S-expression

An S-expression is any product, or ratio of products, of the square superparticulars Sk, which are defined as the fractions of the form ⁠k²/k² − 1⁠. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the commas that appear frequently in JI and temperaments.

Quick rules of S-expressions

As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are tempered out. The linked sections provide deeper information into each comma family.

• Square superparticulars: Sk, superparticular fractions of the form ⁠k²/k² − 1⁠.
  Tempering out Sk equates ⁠k + 1/k⁠ with ⁠k/k − 1⁠ and splits ⁠k + 1/k − 1⁠ in two.
• Triangle-particulars: Sk⋅S(k + 1), superparticular fractions of the form ⁠k(k + 1)/2/(k − 1)(k + 2)/2⁠.
  Tempering out Sk⋅S(k + 1) equates ⁠k + 2/k + 1⁠ with ⁠k/k − 1⁠, and ⁠k + 2/k⁠ with ⁠k + 1/k − 1⁠.
• Lopsided commas: (Sk)²⋅S(k + 1) and (Sk)²⋅S(k − 1).
  Tempering out the former equates ⁠k + 2/k⁠ with (⁠k/k − 1⁠)² and ⁠k + 2/k − 1⁠ with (⁠k/k − 1⁠)³, and tempering out the latter equates ⁠k/k − 2⁠ with (⁠k + 1/k⁠)² and ⁠k + 1/k − 2⁠ with (⁠k + 1/k⁠)³.
• Ultraparticulars: Sk/S(k + 1). Tempering this out splits ⁠k + 2/k − 1⁠ into (⁠k + 1/k⁠)³.
• Semiparticulars: Sk/S(k + 2). Tempering this out splits ⁠k + 3/k − 1⁠ into (⁠k + 2/k⁠)².

Sk (square-particulars)

A square superparticular, or square-particular for short, is a superparticular interval whose numerator is a square number, which is to say, a superparticular of the form

$$ \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k} $$

which is square-superparticular k for a given integer k > 1. A suggested shorthand for this interval is Sk for the k-th square superparticular, where the S stands for second-order/square superparticular. This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of regular temperaments. Note that S2 = 4/3 is the first musically meaningful square-particular, as S1 = 1/0.

Also note that we use the notation Sk^p to mean (Sk)^p rather than S(k^p) for convenience in the practical analysis of regular temperaments using S-expressions.

Significance

Square-superparticulars are important structurally because they are the intervals between consecutive superparticular intervals while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (k + 1)-th harmonic, as well as the potential representational sacrifices that must be made from that point onwards.

Specifically, tempering Sk out makes the harmonic segment centered around k have equal steps; e.g. tempering out S9 = (9/8)/(10/9) = 81/80 equalizes 8:9:10, as in meantone.

In other words, understanding the mappings of Sk in a given temperament (especially an equal temperament) is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents or tries to represent the harmonic series.

Table of square-particulars

31-limit square-particulars

S-expression	Interval relation	Ratio	Subgroup
S2	(2/1)/(3/2)	4/3	2.3
S3	(3/2)/(4/3)	9/8	2.3
S4	(4/3)/(5/4)	16/15	2.3.5
S5	(5/4)/(6/5)	25/24	2.3.5
S6	(6/5)/(7/6)	36/35	2.3.5.7
S7	(7/6)/(8/7)	49/48	2.3.7
S8	(8/7)/(9/8)	64/63	2.3.7
S9	(9/8)/(10/9)	81/80	2.3.5
S10	(10/9)/(11/10)	100/99	2.3.5.11
S11	(11/10)/(12/11)	121/120	2.3.5.11
S12	(12/11)/(13/12)	144/143	2.3.11.13
S13	(13/12)/(14/13)	169/168	2.3.7.13
S14	(14/13)/(15/14)	196/195	2.3.5.7.13
S15	(15/14)/(16/15)	225/224	2.3.5.7
S16	(16/15)/(17/16)	256/255	2.3.5.17
S17	(17/16)/(18/17)	289/288	2.3.17
S18	(18/17)/(19/18)	324/323	2.3.17.19
S19	(19/18)/(20/19)	361/360	2.3.5.19
S20	(20/19)/(21/20)	400/399	2.3.5.7.19
S21	(21/20)/(22/21)	441/440	2.3.5.7.11
S22	(22/21)/(23/22)	484/483	2.3.7.11.23
S23	(23/22)/(24/23)	529/528	2.3.11.23
S24	(24/23)/(25/24)	576/575	2.3.5.23
S25	(25/24)/(26/25)	625/624	2.3.5.13
S26	(26/25)/(27/26)	676/675	2.3.5.13
S27	(27/26)/(28/27)	729/728	2.3.7.13
S28	(28/27)/(29/28)	784/783	2.3.7.29
S29	(29/28)/(30/29)	841/840	2.3.5.7.29
S30	(30/29)/(31/30)	900/899	2.3.5.29.31
S31	(31/30)/(32/31)	961/960	2.3.5.31
S32	(32/31)/(33/32)	1024/1023	2.3.11.31
S33	(33/32)/(34/33)	1089/1088	2.3.11.17
S34	(34/33)/(35/34)	1156/1155	2.3.5.7.11.17
S35	(35/34)/(36/35)	1225/1224	2.3.5.7.17
S39	(39/38)/(40/39)	1521/1520	2.3.5.13.19
S45	(45/44)/(46/45)	2025/2024	2.3.5.11.23
S49	(49/48)/(50/49)	2401/2400	2.3.5.7
S50	(50/49)/(51/50)	2500/2499	2.3.5.7.17
S51	(51/50)/(52/51)	2601/2600	2.3.5.13.17
S55	(55/54)/(56/55)	3025/3024	2.3.5.7.11
S56	(56/55)/(57/56)	3136/3135	2.3.5.7.11.19
S57	(57/56)/(58/57)	3249/3248	2.3.7.19.29
S63	(63/62)/(64/63)	3969/3968	2.3.7.31
S64	(64/63)/(65/64)	4096/4095	2.3.5.7.13
S65	(65/64)/(66/65)	4225/4224	2.3.5.11.13
S69	(69/68)/(70/69)	4761/4760	2.3.5.7.17.23
S76	(76/75)/(77/76)	5776/5775	2.3.5.7.11.19
S77	(77/76)/(78/77)	5929/5928	2.3.7.11.13.19
S91	(91/90)/(92/91)	8281/8280	2.3.5.7.13.23
S92	(92/91)/(93/92)	8464/8463	2.3.7.13.23.31
S99	(99/98)/(100/99)	9801/9800	2.3.5.7.11
S115	(115/114)/(116/115)	13225/13224	2.3.5.19.23.29
S116	(116/115)/(117/116)	13456/13455	2.3.5.13.23.29
S120	(120/119)/(121/120)	14400/14399	2.3.5.7.11.17
S125	(125/124)/(126/125)	15625/15624	2.3.5.7.31
S144	(144/143)/(145/144)	20736/20735	2.3.5.11.13.29
S153	(153/152)/(154/153)	23409/23408	2.3.7.11.17.19
S154	(154/153)/(155/154)	23716/23715	2.3.5.7.11.17.31
S155	(155/154)/(156/155)	24025/24024	2.3.5.7.11.13.31
S161	(161/160)/(162/161)	25921/25920	2.3.5.7.23
S169	(169/168)/(170/169)	28561/28560	2.3.5.7.13.17
S170	(170/169)/(171/170)	28900/28899	2.3.5.13.17.19
S175	(175/174)/(176/175)	30625/30624	2.3.5.7.11.29
S208	(208/207)/(209/208)	43264/43263	2.3.11.13.19.23
S209	(209/208)/(210/209)	43681/43680	2.3.5.7.11.13.19
S231	(231/230)/(232/231)	53361/53360	2.3.5.7.11.23.29
S289	(289/288)/(290/289)	83521/83520	2.3.5.17.29
S323	(323/322)/(324/323)	104329/104328	2.3.7.13.19.23
S324	(324/323)/(325/324)	104976/104975	2.3.5.13.17.19
S341	(341/340)/(342/341)	116281/116280	2.3.5.11.17.19.31
S342	(342/341)/(343/342)	116964/116963	2.3.7.11.19.31
S351	(351/350)/(352/351)	123201/123200	2.3.5.7.11.13
S391	(391/390)/(392/391)	152881/152880	2.3.5.7.13.17.23
S441	(441/440)/(442/441)	194481/194480	2.3.5.7.11.13.17
S494	(494/493)/(495/494)	244036/244035	2.3.5.11.13.17.19.29
S495	(495/494)/(496/495)	245025/245024	2.3.5.11.13.19.31
S528	(528/527)/(529/528)	278784/278783	2.3.11.17.23.31
S551	(551/550)/(552/551)	303601/303600	2.3.5.11.19.23.29
S714	(714/713)/(715/714)	509796/509795	2.3.5.7.11.13.17.23.31
S783	(783/782)/(784/783)	613089/613088	2.3.7.17.23.29
S1275	(1275/1274)/(1276/1275)	1625625/1625624	2.3.5.7.11.13.17.29
S1519	(1519/1518)/(1520/1519)	2307361/2307360	2.3.5.7.11.19.23.31
S1520	(1520/1519)/(1521/1520)	2310400/2310399	2.3.5.7.13.19.31
S2001	(2001/2000)/(2002/2001)	4004001/4004000	2.3.5.7.11.13.23.29
S2024	(2024/2023)/(2025/2024)	4096576/4096575	2.3.5.7.11.17.23
S2431	(2431/2430)/(2432/2431)	5909761/5909760	2.3.5.11.13.17.19
S3249	(3249/3248)/(3250/3249)	10556001/10556000	2.3.5.7.13.19.29
S9801	(9801/9800)/(9802/9801)	96059601/96059600	2.3.5.7.11.13.29
S13311	(13311/13310)/(13312/13311)	177182721/177182720	2.3.5.11.13.17.29
S13455	(13455/13454)/(13456/13455)	181037025/181037024	2.3.5.7.13.23.29.31

