User:Lériendil/On the primes

The way each of the primes tends to behave within xenharmony is largely defined by the numbers that neighbor it. Every prime (bar 2) splits some superparticular in half, and that prime's neighbors are twice the numerator and denominator of that superparticular. Other isoharmonic sequences, such as those which differ by 2, 3, or 4, can also be used, but are much less likely to split intervals that belong to significantly lower prime-limits. However, the process of splitting superparticulars in half induces primes 2 and 3 necessarily, and so isoharmonies of deltas 2 and 3 are fundamental to no-twos or no-threes structures.

Modularity classes

Primes belonging to different residue classes modulo 4, 6, and 10 have meaningfully different properties in relation to splitting isoharmonies.

Modulo 4, primes (apart from 2) can be 4k+1 or 4k-1.

• 4k+1 primes (5, 13, 17, ...) we call "upper".
• 4k-1 primes (3, 7, 11, 19, ...) we call "lower".
• Upper × upper = lower × lower = upper. Upper × lower = lower.

Modulo 6, primes (apart from 2 and 3) can be 6k+1 or 6k-1.

• 6k+1 primes (7, 13, 19, ...) we call "cold".
• 6k-1 primes (5, 11, 17, ...) we call "hot".
• 3 is neutral, but can be grouped with the cold primes by convention.
• Cold × cold = hot × hot = cold. Cold × hot = hot.

Modulo 10, primes (apart from 2 and 5) can be 10k+1, 10k+3, 10k-3, or 10k-1.

• 10k±1 primes (11, 19, 29, 31, ...) we call "telluric".
• 10k±3 primes (3, 7, 13, 17, 23, ...) we call "thalassic".
• 5 is neutral, but can be grouped with the telluric primes by convention.
• Telluric × telluric = thalassic × thalassic = telluric. Telluric × thalassic = thalassic.

Table of the primes by neighbor

Central relations

Delta-1

All subgroups to which the prime is added include 2 and 3 as necessary elements. Recorded in this table are only the additional primes beyond 2 and 3 needed to express each prime's neighbors. As every prime bar 2 and 3 neighbors a multiple of six, that neighbor (the "sixfold neighbor") is listed first.

Prime		Adjacent integers			S-expression of split superparticular
Value	Modularity	Sixfold neighbor's factorization	Nonsixfold neighbor's factorization	Subgroup
5	H, U, T*	2×3	2×2	2.3	S12: 3/2
7	C, L, Th	2×3	2×22	2.3	S2: 4/3
11	H, L, T	2×3×2	2×5	2.3.5	S13: 6/5
13	C, U, Th	2×3×2	2×7	2.3.7
17	H, U, Th	2×3×3	2×23	2.3	S3: 9/8
19	C, L, T	2×3×3	2×2×5	2.3.5	S14: 10/9
23	H, L, Th	2×3×22	2×11	2.3.11
29	H, U, T	2×3×5	2×2×7	2.3.5.7	S15: 15/14
31	C, L, T	2×3×5	2×24	2.3.5	S4: 16/15
37	C, U, Th	2×3×2×3	2×19	2.3.19
41	H, U, T	2×3×7	2×22×5	2.3.5.7	S16: 21/20
43	C, L, Th	2×3×7	2×2×11	2.3.7.11
47	H, L, Th	2×3×23	2×23	2.3.23
53	H, U, Th	2×3×32	2×2×13	2.3.13

Delta-2 (no-twos)

All subgroups to which the prime is added include 3 as a necessary element. Additionally, these subgroups include primes' twin counterparts. Twin primes will be excluded from the coloring based on prime-limits here; isolated primes (23, 37, 47, ...) will be specially highlighted in bold.

Prime		2-adjacent integers
Value	Modularity	Threefold neighbor's factorization	Nonthreefold neighbor's factorization	Subgroup
5	H, U, T*	3	7	3.7
7	C, L, Th	3×3	5	3.5
11	H, L, T	3×3	13	3.13
13	C, U, Th	3×5	11	3.5.11
17	H, U, Th	3×5	19	3.5.19
19	C, L, T	3×7	17	3.7.17
23	H, L, Th	3×7	52	3.5.7
29	H, U, T	3×32	31	3.31
31	C, L, T	3×11	29	3.11.29
37	C, U, Th	3×13	5×7	3.5.7.13
41	H, U, T	3×13	43	3.13.43
43	C, L, Th	3×3×5	41	3.5.41
47	H, L, Th	3×3×5	72	3.5.7
53	H, U, Th	3×17	5×11	3.5.11.17

Delta-3 (no-threes)

All subgroups to which the prime is added include 2 as a necessary element.

Prime		3-adjacent integers
Value	Modularity	Fourfold neighbor's factorization	Nonfourfold neighbor's factorization	Subgroup
5	H, U, T*	22×2	2	2
7	C, L, Th	22	2×5	2.5
11	H, L, T	22×2	2×7	2.7
13	C, U, Th	22×22	2×5	2.5
17	H, U, Th	22×5	2×7	2.5.7
19	C, L, T	22×22	2×11	2.11
23	H, L, Th	22×5	2×13	2.5.13
29	H, U, T	22×23	2×13	2.13
31	C, L, T	22×7	2×17	2.7.17
37	C, U, Th	22×2×5	2×17	2.5.17
41	H, U, T	22×11	2×19	2.11.19
43	C, L, Th	22×2×5	2×23	2.23
47	H, L, Th	22×11	2×52	2.5.11
53	H, U, Th	22×2×7	2×52	2.5.7