## Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

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### Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

This arose as a question from @cmloegcmluin on facebook. And is a summary of information gleaned from the reference he found thanks to Google Books: Theoretic Arithmetic, in Three Books by Thomas Taylor, 1816. The relevant information begins on page 34.

"Superparticular" and "epimoric" are terms of Latin and Greek origin respectively, meaning "greater by one part". Some examples of superparticular/epimoric ratios, when reduced to lowest terms, are 3/2, 4/3, 5/4, 6/5. 7/6 ... . Therefore ratios such as 6/4 and 40/30 are also superparticular/epimoric. But 2/1 is not.

Purported definitions such as: "Ratios of the form (n+1)/n where n is a natural number", are not correct. They falsely include 2/1 and falsely reject 6/4.

A ratio is superparticular or epimoric if and only if, when reduced to lowest terms a/b,
a div b = 1, and
a mod b = 1,
where a and b are natural numbers, and
a div b is the quotient from the integer division of a by b, where
a mod b is the remainder.

Here is a grid summarising the Latin terminology for various classes of ratio whose lowest terms are a/b. It consists of 11 classes which do not overlap and which cover all ratios. These consist of 1 class of equality, 5 classes of greater inequality and (indirectly) 5 classes of lesser inequality.

            remainder
a mod b =
quotient           0                 1                >1
a div b =  +-----------------+-----------------+-----------------------------------+
0 | sub<class-of-inverse-ratio>                                           |
|                                                                       |
+-----------------+-----------------+-----------------------------------+
1 | equal           | superparticular | superpartient                     |
|                 |                 |                                   |
+-----------------+-----------------+-----------------------------------+
>1 | multiple        | multiple        | multiple                          |
|                 | superparticular | superpartient                     |
|                 |                 |                                   |
|                 |                 |                                   |
|                 |                 |                                   |
+-----------------+-----------------+-----------------------------------+


Now the Latin with specific names for values greater than 1.

            remainder
a mod b =
quotient           0                 1                 2                 3
a div b =  +-----------------+-----------------+-----------------+-----------------+
0 | sub<class-of-inverse-ratio>                                             . . .
|
+-----------------+-----------------+-----------------+-----------------+
1 | equal           | superparticular | superbipartient | supertripartient| . . .
|                 |                 |                 |                 |
+-----------------+-----------------+-----------------+-----------------+
2 | double          | double          | double          | double          | . . .
|                 | superparticular | superbipartient | supertripartient|
+-----------------+-----------------+-----------------+-----------------+
3 | triple          | triple          | triple          | triple          | . . .
|                 | superparticular | superbipartient | supertripartient|
+-----------------+-----------------+-----------------+-----------------+
.                 .                 .                 .          .
.                 .                 .                 .            .
.                 .                 .                 .              .


And the corresponding Greek-derived terminology.

            remainder
a mod b =
quotient           0                 1                 >1
a div b =  +-----------------+-----------------+-----------------------------------+
0 | sub<class-of-inverse-ratio>                                           |
|                                                                       |
+-----------------+-----------------+-----------------------------------+
1 | equal           | epimoric        | epipolymoric                      |
|                 |                 |                                   |
+-----------------+-----------------+-----------------------------------+
>1 | multiple        | multiple        | multiple                          |
|                 | epimoric        | epipolymoric                      |
|                 |                 |                                   |
|                 |                 |                                   |
|                 |                 |                                   |
+-----------------+-----------------+-----------------------------------+


Now the Greek with specific names for values greater than 1.

            remainder
a mod b =
quotient           0                 1                 2                 3
a div b =  +-----------------+-----------------+-----------------+-----------------+
0 | sub<class-of-inverse-ratio>                                             . . .
|
+-----------------+-----------------+-----------------+-----------------+
1 | equal           | epimoric        | epidimoric      | epitrimoric     | . . .
|                 |                 |                 |                 |
+-----------------+-----------------+-----------------+-----------------+
2 | double          | double          | double          | double          | . . .
|                 | epimoric        | epidimoric      | epitrimoric     |
+-----------------+-----------------+-----------------+-----------------+
3 | triple          | triple          | triple          | triple          | . . .
|                 | epimoric        | epidimoric      | epitrimoric     |
+-----------------+-----------------+-----------------+-----------------+
.                 .                 .                 .          .
.                 .                 .                 .            .
.                 .                 .                 .              .