Alternatives to tempering out square-particulars

It is common to temper out square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher-accuracy or structural reasons it can be more beneficial to instead temper out differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas Uk = ⁠Sk/S(k + 1)⁠, we get ultraparticulars. Ultraparticulars have a secondary and mathematically equivalent consequence: by tempering out ⁠Sk/S(k + 1)⁠, ⁠k + 2/k + 1⁠ and ⁠k/k − 1⁠ are made equidistant from ⁠k + 1/k⁠, which means that another expression for ⁠Sk/S(k + 1)⁠ is the following:

$$ {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3} $$

This means you can read the k and k + 1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in the table of ultraparticulars).

Furthermore, defining another sequence of commas with formula ⁠Sk/S(k + 2)⁠ leads to semiparticulars, which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering out consecutive ultraparticulars.

Sk⋅S(k + 1) (triangle-particulars)

Significance

1. Every triangle-particular is superparticular, so these are efficient commas. (See also the #Short proof of the superparticularity of triangle-particulars.)
2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form. In particular, every triangle-particular is the difference between two nearly-adjacent superparticular intervals ⁠k + 2/k + 1⁠ and ⁠k/k − 1⁠.
3. Tempering out any two consecutive square-particulars Sk and S(k + 1) implies tempering out a triangle-particular, so these are common commas. (See also: lopsided commas.)
4. If we temper out Sk⋅S(k + 1) but not Sk or S(k + 1), then one or more intervals of ⁠k/k − 1⁠, ⁠k + 1/k⁠, and ⁠k + 2/k + 1⁠ must be mapped inconsistently, because:

If ⁠k + 1/k⁠ is mapped above ⁠k + 2/k + 1⁠ ~ ⁠k/k − 1⁠ we have ⁠k + 1/k⁠ > ⁠k/k − 1⁠ and if it is mapped below we have ⁠k + 1/k⁠ < ⁠k + 2/k + 1⁠.

(Generalisations of this and their implications for consistency are discussed in the section covering 1/n-square-particulars.)

Meaning

If we equate ⁠k + 2/k + 1⁠ with ⁠k/k − 1⁠ by tempering out their difference, then multiply both sides by ⁠k + 1/k⁠, we have:

$$ \left(\frac{k + 2}{k + 1}\right)\left(\frac{k + 1}{k}\right) = \left(\frac{k + 1}{k}\right)\left(\frac{k}{k - 1}\right) $$

which simplifies to:

$$ \frac{k + 2}{k} = \frac{k + 1}{k - 1} $$

This means that if we temper out:

$${\rm S}k \cdot {\rm S}(k+1) = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)} $$

and this equivalence is achieved. Note that there is little to no reason to not also temper out Sk and S(k + 1) individually unless other considerations seem to force your hand.

Short proof of the superparticularity of triangle-particulars

$$ S(k) \cdot S(k + 1) = \frac{\frac{k}{k - 1}}{\frac{k + 2}{k + 1}} = \frac{k(k + 1)}{(k - 1)(k + 2)} = \frac{k^2 + k}{k^2 + k - 2} $$

Then notice that k² + k is always a multiple of 2; therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could be reasoned that it would likely be half as accurate as tempering out either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals are triangular numbers, hence the alternative name triangle-particular.

Table of triangle-particulars

For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of inconsistency of mappings when tempered out for the above reason.

