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== 1-cubic-limit ==
The '''1-cubic-limit''', '''1-Chebyshev-limit''' or '''1-exponential-limit''' consists of all intervals whose prime factorizations contain a maximum exponent of 1. The reduced 1-cubic-limit contains all intervals whose prime factorizations contain a maximum exponent of 1 once all powers of two are removed. Geometrically, the 1-cubic-limit consists of all intervals in an idealized infinite-dimensional hypercube centered on the unison [[1/1]] (hence "cubic limit"), or those with a Chebyshev distance of at most 1 from 1/1.

The reduced 1-cubic-limit contains:

- The [[7-odd-limit]].

- The reduced [[1-octahedral-limit]].

Below is a list of all octave-reduced 7-limit intervals in the reduced 1-cubic-limit:
{| class="wikitable"
|+
!Ratio
!Cents
!In non-reduced limit?
|-
|35/24
|653.18
|No
|-
|35/32
|155.14
|No
|-
|105/64
|857.09
|No
|-
|7/6
|266.87
|Yes
|-
|7/4
|968.83
|No
|-
|21/16
|470.78
|No
|-
|28/15
|1080.56
|No
|-
|7/5
|582.51
|Yes
|-
|21/20
|84.47
|No
|-
|5/3
|884.36
|Yes
|-
|5/4
|386.31
|No
|-
|15/8
|1088.27
|No
|-
|4/3
|498.04
|No
|-
|1/1
|0.00
|Yes
|-
|3/2
|701.96
|Yes
|-
|16/15
|111.73
|No
|-
|8/5
|813.69
|No
|-
|5/3
|884.36
|Yes
|-
|40/21
|1115.53
|No
|-
|10/7
|617.49
|Yes
|-
|15/14
|119.44
|Yes
|-
|32/21
|729.22
|No
|-
|8/7
|231.17
|No
|-
|12/7
|933.13
|No
|-
|128/105
|342.91
|No
|-
|64/35
|1044.86
|No
|-
|48/35
|546.82
|No
|}
The size of the 1-cubic-limit is infinite, but can be limited by restricting it to a subgroup. The size of a cubic limit n given a subgroup size m is (2n+1)^m.
{| class="wikitable"
|+
!Subgroup primes (excluding 2 if reduced)
!Reduced size
|-
|1
|3
|-
|2
|9
|-
|3
|27
|-
|4
|81
|-
|5
|243
|-
|6
|729
|}

== 1-octahedral-limit ==
The '''1-octahedral-limit''', '''1-taxicab-limit''', '''1-Manhattan-limit''' or '''1-cardinal-limit''' consists of all intervals whose prime factorizations contain one element, counting repeated primes. Equivalently, it contains all the prime harmonics and subharmonics. The reduced 1-octahedral-limit consists of all intervals whose prime factorizations contain one element, counting repeated primes, but once all powers of 2 are removed. Geometrically, the 1-octahedral-limit consists of all intervals in an idealized infinite-dimensional hyper-octahedron centered on the unison [[1/1]] (hence "octahedral limit"), or those with a Manhattan distance of at most 1 from 1/1.

The reduced 1-octahedral-limit contains:

- [[3-odd-limit|The 3-odd-limit]].

Below is a list of all octave-reduced 7-limit intervals in the reduced 1-octahedral-limit:
{| class="wikitable"
|+
!Ratio
!Cents
!In non-reduced limit?
|-
|7/4
|968.83
|No
|-
|5/4
|386.31
|No
|-
|3/2
|701.96
|No
|-
|1/1
|0.00
|Yes
|-
|4/3
|498.04
|No
|-
|8/5
|813.69
|No
|-
|8/7
|231.17
|No
|}
The size of the 1-octahedral-limit is infinite, but can be limited by restricting it to a subgroup. The size of an octahedral limit n given a subgroup size m is the Delannoy number D(n, m).
{| class="wikitable"
!Subgroup primes (excluding 2 if reduced)
!Reduced size
|-
|1
|3
|-
|2
|5
|-
|3
|7
|-
|4
|9
|-
|5
|11
|-
|6
|13
|}

== 2-octahedral-limit ==
The '''2-octahedral-limit''', '''2-taxicab-limit''', '''2-Manhattan-limit''' or '''2-cardinal-limit''' consists of all intervals whose prime factorizations contain two elements, counting repeated primes. Equivalently, it contains all the prime and semiprime harmonics and subharmonics, as well as ratios of primes. The reduced 2-octahedral-limit consists of all intervals whose prime factorizations contain two elements, counting repeated primes, but once all powers of 2 are removed. Geometrically, the 2-octahedral-limit consists of all intervals in an idealized infinite-dimensional hyper-octahedron centered on the unison [[1/1]] (hence "octahedral limit"), or those with a Manhattan distance of at most 2 from 1/1.

The reduced 2-octahedral-limit contains:

- The [[7-odd-limit|7-odd-limit.]]

- The reduced [[1-octahedral-limit]].

Below is a list of all octave-reduced 7-limit intervals in the reduced 2-octahedral-limit:
{| class="wikitable"
|+
!Ratio
!Cents
|-
|49/32
|737.65
|-
|35/32
|155.14
|-
|21/16
|470.78
|-
|7/4
|968.83
|-
|7/6
|266.87
|-
|7/5
|582.51
|-
|25/16
|772.63
|-
|15/8
|1088.27
|-
|5/4
|386.31
|-
|5/3
|884.36
|-
|9/8
|203.91
|-
|3/2
|701.96
|-
|1/1
|0.00
|-
|4/3
|498.04
|-
|16/9
|996.09
|-
|6/5
|315.64
|-
|8/5
|813.69
|-
|16/15
|111.73
|-
|32/25
|427.37
|-
|10/7
|617.49
|-
|12/7
|933.13
|-
|8/7
|231.17
|-
|32/21
|729.22
|-
|64/35
|1044.86
|-
|64/49
|462.35
|}
The size of the 2-octahedral-limit is infinite, but can be limited by restricting it to a subgroup. The size of an octahedral limit n given a subgroup size m is the Delannoy number D(n, m).
{| class="wikitable"
!Subgroup primes (excluding 2 if reduced)
!Reduced size
|-
|1
|5
|-
|2
|13
|-
|3
|25
|-
|4
|41
|-
|5
|61
|-
|6
|85
|}

User:VectorGraphics/Cubic and octahedral limits - Revision history

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