"Epipolymoric", "epidimoric", "epitrimoric" etc, are my own coinage. Also, I suggest that the corresponding Latin terms ("superpartient", "superbipartient", "supertripartient" etc) might be anglicised to "supermultiparticular", "superbiparticular", "supertriparticular" etc, to eliminate the need for "-partient".

cmloegcmluin
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### Re: Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

sorry for the roughness of the note but I’m typing it out at the gym. If I understand correctly, then here are some examples:

1/1 div 1 mod 0 subparticular
2/1 div 2 mod 0 double subparticular
3/1 div 3 mod 0 triple subparticular
4/1 div 4 mod 0 quadruple subparticular
...

3/2 div 1 mod 1 superparticular
5/2 div 2 mod 1 double superparticular
7/2 div 3 mod 1 triple superparticular
9/2 div 4 mod 1 quadruple superparticular
...

4/3 div 1 mod 1 superparticular
5/3 div 1 mod 2 superbiparticular
7/3 div 2 mod 1 double superparticular
8/3 div 2 mod 2 double superbiparticular
10/3 div 3 mod 1 triple superparticular
...

5/4 div 1 mod 1 superparticular
7/4 div 1 mod 3 supertriparticular
9/4 div 2 mod 1 double superparticular
11/4 div 2 mod 3 double supertriparticular
...

6/5 div 1 mod 1 superparticular
7/5 div 1 mod 2 superbiparticular
8/5 div 1 mod 3 supertriparticular
9/5 div 1 mod 4 superquadriparticular
11/5 div 2 mod 1 double superparticular
...

superparticulars: 3/2, 4/3, 5/4, 6/5...
superbiparticulars: 5/3, 7/5, 9/7, 11/9...
supertriparticulars: 7/4, 8/5, 10/7, 11/8, 13/10...

double superparticulars: 5/2, 7/3, 9/4, 11/5...
double superbiparticulars: 8/3, 12/5, 16/7, 20/9...
double supertriparticulars: 11/4, 13/5, 17/7, 19/8, 23/10...

triple superparticulars: 7/2, 10/3, 13/4, 16/5...
triple superbiparticulars: 11/3, 17/5, 23/7, 29/9...
triple supertriparticulars: 15/4, 18/5, 24/7, 27/8, 33/10...

Dave Keenan
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### Re: Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

@cmloegcmluin, Thank you for the examples.

Assuming you have consciously adopted my suggested anglicised "super<n>particular" in place of the strict Latin "super<n>partient" for n>1, then you have them all correct except the first four. With those, you seem to have confused mod=0 with div=0. When mod=0 the ratio is either equal or multiple. More specifically we have:

1/1, div=1, mod=0, equal
2/1, div=2, mod=0, double
3/1, div=3, mod=0, triple
4/1, div=4, mod=0, quadruple

According to Taylor, 1816, when div=0 we literally prepend "sub" to the name of the class of the inverse ratio. We do not replace "super" with "sub". See pages 35 and 47, but note the inconsistency of "multiple subsuperparticular" versus "submultiple superparticular". I've been using the latter version. For example:

1/2, div=0, but 2/1, div=2, mod=0, is double, so 1/2 is subdouble
2/3, div=0, but 3/2, div=1, mod=1, is superparticular, so 2/3 is subsuperparticular
3/5, div=0, but 5/3, div=1, mod=2, is superbiparticular, so 3/5 is subsuperbiparticular
2/5, div=0, but 5/2, div=2, mod=1, is double superparticular, so 2/5 is subdouble superparticular

A web search on "subsuperparticular" found the following snippet. Unfortunately I have been unable to load the actual page. But note that the snippet agrees with the page 35 version in Taylor, giving "multiple subsuperparticular".