31-limit triangle-particulars

S-expression	Interval relation	Ratio	Subgroup
S2⋅S3	(3/1)/(2/1)	3/2	2.3
S3⋅S4	(3/2)/(5/4)	6/5	2.3.5
S4⋅S5	(4/3)/(6/5)	10/9	2.3.5
S5⋅S6	(5/4)/(7/6)	15/14	2.3.5.7
S6⋅S7	(6/5)/(8/7)	21/20	2.3.5.7
S7⋅S8	(7/6)(9/8)	28/27	2.3.7
S8⋅S9	(8/7)/(10/9)	36/35	2.3.5.7
S9⋅S10	(9/8)/(11/10)	45/44	2.3.5.11
S10⋅S11	(10/9)/(12/11)	55/54	2.3.5.11
S11⋅S12	(11/10)/(13/12)	66/65	2.3.5.11.13
S12⋅S13	(12/11)/(14/13)	78/77	2.3.7.11.13
S13⋅S14	(13/12)/(15/14)	91/90	2.3.5.7.13
S14⋅S15	(14/13)/(16/15)	105/104	2.3.5.7.13
S15⋅S16	(15/14)/(17/16)	120/119	2.3.5.7.17
S16⋅S17	(16/15)/(18/17)	136/135	2.3.5.17
S17⋅S18	(17/16)/(19/18)	153/152	2.3.17.19
S18⋅S19	(18/17)/(20/19)	171/170	2.3.5.17.19
S19⋅S20	(19/18)/(21/20)	190/189	2.3.5.7.19
S20⋅S21	(20/19)/(22/21)	210/209	2.3.5.7.11.19
S21⋅S22	(21/20)/(23/22)	231/230	2.3.5.7.11.23
S22⋅S23	(22/21)/(24/23)	253/252	2.3.5.7.11.23
S23⋅S24	(23/22)/(25/24)	276/275	2.3.5.11.23
S24⋅S25	(24/23)/(26/25)	300/299	2.3.5.13.23
S25⋅S26	(25/24)/(27/26)	325/324	2.3.5.13
S26⋅S27	(26/25)/(28/27)	351/350	2.3.5.7.13
S27⋅S28	(27/26)/(29/28)	378/377	2.3.5.7.13.29
S28⋅S29	(28/27)/(30/29)	406/405	2.3.5.7.29
S29⋅S30	(29/28)/(31/30)	435/434	2.3.5.7.29.31
S30⋅S31	(30/29)/(32/31)	465/464	2.3.5.29.31
S31⋅S32	(31/30)/(33/32)	496/495	2.3.5.11.31
S32⋅S33	(32/31)/(34/33)	528/527	2.3.11.17.31
S33⋅S34	(33/32)/(35/34)	561/560	2.3.5.7.11.17
S34⋅S35	(34/33)/(36/35)	595/594	2.3.5.7.11.17
S49⋅S50	(49/48)/(51/50)	1225/1224	2.3.5.7.17
S50⋅S51	(50/49)/(52/51)	1275/1274	2.3.5.7.13.17
S55⋅S56	(55/54)/(57/56)	1540/1539	2.3.5.7.11.19
S63⋅S64	(63/62)/(65/64)	2016/2015	2.3.5.7.13.31
S64⋅S65	(64/63)/(66/65)	2080/2079	2.3.5.7.11.13
S76⋅S77	(76/75)/(78/77)	2926/2925	2.3.5.7.11.13.19
S91⋅S92	(91/90)/(93/92)	4186/4185	2.3.5.7.13.23.31
S115⋅S116	(115/114)/(117/116)	6670/6669	2.3.5.13.19.23.29
S153⋅S154	(153/152)/(155/154)	11781/11780	2.3.5.7.11.17.19.31
S154⋅S155	(154/153)/(156/155)	11935/11934	2.3.5.7.11.13.17.31
S169⋅S170	(169/168)/(171/170)	14365/14364	2.3.5.7.13.17.19
S208⋅S209	(208/207)/(210/209)	21736/21735	2.3.5.7.11.13.19
S323⋅S324	(323/322)/(325/324)	52326/52325	2.3.5.7.13.17.19.23
S341⋅S342	(341/340)/(343/342)	58311/58310	2.3.5.7.11.17.19.31
S494⋅S495	(494/493)/(496/495)	122265/122264	2.3.5.7.11.13.19.29.31
S1519⋅S1520	(1519/1518)/(1521/1520)	1154440/1154439	2.3.5.7.11.13.19.23.31

Sk⋅S(k + 1)⋅…⋅S(k + n − 1) (1/n-square-particulars)

Significance

1/n-square-particulars are a generalization of square- and 1/2-square-particulars to an interval whose S-expression can be written in the form of a product of n consecutive square-particulars (including Sk but not including S(k + n)) and which can therefore be written as the ratio between the two superparticulars ⁠k/k − 1⁠ and ⁠k + n/k + n − 1⁠.

In other words, each and every S-expression of a comma as a 1/n-square-particular corresponds exactly to expressing it as the ratio between two superparticular intervals, with n distance between them. For example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to ordinary square-particulars (in this case S10).

These commas are important in a few ways:

1. As a generalization of important special cases n = 0, 1, 2 (which are almost all superparticular; the only case where they are not is that n = 3 (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.
2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we do not want to temper out, for example ⁠81/80/91/90⁠ = S81⋅S82⋅…⋅S90 = 729/728 = S27. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for S81⋅S82⋅ …⋅S90 and more generally Sa..b for Sa⋅S(a + 1)⋅…⋅Sb.
3. They often correspond to "nontrivial" equivalences that need to be dug up, which are not obvious from their expression as a ratio of two superparticular intervals, for example, S33⋅S34⋅S35, suggesting they are a goldmine for valuable tempering opportunities.
4. Their expressions naturally make them implied by tempering out consecutive square-particulars, so if they are present and that the individual square-particulars are not tempered out, if you want to extend the temperament and/or reduce its rank (tempering it down) and/or hope to make the temperament more efficient, you can try tempering out the observed square-particulars that a vanishing 1/n-square-particular is composed of – although this is not always possible. There is also good theoretical motivation for wanting to do this, as the next section will discuss.
5. They are relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently. As such, they are relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.

