Harmonisma: Difference between revisions

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As it happens, the difference between 11/9 and 13/11 is 121/117, a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.

In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273c) and 91/88 (58.036c) at 364/363 apart (4.763c); 91/88 and 121/117 at 10648/10647 apart (0.163c); 121/117 and 28/27 at 62.961c at 364/363 (4.763c) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688c) apart. Thus 896/891 = (352/351 x 364/363), and also (364/363 x 10648/10647 x 364/363).

In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273c) and 91/88 (58.036c) at 364/363 apart (4.763c); 91/88 and 121/117 at 10648/10647 apart (0.163c); 121/117 and 28/27 at 62.961c at 364/363 (4.763c) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688c) apart. Thus 896/891 = (352/351 x 364/363), and also (364/363 x 10648/10647 x 364/363). A difference of 352/351 (4.925c) or (364/363 x 10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.

Revision as of 01:13, 10 November 2022 view source Mschulter1325 (talk \| contribs) autopatrolled, emailconfirmed 31 edits m Deleted my own unpredictably misformatted diagram; added text to make same points in place of diagram. Tags: Visual edit Mobile edit Mobile web edit ← Older edit		Revision as of 01:18, 10 November 2022 view source Mschulter1325 (talk \| contribs) autopatrolled, emailconfirmed 31 edits Added a sentence to account for 352/351 in prose replacing my diagram which I had problems formatting.~~~~ Tags: Visual edit Mobile edit Mobile web edit Newer edit →
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	As it happens, the difference between 11/9 and 13/11 is 121/117, a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.		As it happens, the difference between 11/9 and 13/11 is 121/117, a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.

	In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273c) and 91/88 (58.036c) at 364/363 apart (4.763c); 91/88 and 121/117 at 10648/10647 apart (0.163c); 121/117 and 28/27 at 62.961c at 364/363 (4.763c) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688c) apart. Thus 896/891 = (352/351 x 364/363), and also (364/363 x 10648/10647 x 364/363).		In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273c) and 91/88 (58.036c) at 364/363 apart (4.763c); 91/88 and 121/117 at 10648/10647 apart (0.163c); 121/117 and 28/27 at 62.961c at 364/363 (4.763c) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688c) apart. Thus 896/891 = (352/351 x 364/363), and also (364/363 x 10648/10647 x 364/363). A difference of 352/351 (4.925c) or (364/363 x 10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.