https://www.maa.org/press/periodicals/c ... f-boethius
The numerical listing at the bottom of the righthand page denotes the ten categories into which a comparison of unequals falls: the greater into five classes bearing such names as superpartient, and the less into five classes with similarly unusual names, such as multiple subsuperparticular. Boethian arithmetic books remained popular throughout ...
Fortunately, we rarely need to distinguish multiple superparticular ratios from their inverses, in music theory. But I argue that no matter how Boethius might have written it, putting "sub" at the beginning makes more sense in English, since the inversion applies to the multiplicity just as much as it does to the superparticularity, and if the "sub" comes at the beginning it can be read as applying to both, but not so if "sub" were to come in the middle.

Dave Keenan
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### Re: Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

Examples shown with the Greek-derived names.
      .                .                      .                      .                 .
.                .                      .                      .               .
.                .                      .                      .             .
+-----------+----------------------+----------------------+----------------------+
| subtriple | subtriple            | subtriple            | subtriple            |
|           | epimoric             | epidimoric           | epitrimoric          | . . .
| 1/3       | 2/7 3/10 4/13 5/16 … | 3/11 5/17 7/23 9/29 …| 4/15 5/18 7/24 8/27 …|
+-----------+----------------------+----------------------+----------------------+
| subdouble | subdouble            | subdouble            | subdouble            |
|           | epimoric             | epidimoric           | epitrimoric          | . . .
| 1/2       | 2/5 3/7 4/9 5/11 …   | 3/8 5/12 7/16 9/20 … | 4/11 5/13 7/17 8/19 …|
+-----------+----------------------+----------------------+----------------------+
|           | subepimoric          | subepidimoric        | subepitrimoric       |
|           |                      |                      |                      | . . .
| equal     | 2/3 3/4 4/5 5/6 …    | 3/5 5/7 7/9 9/11 …   | 4/7 5/8 7/10 8/11 …  |
|           +----------------------+----------------------+----------------------+
| 1/1       | epimoric             | epidimoric           | epitrimoric          |
|           |                      |                      |                      | . . .
|           | 3/2 4/3 5/4 6/5 …    | 5/3 7/5 9/7 11/9 …   | 7/4 8/5 10/7 11/8 …  |
+-----------+----------------------+----------------------+----------------------+
| double    | double               | double               | double               |
|           | epimoric             | epidimoric           | epitrimoric          | . . .
| 2/1       | 5/2 7/3 9/4 11/5 …   | 8/3 12/5 16/7 20/9 … | 11/4 13/5 17/7 19/8 …|
+-----------+----------------------+----------------------+----------------------+
| triple    | triple               | triple               | triple               |
|           | epimoric             | epidimoric           | epitrimoric          | . . .
| 3/1       | 7/2 10/3 13/4 16/5 … | 11/3 17/5 23/7 29/9 …| 15/4 18/5 24/7 27/8 …|
+-----------+----------------------+----------------------+----------------------+
.                .                      .                      .             .
.                .                      .                      .               .
.                .                      .                      .                 .

Maybe someone else can show what pattern these Boethian categories make on the Stern-Brocot tree.
[Edit: Fixed broken link]

Dave Keenan
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### Re: Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