Implications for consistency

1/n-square-particulars, which is to say, commas which can be written in the form of a product of n consecutive square-particulars (including Sk but not including S(k + n)) and which can therefore be written as the ratio between the two superparticulars ⁠k/k − 1⁠ and ⁠k + n/k + n − 1⁠ have implications for the consistency of the (k + n)-odd-limit when tempered out. Specifically:

If a temperament tempers out a 1/n-square-particular of the form Sk⋅S(k + 1)⋅…⋅S(k + n − 1), it must temper out all of the n square-particulars that compose it, which is to say it must also temper all of Sk, S(k + 1), …, S(k + n − 1) to make the (k + n)-odd-limit consistent. If it does not, it is necessarily inconsistent due to the lack of monotonicity in the segment. A proof is as follows:

Proof

Consider the following sequence of superparticular intervals, all of which in the (k + n)-odd-limit:

[math]\displaystyle{ \displaystyle \frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1} }[/math]

Because of tempering out Sk⋅S(k + 1)⋅…⋅S(k + n − 1), we require that ⁠k + n/k + n − 1⁠ = ⁠k/k − 1⁠ consistently. Therefore, if any superparticular ⁠x/x − 1⁠ imbetween (meaning k + n > x > k) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:

[math]\displaystyle{ \displaystyle\begin{align} \operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\ \operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right) \end{align} }[/math]

Therefore any superparticular interval ⁠x/x − 1⁠ between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the (k + n)-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering Sk*S(k + 1)*…*S(k + n − 1) but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the (k − 1)-odd-limit. [math]\displaystyle{ \square }[/math]

S(k − 1)⋅Sk⋅S(k + 1) (1/3-square-particulars)

This section concerns commas of the form S(k − 1)⋅Sk⋅S(k + 1) = ⁠(k − 1)/(k − 2)/(k + 2)/(k + 1⁠) which therefore do not directly involve the k-th harmonic. These, along with square-particulars and 1⁄2-square-particulars (a.k.a. triangle-particulars), are a special case of 1/n-square-particulars.

Significance

1. Two-thirds of all 1⁄3-square-particulars are superparticular and the other third are throdd-particular, so these are efficient commas. (See also the #Proof of simplification of 1/3-square-particulars.)
2. They are often implied in a variety of ways by combinations of other commas discussed on this page.
3. Their omission of direct relation to the k-th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is semiparticulars.)

Proof of simplification of 1/3-square-particulars

We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:

[math]\displaystyle{ \displaystyle \begin{align} S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\ &= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\ &= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\ &= \frac{k^2 - 1}{k^2 - 4} \end{align} }[/math]

If k = 3n + 1 then:

[math]\displaystyle{ \displaystyle S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1} }[/math]

If k = 3n + 2 then:

[math]\displaystyle{ \displaystyle S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n} }[/math]

If k = 3n then:

[math]\displaystyle{ \displaystyle S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 - 1}{9n^2 - 4} }[/math]

In other words, what this shows is all 1⁄3-square-particulars of the form S(k − 1)⋅Sk⋅S(k + 1) are superparticular iff k is throdd (not a multiple of 3), and all 1⁄3-square-particulars of the form S(3k − 1)⋅S(3k)⋅S(3k + 1) are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff k is threven and superparticular iff k is throdd). [math]\displaystyle{ \square }[/math]

Tables of 1/n-square-particulars

41-limit 1⁄3-square-particulars

S-expression	Interval relation	Ratio	Prime limit
S2⋅S3⋅S4	(2/1)/(5/4)	8/5	5
S3⋅S4⋅S5	(3/2)/(6/5)	5/4	5
S4⋅S5⋅S6	(4/3)/(7/6)	8/7	7
S5⋅S6⋅S7	(5/4)/(8/7)	35/32	7
S6⋅S7⋅S8	(6/5)/(9/8)	16/15	5
S7⋅S8⋅S9	(7/6)/(10/9)	21/20	7
S8⋅S9⋅S10	(8/7)/(11/10)	80/77	11
S9⋅S10⋅S11	(9/8)/(12/11)	33/32	11
S10⋅S11⋅S12	(10/9)/(13/12)	40/39	13
S11⋅S12⋅S13	(11/10)/(14/13)	143/140	13
S12⋅S13⋅S14	(12/11)/(15/14)	56/55	11
S13⋅S14⋅S15	(13/12)/(16/15)	65/64	13
S14⋅S15⋅S16	(14/13)/(17/16)	224/221	17
S15⋅S16⋅S17	(15/14)/(18/17)	85/84	17
S16⋅S17⋅S18	(16/15)/(19/18)	96/95	19
S17⋅S18⋅S19	(17/16)/(20/19)	323/320	19
S18⋅S19⋅S20	(18/17)/(21/20)	120/119	17
S19⋅S20⋅S21	(19/18)/(22/21)	133/132	19
S20⋅S21⋅S22	(20/19)/(23/22)	440/437	23
S21⋅S22⋅S23	(21/20)/(24/23)	161/160	23
S22⋅S23⋅S24	(22/21)/(25/24)	176/175	11
S23⋅S24⋅S25	(23/22)/(26/25)	575/572	23
S24⋅S25⋅S26	(24/23)/(27/26)	208/207	23
S25⋅S26⋅S27	(25/24)/(28/27)	225/224	7
S26⋅S27⋅S28	(26/25)/(29/28)	728/725	29
S27⋅S28⋅S29	(27/26)/(30/29)	261/260	29
S28⋅S29⋅S30	(28/27)/(31/30)	280/279	31
S29⋅S30⋅S31	(29/28)/(32/31)	899/896	31
S30⋅S31⋅S32	(30/29)/(33/32)	320/319	29
S31⋅S32⋅S33	(31/30)/(34/33)	341/340	31
S32⋅S33⋅S34	(32/31)/(35/34)	1088/1085	31
S33⋅S34⋅S35	(33/32)/(36/35)	385/384	11
S34⋅S35⋅S36	(34/33)/(37/36)	408/407	37
S35⋅S36⋅S37	(35/34)/(38/37)	1295/1292	37
S36⋅S37⋅S38	(36/35)/(39/38)	456/455	19
S37⋅S38⋅S39	(37/36)/(40/39)	481/480	37
S38⋅S39⋅S40	(38/37)/(41/40)	1520/1517	41
S39⋅S40⋅S41	(39/38)/(42/41)	533/532	41
S42⋅S43⋅S44	(42/41)/(45/44)	616/615	41
S46⋅S47⋅S48	(46/45)/(49/48)	736/735	23
S49⋅S50⋅S51	(49/48)/(52/51)	833/832	17
S52⋅S53⋅S54	(52/51)/(55/54)	936/935	17
S55⋅S56⋅S57	(55/54)/(58/57)	1045/1044	29
S63⋅S64⋅S65	(63/62)/(66/65)	1365/1364	31
S66⋅S67⋅S68	(66/65)/(69/68)	1496/1495	23
S75⋅S76⋅S77	(75/74)/(78/77)	1925/1924	37
S78⋅S79⋅S80	(78/77)/(81/80)	2080/2079	13
S82⋅S83⋅S84	(82/81)/(85/84)	2296/2295	41
S85⋅S86⋅S87	(85/84)/(88/87)	2465/2464	29
S88⋅S89⋅S90	(88/87)/(91/90)	2640/2639	29
S93⋅S94⋅S95	(93/92)/(96/95)	2945/2944	31
S96⋅S97⋅S98	(96/95)/(99/98)	3136/3135	19
S112⋅S113⋅S114	(112/111)/(115/114)	4256/4255	37
S117⋅S118⋅S119	(117/116)/(120/119)	4641/4640	29
S121⋅S122⋅S123	(121/120)/(124/123)	4961/4960	41
S133⋅S134⋅S135	(133/132)/(136/135)	5985/5984	19
S145⋅S146⋅S147	(145/144)/(148/147)	7105/7104	37
S153⋅S154⋅S155	(153/152)/(156/155)	7905/7904	31
S162⋅S163⋅S164	(162/161)/(165/164)	8856/8855	41
S187⋅S188⋅S189	(187/186)/(190/189)	11781/11780	31
S205⋅S206⋅S207	(205/204)/(208/207)	14145/14144	41
S222⋅S223⋅S224	(222/221)/(225/224)	16576/16575	37
S243⋅S244⋅S245	(243/242)/(246/245)	19845/19844	41
S253⋅S254⋅S255	(253/252)/(256/255)	21505/21504	23
S273⋅S274⋅S275	(273/272)/(276/275)	25025/25024	23
S286⋅S287⋅S288	(286/285)/(289/288)	27456/27455	19
S287⋅S288⋅S289	(287/286)/(290/289)	82943/82940	41
S297⋅S298⋅S299	(297/296)/(300/299)	29601/29600	37
S320⋅S321⋅S322	(320/319)/(323/322)	103040/103037	29
S361⋅S362⋅S363	(361/360)/(364/363)	43681/43680	19
S375⋅S376⋅S377	(375/374)/(378/377)	47125/47124	29

