User:2^67-1/Sandbox

Testing

The blackwood-dicot-semaphore equivalence continuum is a continuum of 7-limit rank-3 temperaments describing the set of all 7-limit rank-3 temperaments supported by 10edo. Any rank-2 temperament supported by 10edo can thus be represented by a line between two points in this continuum.

All temperaments in the continuum satisfy (25/24)^p(49/48)^q ~ 256/243, equating a stack of dicot commas (25/24) and semaphore commas (49/48) with the blackwood comma (256/243).

The blackwood comma is the characteristic 3-limit comma tempered out in 10edo.

User:Godtone notes the following JIP's, each one corresponding to a 1D continua contained therein, for which increasingly efficient approximations generally represents increasingly efficient 7-limit temperaments:

• log₂(256/243) / log₂(25/24 * 49/48) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
• log₂(256/243) / log₂(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)
• log₂(256/243) / log₂(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)
• log₂(256/243) / log₂(2401/2400) = 125.1... ; this is the JIP of p=-1, q=2 (equiv. to p=1, q=-2)
• log₂(256/243) / log₂(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
• log₂(256/243) / log₂(25/24 * 50/49) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)

Importantly, each JIP corresponds to a rational, so that, for example, (p, q) = (1, -2) is equivalent to (p, q) = (2, -4) and to (p, q) = (-1, 2) but not to (1, 2). Note that all these JIPs lie on the JIL (just intonation line).

Also note that continua separated by 2401/2400 are meaningfully different, but due to the efficiency of 2401/2400, one may want to examine the continuum of all 7-limit temperaments supported by 10edo for which 2401/2400 is tempered.

Selected temperaments with integer p and q

p	q	Temperament	Comma		Added by someone else?
			Ratio	Monzo
0	0	Blackwood	256/243	[8 -5 0 0⟩	n
0	1	Archytas (squared)	4096/3969	[12 -4 0 -2⟩	n
0	2	Buzzard	65536/64827	[16 -3 0 -4⟩	n
0	2.5 = 5/2	Slendrismic	68719476736/68641485507	[36 -5 0 -10⟩	Y
0	2.6 = 13/5	2.3.7 5 & 171	(28 digits)	[-92 12 0 26⟩	Y
0	2.6 = 8/3	2.3.7 5 & 212	72680419155717387/72057594037927936	[-56 7 0 16⟩	n
0	3	Slendric (squared)	1058841/1048576	[-20 2 0 6⟩	n
0	∞	Semaphore	49/48	[-4 -1 0 2⟩	n
1	0	Srutal	2048/2025	[11 -4 -2 0⟩	n
1.25 = 5/4	0	Quintosec	140737488355328/140126044921875	[47 -15 -10 0⟩	n
1.3 = 13/10	0	2.3.5 10 & 171	(36 digits)	[-119 37 26 0⟩	n
1.3 = 4/3	0	Submajor	69198046875/68719476736	[-36 11 8⟩	Y
1.5 = 3/2	0	2.3.5 Miracle	34171875/33554432	[-25 7 6⟩	Y
2	0	Negri	16875/16384	[-14 3 4 0⟩	n
∞	0	Dicot	25/24	[-3 -1 2 0⟩	n
1	1	Mirwomo	33075/32768	[15 -3 -2 -2⟩	n
0.857142 = 6/7	0.857142 = 6/7	10 & 77 & 308	(30 digits)	[-98 23 12 12⟩	Y
1	∞	Jubilic	50/49	[1 0 2 -2⟩	n
∞	1	Jubilic	50/49	[1 0 2 -2⟩	n
∞	2	Breedsmic	2401/2400	[-5 -1 -2 4⟩	n