User:Overthink/Miscellaneous ideas - Revision history

Overthink: Move this here from user page

2026-02-06T05:37:38Z

Move this here from user page

← Older revision		Revision as of 05:37, 6 February 2026
Line 38:		Line 38:

	<nowiki/>*By flat, this means being flat of the fifth of [[argent tuning]], or 702.944 cents, rather than the just fifth. Note that in the formula, the argent fifth corresponds to itself.		<nowiki/>*By flat, this means being flat of the fifth of [[argent tuning]], or 702.944 cents, rather than the just fifth. Note that in the formula, the argent fifth corresponds to itself.

			== Miscellaneous thoughts ==
			The 10-form structure (temperaments include pajara and miracle) should really be explored more, and also the 14-form (temperaments include octacot, squares, and harry) and possibly also 15-form (temperaments include tritikleismic and valentine).

Overthink: /* Meantone-superpyth fifth pairs */ added just 3/2

2025-12-06T19:20:14Z

Meantone-superpyth fifth pairs: added just 3/2

← Older revision		Revision as of 19:20, 6 December 2025
Line 32:		Line 32:

	[[53edo\|31\53]] ~ [[75edo\|44\75]]		[[53edo\|31\53]] ~ [[75edo\|44\75]]

			(Just 3/2) 701.955{{c}} ~ 703.932{{c}}

	[[41edo\|24\41]] ~ [[29edo\|17\29]]		[[41edo\|24\41]] ~ [[29edo\|17\29]]

	<nowiki/>*By flat, this means being flat of the fifth of [[argent tuning]], or 702.944 cents, rather than the just fifth. Note that in the formula, the argent fifth corresponds to itself.		<nowiki/>*By flat, this means being flat of the fifth of [[argent tuning]], or 702.944 cents, rather than the just fifth. Note that in the formula, the argent fifth corresponds to itself.

Overthink: created page with one idea

2025-12-06T19:08:51Z

created page with one idea

New page

This is a page containing ideas that are not (yet) developed enough to get its own page.
== Meantone-superpyth fifth pairs ==
Let the perfect fifth in a tuning be <math>f/o</math> octaves. If a fifth is flat*, the corresponding sharp fifth is <math>(2o-2f)/(2o-f)</math> octaves, or <math>(2o-2f)</math> steps of <math>(2o-f)</math>-edo, and the same formula is used to obtain the corresponding flat fifth of a sharp one. As an example, the perfect fifth of [[12edo]] is <math>7/12</math> octaves, so <math>f=7</math> and <math>o=12</math>. The corresponding sharp fifth is <math>(2\cdot12-2\cdot7)</math> steps of <math>(2\cdot12-7)</math>-edo, or 10 steps of [[17edo]]. We apply the same formula on 10\17 to get <math>(2\cdot17-2\cdot10)/(2\cdot17-10)=14/24=7/12</math> octaves.

We can check that the formula can be applied twice to get back the original fifth as follows: If our original fifth was <math>f_1/o_1</math> octaves, the corresponding fifth will be <math>(2o_1-2f_1)/(2o_1-f_1)</math> octaves. We plug this into the formula with <math>f_2=2o_1-2f_1</math> and <math>o_2=2o_1-f_1</math>, and we get <math>(2(2o_1-f_1)-2(2o_1-2f_1))/(2(2o_1-f_1)-(2o_1-2f_1))=(2f_1)/(2o_1)=f_1/o_1</math> octaves.

This formula can also work if the fifth is not a rational fraction of the octave; if we let the fifth be <math>a</math> octaves, the corresponding fifth is <math>(2-2a)/(2-a)</math> octaves.

Some examples of corresponding fifth pairs are below.

[[7edo|4\7]] ~ [[5edo|3\5]]

[[26edo|15\26]] ~ [[37edo|22\37]]

[[45edo|26\45]] ~ [[32edo|19\32]]

[[19edo|11\19]] ~ [[27edo|16\27]]

[[69edo|40\69]] ~ [[49edo|29\49]]

[[50edo|29\50]] ~ [[71edo|42\71]]

[[31edo|18\31]] ~ [[22edo|13\22]]

[[43edo|25\43]] ~ [[61edo|36\61]]

[[55edo|32\55]] ~ [[39edo|23\39]]

[[12edo|7\12]] ~ [[17edo|10\17]]

[[65edo|38\65]] ~ [[46edo|27\46]]

[[53edo|31\53]] ~ [[75edo|44\75]]

[[41edo|24\41]] ~ [[29edo|17\29]]

<nowiki/>*By flat, this means being flat of the fifth of [[argent tuning]], or 702.944 cents, rather than the just fifth. Note that in the formula, the argent fifth corresponds to itself.