23-limit 1⁄4-square particulars

S-expression	Interval relation	Ratio	Prime limit
S2⋅S3⋅S4⋅S5	(2/1)/(6/5)	5/3	5
S3⋅S4⋅S5⋅S6	(3/2)/(7/6)	9/7	7
S4⋅S5⋅S6⋅S7	(4/3)/(8/7)	7/6	7
S5⋅S6⋅S7⋅S8	(5/4)/(9/8)	10/9	5
S6⋅S7⋅S8⋅S9	(6/5)/(10/9)	27/25	5
S7⋅S8⋅S9⋅S10	(7/6)/(11/10)	35/33	11
S8⋅S9⋅S10⋅S11	(8/7)/(12/11)	22/21	11
S9⋅S10⋅S11⋅S12	(9/8)/(13/12)	27/26	13
S10⋅S11⋅S12⋅S13	(10/9)/(14/13)	65/63	13
S11⋅S12⋅S13⋅S14	(11/10)/(15/14)	77/75	11
S12⋅S13⋅S14⋅S15	(12/11)/(16/15)	45/44	11
S13⋅S14⋅S15⋅S16	(13/12)/(17/16)	52/51	17
S14⋅S15⋅S16⋅S17	(14/13)/(18/17)	119/117	17
S15⋅S16⋅S17⋅S18	(15/14)/(19/18)	135/133	19
S16⋅S17⋅S18⋅S19	(16/15)/(20/19)	76/75	19
S17⋅S18⋅S19⋅S20	(17/16)/(21/20)	85/84	17
S18⋅S19⋅S20⋅S21	(18/17)/(22/21)	189/187	17
S19⋅S20⋅S21⋅S22	(19/18)/(23/22)	209/207	23
S20⋅S21⋅S22⋅S23	(20/19)/(24/23)	115/114	23
S21⋅S22⋅S23⋅S24	(21/20)/(25/24)	126/125	7
S22⋅S23⋅S24⋅S25	(22/21)/(26/25)	275/273	13
S23⋅S24⋅S25⋅S26	(23/22)/(27/26)	299/297	23
S24⋅S25⋅S26⋅S27	(24/23)/(28/27)	162/161	23
S35⋅S36⋅S37⋅S38	(35/34)/(39/38)	665/663	19
S36⋅S37⋅S38⋅S39	(36/35)/(40/39)	351/350	13
S45⋅S46⋅S47⋅S48	(45/44)/(49/48)	540/539	11
S46⋅S47⋅S48⋅S49	(46/45)/(50/49)	1127/1125	23
S51⋅S52⋅S53⋅S54	(51/50)/(55/54)	1377/1375	17
S52⋅S53⋅S54⋅S55	(52/51)/(56/55)	715/714	17
S65⋅S66⋅S67⋅S68	(65/64)/(69/68)	1105/1104	23
S66⋅S67⋅S68⋅S69	(66/65)/(70/69)	2277/2275	23
S77⋅S78⋅S79⋅S80	(77/76)/(81/80)	1540/1539	19
S81⋅S82⋅S83⋅S84	(81/80)/(85/84)	1701/1700	17
S92⋅S93⋅S94⋅S95	(92/91)/(96/95)	2185/2184	23
S96⋅S97⋅S98⋅S99	(96/95)/(100/99)	2376/2375	19
S221⋅S222⋅S223⋅S224	(221/220)/(225/224)	12376/12375	17

23-limit 1⁄5-square particulars

S-expression	Interval relation	Ratio	Prime limit
S2⋅S3⋅S4⋅S5⋅S6	(2/1)/(7/6)	12/7	7
S3⋅S4⋅S5⋅S6⋅S7	(3/2)/(8/7)	21/16	7
S4⋅S5⋅S6⋅S7⋅S8	(4/3)/(9/8)	32/27	3
S5⋅S6⋅S7⋅S8⋅S9	(5/4)/(10/9)	9/8	3
S6⋅S7⋅S8⋅S9⋅S10	(6/5)/(11/10)	12/11	11
S7⋅S8⋅S9⋅S10⋅S11	(7/6)/(12/11)	77/72	11
S8⋅S9⋅S10⋅S11⋅S12	(8/7)/(13/12)	96/91	13
S9⋅S10⋅S11⋅S12⋅S13	(9/8)/(14/13)	117/112	13
S10⋅S11⋅S12⋅S13⋅S14	(10/9)/(15/14)	28/27	7
S11⋅S12⋅S13⋅S14⋅S15	(11/10)/(16/15)	33/32	11
S12⋅S13⋅S14⋅S15⋅S16	(12/11)/(17/16)	192/187	17
S13⋅S14⋅S15⋅S16⋅S17	(13/12)/(18/17)	221/216	17
S14⋅S15⋅S16⋅S17⋅S18	(14/13)/(19/18)	252/247	19
S15⋅S16⋅S17⋅S18⋅S19	(15/14)/(20/19)	57/56	19
S16⋅S17⋅S18⋅S19⋅S20	(16/15)/(21/20)	64/63	7
S17⋅S18⋅S19⋅S20⋅S21	(17/16)/(22/21)	357/352	17
S18⋅S19⋅S20⋅S21⋅S22	(18/17)/(23/22)	396/391	23
S19⋅S20⋅S21⋅S22⋅S23	(19/18)/(24/23)	437/432	23
S20⋅S21⋅S22⋅S23⋅S24	(20/19)/(25/24)	96/95	19
S21⋅S22⋅S23⋅S24⋅S25	(21/20)/(26/25)	105/104	13
S22⋅S23⋅S24⋅S25⋅S26	(22/21)/(27/26)	572/567	13
S23⋅S24⋅S25⋅S26⋅S27	(23/22)/(28/27)	621/616	23
S28⋅S29⋅S30⋅S31⋅S32	(28/27)/(33/32)	896/891	11
S34⋅S35⋅S36⋅S37⋅S38	(34/33)/(39/38)	1292/1287	19
S35⋅S36⋅S37⋅S38⋅S39	(35/34)/(40/39)	273/272	17
S40⋅S41⋅S42⋅S43⋅S44	(40/39)/(45/44)	352/351	13
S45⋅S46⋅S47⋅S48⋅S49	(45/44)/(50/49)	441/440	11
S46⋅S47⋅S48⋅S49⋅S50	(46/45)/(51/50)	460/459	23
S50⋅S51⋅S52⋅S53⋅S54	(50/49)/(55/54)	540/539	11
S51⋅S52⋅S53⋅S54⋅S55	(51/50)/(56/55)	561/560	17
S52⋅S53⋅S54⋅S55⋅S56	(52/51)/(57/56)	2912/2907	19
S64⋅S65⋅S66⋅S67⋅S68	(64/63)/(69/68)	4352/4347	23
S65⋅S66⋅S67⋅S68⋅S69	(65/64)/(70/69)	897/896	23
S76⋅S77⋅S78⋅S79⋅S80	(76/75)/(81/80)	1216/1215	19
S91⋅S92⋅S93⋅S94⋅S95	(91/90)/(96/95)	1729/1728	19
S100⋅S101⋅S102⋅S103⋅S104	(100/99)/(105/104)	2080/2079	13
S115⋅S116⋅S117⋅S118⋅S119	(115/114)/(120/119)	2737/2736	23
S121⋅S122⋅S123⋅S124⋅S125	(121/120)/(126/125)	3025/3024	11
S171⋅S172⋅S173⋅S174⋅S175	(171/170)/(176/175)	5985/5984	19

