doesn't just apply to steps, though, but to all intervals! I'll explain using porcupine as an example.
>
> This dictates that porcupine[7] should be notated such that A-B, B-C... F-G are all porcupine generators, and G-A is the large step because it wraps back around.
>
> Now, what about thirds? It's simple: if the letters are in the usual order in the alphabet (A-C, B-D, C-E, D-F, E-G), it's a minor third; whereas if the letters "wrap around" back to the beginning of the alphabet (F-A, G-B), it's a major third.
>
> Similarly, if the letters of a fourth are in alphabet order (A-D, B-E, C-F, D-G), it's a porcupine minor fourth (~4/3), whereas if the letters wrap around (E-A, F-B, G-C), it's a porcupine major fourth (~11/8).
>
> If you are have standard notation in mind, all of this will be just as confusing as any other assignment of nominals. But if you forget standard notation and just think of the alphabet, I think this is a great mnemonic with the potential to be intuitive and fast.
>
> Other examples:
>
> For slendric[5], A B C D E A where E-A is the large step (~7/6). Three-step intervals that don't wrap are 3/2 whereas ones that wrap are 32/21, a comma wide of 3/2.
>
> For machine[6], A B C D E F A where F-A is the small step. Thirds that don't wrap are major (~9/7) whereas thirds that wrap are minor (~11/9). Three-step intervals that don't wrap are 16/11 whereas ones that do wrap are 11/8.
>
> For tetracot[7], A B C D E F G A where G-A is the small step. Thirds that don't wrap are 11/9 whereas thirds that wrap are 6/5. Fifths that don't wrap are 3/2 whereas fifths that wrap are narrow.
>
> For bleu[9], A B C D E F G H I A where I-A is the small step. Two-step intervals that don't wrap are 7/6 whereas ones that wrap are 8/7; three-step intervals that don't wrap are 9/7 whereas ones that wrap are 11/9; four-step intervals that don't wrap are 11/8 whereas ones that wrap are 4/3.
>
> Keenan
>