Fractions of a comma for European tunings and temperaments c. 1500-1900

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michaelkudirka
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Real Name: Michael Kudirka

Fractions of a comma for European tunings and temperaments c. 1500-1900

Post by michaelkudirka »

Just got off an extremely informative video chat with Doug Blumeyer after meeting him at the recent Microtonal Adventures Festival in Bellingham, WA.

As someone who works mostly in Western European tunings and temperaments, I'm very interested in starting to use Saggital notation in my presentations. I think that the saggitals will be great for efficiently showing the adjustment of the Pythagorean backbone of western music as it moved through several stages of evolution after the Middle Ages from 1/4-comma meantone, through 1/5-comma, and 1/6-comma, and the explosion of circulating temperaments in the 18th century.

My question is this: it seems to me that the saggitals are indicating various important commas, but in the tunings I use, the adjustments to the chained 3/2s are (almost) always made by some fraction of a comma. Is there a way to use a saggital to indicate a fraction of a comma such as the syntonic?

Thanks!
Michael Kudirka - new member of the forum

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Dave Keenan
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Re: Fractions of a comma for European tunings and temperaments c. 1500-1900

Post by Dave Keenan »

Hi Michael. Welcome to the forum. Good questions.

What fractions of the syntonic comma do you need to represent relative to the Pythagorean fifths?

The primary role or definition of a sagittal symbol is always a whole comma of some kind, but it is an important aspect of Sagittal that we redefine symbols for specific purposes, that are close in size in the tuning or temperament under consideration. The symbol :/|: has the whole syntonic comma as its primary role. In figure 11 on page 20 of https://sagittal.org/sagittal.pdf you can see :|(: used to represent 1/4-syntonic-comma. And we have the accent mark :': to represent the schisma, which is close to 1/11th-comma, and might be defined as 1/10th comma or 1/12-comma depending on what you combine it with, so that you could symbolise 1/5 and 1/6 commas.

The schisma accent can be added or subtracted from other symbols. So :'::|(: might represent 1/3-comma and :.::|(: 1/6-comma. :'::/|: represents the Pythagorean comma (by definition). If there is no other symbol combined with the schisma accent, it should be followed by a bare shaft symbol to indicate its direction. :'::|: for 1/11-comma up, and :.::!: for 1/11-comma down. And :'::'::|: could represent 1/5-comma.

For 1/2 and 3/4 commas, you would not need to use the other symbols shown in figure 11. You could just use :|(::|(: and :|(::|(::|(:. And when combined with whole comma symbols, the convention is that symbols that represent smaller alterations go to the left (further from the notehead) :|(::/|: .

Thanks to @ndentonprotsack, we have recently developed the "Stoic" set of symbols that merge such sequences into a single symbol, :|((: :|(((: :/|((: . See this thread: viewtopic.php?p=4525#p4525

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