Sk/S(k + 1) (ultraparticulars)

Motivational example

Often it is desirable to make consecutive superparticular intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.

For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate ⁠5/4/6/5⁠ = 25/24 = S5 with ⁠6/5/7/6⁠ = 36/35 = S6, hence tempering out ⁠S5/S6⁠ = ⁠25/24/36/35⁠ = 875/864, but it is actually often not necessary to know the specific numbers. Often familiarizing yourself with and understanding the "Sk" notation will give you a lot of insight, as we will see.

Back to our example: we know that S5 ~ S6 because we are tempering out S5/S6; from this we can deduce that the intervals must be arranged like this: 7/6 ← S5~S6 → 6/5 ← S5~S6 → 5/4.

From this, it can be deduced that (6/5)³ ~ 7/4, because one of the 6/5's can be lowered by S6 to 7/6 and another of the 6/5's can be raised by S5 to 5/4. Then because we have tempered S5 and S6 together, we have lowered and raised by the same amount, so the result of (7/6)⋅(6/5)⋅(5/4) = 7/4 must be the same as the result of (6/5)⋅(6/5)⋅(6/5) in this temperament.

The reader is encouraged to familiarize themself with the structure of this argument, as it generalizes to arbitrary Sk (→ [[S-expression/Advanced results #Mathematical derivations|]]); the algebraic proof is tedious, but the intuition is the same:

$$ \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1} $$

… implies that three ⁠k + 1/k⁠ give ⁠k + 2/k − 1⁠ iff we temper out ⁠Sk/S(k + 1)⁠. [math]\displaystyle{ \square }[/math]

Significance

1. Tempering out any two consecutive square-particulars Sk and S(k + 1) will naturally imply tempering out the ultraparticular between them, ⁠Sk/S(k + 1)⁠, meaning they are very common implicit commas.
2. Tempering out any two consecutive ultraparticulars will imply tempering out the semiparticular, which is their product. A rather-interesting arithmetic of square-particular and related commas exists. This arithmetic can be described compactly with S-expressions, which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.^{[clarification needed]}
3. Tempering out the ultraparticular Sk/S(k + 1) along with either the corresponding 1/2-square-particular Sk⋅S(k + 1) or one of the two corresponding lopsided commas Sk²⋅S(k + 1) or Sk⋅S(k + 1)² implies tempering both of Sk and S(k + 1) individually, and vice versa, so that there is a total of five equivalences—corresponding to five infinite families of commas—for every such Sk and S(k + 1). This only gets better^{[clarification needed]} if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another motivation of the use of S-expressions. For example, {S16, S17} → {S16⋅S17, S16/S17, S16²⋅S17, S16⋅S17²}, and any of the two commas in the latter set imply all the other commas too.