Examples shown with the Latin-derived names.
      .                .                      .                      .                 .
.                .                      .                      .               .
.                .                      .                      .             .
+-----------+----------------------+----------------------+----------------------+
| subtriple | subtriple            | subtriple            | subtriple            |
|           | superparticular      | superbiparticular    | supertriparticular   | . . .
| 1/3       | 2/7 3/10 4/13 5/16 … | 3/11 5/17 7/23 9/29 …| 4/15 5/18 7/24 8/27 …|
+-----------+----------------------+----------------------+----------------------+
| subdouble | subdouble            | subdouble            | subdouble            |
|           | superparticular      | superbiparticular    | supertriparticular   | . . .
| 1/2       | 2/5 3/7 4/9 5/11 …   | 3/8 5/12 7/16 9/20 … | 4/11 5/13 7/17 8/19 …|
+-----------+----------------------+----------------------+----------------------+
|           | subsuperparticular   | subsuperbiparticular | subsupertriparticular|
|           |                      |                      |                      | . . .
| equal     | 2/3 3/4 4/5 5/6 …    | 3/5 5/7 7/9 9/11 …   | 4/7 5/8 7/10 8/11 …  |
|           +----------------------+----------------------+----------------------+
| 1/1       | superparticular      | superbiparticular    | supertriparticular   |
|           |                      |                      |                      | . . .
|           | 3/2 4/3 5/4 6/5 …    | 5/3 7/5 9/7 11/9 …   | 7/4 8/5 10/7 11/8 …  |
+-----------+----------------------+----------------------+----------------------+
| double    | double               | double               | double               |
|           | superparticular      | superbiparticular    | supertriparticular   | . . .
| 2/1       | 5/2 7/3 9/4 11/5 …   | 8/3 12/5 16/7 20/9 … | 11/4 13/5 17/7 19/8 …|
+-----------+----------------------+----------------------+----------------------+
| triple    | triple               | triple               | triple               |
|           | superparticular      | superbiparticular    | supertriparticular   | . . .
| 3/1       | 7/2 10/3 13/4 16/5 … | 11/3 17/5 23/7 29/9 …| 15/4 18/5 24/7 27/8 …|
+-----------+----------------------+----------------------+----------------------+
.                .                      .                      .             .
.                .                      .                      .               .
.                .                      .                      .                 .


cmloegcmluin
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### Re: Generalisation of the terms "epimoric" and "superparticular" as applied to ratios

Dave,

Yes, I did indeed go with your suggestion to anglicize -partient endings to -particular.

Thank you for correcting my examples. You're right, I got mod 0 and div 0 switcherooed. Your diagrams are extremely helpful so thank you for those too.

Your Boethius link is broken for me but this one works: http://mathshistory.st-andrews.ac.uk/Bi ... thius.html

And here's the rest of the main text from that page you couldn't get to load (https://www.maa.org/press/periodicals/c ... f-boethius). There's not much more relevant stuff about -particulars:
This image of a well-worn vellum manuscript page is from Boethius’ De arithmetica (ca. 1000). Anicius Manlius Severinus Boetius, commonly known as Boethius (ca. 480-524), was a Roman aristocrat, statesman, and scholar. Literate in Greek, Boethius was a translator and commentator on earlier Greek works and, as such, provided a vital intellectual link between the ancient classical world and the emerging Middle Ages. His De arithmetica is largely a translation of Nicomachus of Gerasa’s De institutione arithmetica libri duo (ca. 100). Boethius considered mathematics as consisting of four parts: arithmetic, music, geometry, and astronomy – the four subjects that formed the medieval quadrivium. Arithmetic, as the foundation of the other three, was the most important of these subjects. His De arithmetica consists of rather esoteric number theory involving complex categorizations of numbers. Modern scholarship shows how such number theory was useful in proportions involving music and architecture (Masi 1983).

The page shown on the right is from Chapter 11 of De arithmetica. It discusses “oddly-even” numbers. To generate such numbers, the reader is instructed to compile a table as shown, where the upper row is comprised of a sequence of the odd numbers: 3, 5, …, 15, and the second row below is formed by the “evenly-even” numbers: 4, 8, 16, …, 256. Then each term of the first sequence, in turn, is multiplied by all terms in the second sequence. The products resulting, 3 x 4 = 12, 3 x 8 = 24, …, 5 x 4 = 20, 5 x 8 = 40, …, and 15 x 256 = 3840, are the “oddly-even” numbers.

On these pages, Boethius discussed equality and inequality. He noted that equals bear the same name, such as denarius, cubit, or foot, whereas unequals are designated by different names, such as teacher and pupil or conqueror and conquered. The numerical listing at the bottom of the righthand page denotes the ten categories into which a comparison of unequals falls: the greater into five classes bearing such names as superpartient, and the less into five classes with similarly unusual names, such as multiple subsuperparticular. Boethian arithmetic books remained popular throughout the Middle Ages and well into the 16th century.