myna temperament notation

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herman.miller
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Joined: Sun Sep 06, 2015 8:27 am

myna temperament notation

Post by herman.miller »

Myna temperament (7-limit 4p&27p or 11-limit 31p&58p) is one that might be a good test case for different notation strategies. With the mapping [<1 -1 0 1 -3] <0 10 9 7 25]>, it's going to need many different sizes of accidentals to fill the gaps between the fifths, and it doesn't get any better if you use the 4-note MOS for the nominals in place of a chain of fifths. So I tried the strategy of notating the simplest JI representation of a tempered interval. This seems to be working out well for the simpler, most common JI intervals, e.g.

[+1, -1> 5/3 B:\!:
[-2, +9> 5/4 F:||\:
[-1, +7> 7/4 C:!):

When you get to [+1, -2>, the simplest representation is 7/5, which requires a schisma accent if you write it in Sagittal as G:.::||): , but A:!!!(: is an alternative that doesn't require an accent. We can ignore the schisma accent if we're using fewer than 89 notes, since 32805/32768 maps to [-23, +89>. This strategy works reasonably well for 7-limit myna until you get to +/- 24, where [-6, +24> is represented as 81/70 and [+7, -24> as 140/81; the JI calculator spreadsheet gives :\ \!:F and :/ /|:B as the Athenian approximations for these pitches. The biggest problem with this approach is that you get redundant symbols for the same tempered comma, e.g.

[-1, +4> 36/35 D:/|):
[+5, -16> 49/27 C:~|): = C [+6, -20> :~|): [-1, +4>

Adding 11-limit intervals gives simpler notations for some notes, e.g.

[-3, +15> 11/6 C:/|\:
[-6, +24> 55/48 E:|\:

On the other hand, some intervals like [-4, +18> bring in Sagittal symbols that are less commonly seen than the 7-limit notation.

[-4, +18> 25/16 A:)||(:
[-4, +18> 11/7 A:)||~:

Based on these options, I chose a single symbol for each comma. The choice of :|\: for [-1, +4> instead of :/|): or :~|): was motivated by flag addition: :/|: + :|\: = :/|\: .

[+15, -58> 896/891 :)|(:
[-8, +31> 81/80 :/|:
[+7, -27> 64/63 :|):
[-1, +4> 55/54 :|\:
[-9, +35> 33/32 :/|\:
[-2, +8> 25/24 :)||(:
[-10, +39> 135/128 :||\:

The one interval that has a couple of options is [+6, -23>, which I'm notating as 28/27: it can be represented approximately as :(|\: (Athenian) or :|\ \: (Promethean). Since I'm not using any other Promethean symbols, I chose to use :(|\: and ignore the schisma accent. So what I ended up with looks like a reasonable notation for myna temperament. Notating temperaments as JI is a good starting point, but you can get better results with some careful observation.

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Dave Keenan
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Re: myna temperament notation

Post by Dave Keenan »

Good work there, Herman! I'm sorry I can't spare the time to investigate enough to suggest any alternatives.

It certainly underscores the point that linear temperament notations are still more art than science.
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