Table of ultraparticulars

S-expression	Cube relation	Ratio	Cents	Subgroup
S2/S3 = (4/3)/(9/8)	(4/1)/(3/2)3	32/27	294.135	2.3
S3/S4 = (9/8)/(16/15)	(5/2)/(4/3)3	135/128	92.179	2.3.5
S4/S5 = (16/15)/(25/24)	(2/1)/(5/4)3	128/125	41.059	2.5
S5/S6 = (25/24)/(36/35)	(7/4)/(6/5)3	875/864	21.902	2.3.5.7
S6/S7 = (36/35)/(49/48)	(8/5)/(7/6)3	1728/1715	13.074	2.3.5.7
S7/S8 = (49/48)/(64/63)	(3/2)/(8/7)3	1029/1024	8.433	2.3.7
S8/S9 = (64/63)/(81/80)	(10/7)/(9/8)3	5120/5103	5.758	2.3.5.7
S9/S10 = (81/80)/(100/99)	(11/8)/(10/9)3	8019/8000	4.107	2.3.5.11
S10/S11 = (100/99)/(121/120)	(4/3)/(11/10)3	4000/3993	3.032	2.3.5.11
S11/S12 = (121/120)/(144/143)	(13/10)/(12/11)3	17303/17280	2.303	2.3.5.11.13
S12/S13 = (144/143)/(169/168)	(14/11)/(13/12)3	24192/24167	1.790	2.3.7.11.13
S13/S14 = (169/168)/(196/195)	(5/4)/(14/13)3	10985/10976	1.419	2.5.7.13
S14/S15 = (196/195)/(225/224)	(16/13)/(15/14)3	43904/43875	1.144	2.3.5.7.13
S15/S16 = (225/224)/(256/255)	(17/14)/(16/15)3	57375/57344	0.936	2.3.5.7.17
S16/S17 = (256/255)/(289/288)	(6/5)/(17/16)3	24576/24565	0.775	2.3.5.17
S17/S18 = (289/288)/(324/323)	(19/16)/(18/17)3	93347/93312	0.649	2.3.17.19
S18/S19 = (324/323)/(361/360)	(20/17)/(19/18)3	116640/116603	0.549	2.3.5.17.19
S19/S20 = (361/360)/(400/399)	(7/6)/(20/19)3	48013/48000	0.469	2.3.5.7.19
S20/S21 = (400/399)/(441/440)	(22/19)/(21/20)3	176000/175959	0.403	2.3.5.7.11.19
S21/S22 = (441/440)/(484/483)	(23/20)/(22/21)3	213003/212960	0.350	2.3.5.7.11.23
S22/S23 = (484/483)/(529/528)	(8/7)/(23/22)3	85184/85169	0.305	2.7.11.23
S23/S24 = (529/528)/(576/575)	(25/22)/(24/23)3	304175/304128	0.268	2.3.5.11.23
S24/S25 = (576/575)/(625/624)	(26/23)/(25/24)3	359424/359375	0.236	2.3.5.13.23
S25/S26 = (625/624)/(676/675)	(9/8)/(26/25)3	140625/140608	0.209	2.3.5.13
S26/S27 = (676/675)/(729/728)	(28/25)/(27/26)3	492128/492075	0.186	2.3.5.7.13
S27/S28 = (729/728)/(784/783)	(29/26)/(28/27)3	570807/570752	0.167	2.3.7.13.29
S28/S29 = (784/783)/(841/840)	(10/9)/(29/28)3	219520/219501	0.150	2.3.5.7.29
S31/S32 = (961/960)/(1024/1023)	(11/10)/(32/31)3	327701/327680	0.111	2.5.11.31
S33/S34 = (1089/1088)/(1156/1155)	(35/32)/(34/33)3	1257795/1257728	0.092	2.3.5.7.11.17
S34/S35 = (1156/1155)/(1225/1224)	(12/11)/(35/34)3	471648/471625	0.084	2.3.5.7.11.17
S37/S38 = (1369/1368)/(1444/1443)	(13/12)/(38/37)3	658489/658464	0.066	2.3.13.19.37
S40/S41 = (1600/1599)/(1681/1680)	(14/13)/(41/40)3	896000/895973	0.052	2.5.7.13.41
S43/S44 = (1849/1848)/(1936/1935)	(15/14)/(44/43)3	1192605/1192576	0.042	2.3.5.7.11.43
S46/S47 = (2116/2115)/(2209/2208)	(16/15)/(47/46)3	1557376/1557345	0.034	2.3.5.23.47
S49/S50 = (2401/2400)/(2500/2499)	(17/16)/(50/49)3	2000033/2000000	0.029	2.5.7.17
S50/S51 = (2500/2499)/(2601/2600)	(52/49)/(51/50)3	6500000/6499899	0.027	2.3.5.7.13.17
S55/S56 = (3025/3024)/(3136/3135)	(19/18)/(56/55)3	3161125/3161088	0.020	2.3.5.7.11.19
S64/S65 = (4096/4095)/(4225/4224)	(22/21)/(65/64)3	5767168/5767125	0.013	2.3.5.7.11.13

The above table is a list of all 23-limit ultraparticulars corresponding to Sk with k < 77, plus ultraparticulars corresponding to dividing a superparticular interval into three equal parts up to 17/16 (or up to 19/18 but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.

📝Note:

This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page.

Note that ultraparticulars are, in general, extremely precise commas so that usually one would not consider tempering them directly rather than through tempering the square-particulars Sk which they are composed of. As an example of this, notice that S10/S11 is the largest ultraparticular categorised as an unnoticeable comma (unnoticeable in the sense of being smaller than the melodic just-noticeable difference), despite only dividing a superparticular as simple and unremarkable as 4/3. For this reason, a cents column has been included to aid an appreciation of their precision. The cent value of a semiparticular is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.

Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably consistent) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3k + 1)/S(3k+ 2) for a positive integer k, because then the superparticular can be expressed as:

$$ \frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3} $$

Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of the bulk of the tempering may be possible through semiparticulars, discussed next.

Sk/S(k + 2) (semiparticulars)

Motivational examples

If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:

• 4/3 is approximately half of 9/5
• 9/7 is approximately half of 5/3 (= 10/6)
• 5/4 is approximately half of 11/7
• 11/9 is approximately half of 3/2 (= 12/8)
• 6/5 is approximately half of 13/9
• 13/11 is approximately half of 7/5 (= 14/10)
• 7/6 is approximately half of 15/11
• 15/13 is approximately half of 4/3 (= 16/12)
• 8/7 is approximately half of 17/13
• 17/15 is approximately half of 9/7 (= 18/14)
• 9/8 is approximately half of 19/15
• 19/17 is approximately half of 5/4 (= 20/16)

These properties show a pattern: take some arbitrary quodd-particular (k + 4)/k; observe that we can split it into (k + 4)/(k + 2) * (k + 2)/k. Now observe that (k + 2)/k > (k + 3)/(k + 1) > (k + 4)/(k + 2); in fact, it can be shown fairly easily that (k + 3)/(k + 1) is the mediant of (k + 4)/(k + 2) and (k + 2)/k. It turns out that making this mediant — (k + 3)/(k + 1) — equal to half of (k + 4)/k is equivalent to tempering S(k + 1)/S(k + 3).

Significance

1. For differences between square-particulars of the form Sk/S(k + 2), the resulting comma is either superparticular or odd-particular, so these are efficient commas. (This terminology also suggests throdd-particular and quodd-particular as generalizations.)
2. Tempering out any two consecutive ultraparticulars implies tempering out a semiparticular, so from two adjacent "thirding" equivalences you get a "halving" equivalence for free.
3. Tempering out any two nearly-consecutive square-particulars (Sk and S(k + 2)) implies tempering out a semiparticular; this is generally much more ideal than tempering out two consecutive Sk because it is of a lot lower damage (see lopsided commas for relatively large commas implied by this higher-damage strategy).
4. On top of the halving equivalence, there is a number of subtler structural implications, discussed below, that may be desirable to the temperament designer.

Meaning

In the below, we use S(k - 1)/S(k + 1) for symmetry around k to make the math visually simpler, but keep in mind it is equivalent to using an offset k. Also keep in mind that k - a (for positive a) is smaller than k, so that k/(k - a) > (k + a)/k (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.

Tempering out S(k - 1)/S(k + 1) implies that (k + 2)/(k - 2) is divisible exactly into two halves of (k + 1)/(k - 1). It also implies that the intervals (k + 2)/k (= s) and k/(k - 2) (= L) are equidistant from (k + 1)/(k - 1) (= M) because, to make them equidistant, we need to temper out:

$$ \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} $$

… and notice that the latter expression is the one we have shown is equal to S(k - 1)/S(k + 1)]] up to an offset k (→ S-expression/Advanced results #Mathematical derivations). In other words, that tempering out S(k - 1)/S(k + 1) results in (k + 1)/(k - 1) being half of (k + 2)/(k - 2) is an implication that it makes (k + 2)/k, (k + 1)/(k - 1), and k/(k - 2) equidistant.

Also note that in the above, (k + 1)/(k - 1) is the mediant of the adjacent two intervals, meaning that division of an interval into two via tempering out a semiparticular is in some sense 'optimal' relative to the complexity. This also means that if k is a multiple of 2, this corresponds to a natural way to split the square superparticular S(k/2) into two parts. For example, if k = 10, then we have (10 + 2)/10, (10 + 1)/(10 - 1), 10/(10 - 2) as equidistant, which simplified is 6/5, 11/9, 5/4, with 11/9 being the mediant of 6/5 and 5/4, and therefore the corresponding superparticular S5 = (5/4)/(6/5) is split into two parts which are tempered together: (5/4)/(11/9) = 45/44 and (11/9)/(6/5) = 55/54. The semiparticular is therefore S(10 - 1)/S(10 + 1) = S9/S11 = 243/242 = (45/44)/(55/54) = ((10 + 2)/(10 - 2))/((10 + 1)/(10 - 1))².

This form of comma has been named semiparticular because most of the time it is superparticular but less often it is odd-particular, and because, when tempered out, they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular. Specifically:

• To find out what a superparticular (a + 1)/a is approximately half of, temper out the semiparticular S(2a)/S(2a+2) and you can observe that (2a + 3)/(2a - 1) is the interval it is approximately half of.
• To find out what an odd-particular (2a + 1)/(2a - 1) is approximately half of, temper out the semiparticular S(2a - 1)/S(2a + 1) and you can observe that (2a + 2)/(2a - 2) = (a + 1)/(a - 1), a superparticular or odd-particular, is the interval it is approximately half of.
• To find out what splits a superparticular (a + 1)/a in half, temper out the semiparticular S(4a + 1)/S(4a + 3) and you can observe that (4a + 3)/(4a + 1), an odd-particular, is the interval that is approximately half of it.
• To find out what splits an odd-particular (2a + 1)/(2a-1) in half, temper out the semiparticular S(4a - 2)/S(4a + 2) and you can observe that (4a - 1)/(4a + 1), an odd-particular, is the interval that is approximately half of it.

Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)² is significant in that it has a special relationship: specifically, consider tempering out (a/b)/(c/d)² so the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of two consecutive superparticulars, whose product is a superparticular or odd-particular, with the two superparticulars directly adjacent to them.

For example, from 9/8 = (18/17)⋅(17/16), we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering out S16/S18 = (19/15)/(9/8)² because we can multiply 9/8 by the vanishing comma (19/15)/(9/8)² to get (19/15)/(9/8) = (19/18)(16/15), or as 13/11 = (13/12)⋅(12/11) we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering out S11/S13 = (7/5)/(13/11)² because we can multiply 13/11 by the vanishing comma (7/5)/(13/11)² to get (7/5)/(13/11) = (14/13)⋅(11/10). Both intervals should be replaced simultaneously as this results in lower error, and note that to replace them individually we must pick the higher-error route of tempering out S16 and S18 or S11 and S13 individually, for which tempering out the semiparticular is then an implied consequence. The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.

Table of semiparticulars

Here follows a table of 23-limit semiparticulars corresponding to square-particulars Sk for k < 96, plus all semiparticulars up to S33/S35 = S99, an exceptional 11-limit comma, plus all semiparticulars dividing superparticular intervals up to 13/12 (corresponding to the 17-limit semiparticular S49/S51) for completeness. This table also shows all semiparticulars corresponding to splitting an odd-particular in two up to 17/15 (although a common strategy is to temper out the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound k < 96 was chosen as it corresponds to another remarkable semiparticular S78/S80 = S351. Perhaps many of the patterns will become clearer if you examine this table:

S-expression	Square relation	Ratio	Subgroup
S2/S4 = (4/3)/(16/15)	(5/1)/(2/1)2	5/4	2.5
S3/S5 = (9/8)/(25/24)	(3/1)/(5/3)2	27/25	2.3.5
S4/S6 = (16/15)/(36/35)	(7/3)/(3/2)2	28/27	2.3.7
S5/S7 = (25/24)/(49/48)	(2/1)/(7/5)2	50/49	2.5.7
S6/S8 = (36/35)/(64/63)	(9/5)/(4/3)2	81/80	2.3.5
S7/S9 = (49/48)/(81/80)	(5/3)/(9/7)2	245/243	3.5.7
S8/S10 = (64/63)/(100/99)	(11/7)/(5/4)2	176/175	2.5.7.11
S9/S11 = (81/80)/(121/120)	(3/2)/(11/9)2	243/242	2.3.11
S10/S12 = (100/99)/(144/143)	(13/9)/(6/5)2	325/324	2.3.5.13
S11/S13 = (121/120)/(169/168)	(7/5)/(13/11)2	847/845	5.7.11.13
S12/S14 = (144/143)/(196/195)	(15/11)/(7/6)2	540/539	2.3.5.7.11
S13/S15 = (169/168)/(225/224)	(4/3)/(15/13)2	676/675	2.3.5.13
S14/S16 = (196/195)/(256/255)	(17/13)/(8/7)2	833/832	2.7.13.17
S15/S17 = (225/224)/(289/288)	(9/7)/(17/15)2	2025/2023	3.5.7.17
S16/S18 = (256/255)/(324/323)	(19/15)/(9/8)2	1216/1215	2.3.5.19
S17/S19 = (289/288)/(361/360)	(5/4)/(19/17)2	1445/1444	2.5.17.19
S18/S20 = (324/323)/(400/399)	(21/17)/(10/9)2	1701/1700	2.3.5.7.17
S19/S21 = (361/360)/(441/440)	(11/9)/(21/19)2	3971/3969	3.7.11.19
S20/S22 = (400/399)/(484/483)	(23/19)/(11/10)2	2300/2299	2.5.11.19.23
S21/S23 = (441/440)/(529/528)	(6/5)/(23/21)2	2646/2645	2.3.5.7.23
S22/S24 = (484/483)/(576/575)	(25/21)/(12/11)2	3025/3024	2.3.5.7.11
S23/S25 = (529/528)/(625/624)	(13/11)/(25/23)2	6877/6875	5.11.13.23
S24/S26 = (576/575)/(676/675)	(27/23)/(13/12)2	3888/3887	2.3.13.23
S25/S27 = (625/624)/(729/728)	(7/6)/(27/25)2	4375/4374	2.3.5.7
S26/S28 = (676/675)/(784/783)	(29/25)/(14/13)2	4901/4900	2.5.7.13.29
S27/S29 = (729/728)/(841/840)	(15/13)/(29/27)2	10935/10933	2.3.5.13.29
S28/S30 = (784/783)/(900/899)	(31/27)/(15/14)2	6076/6075	2.3.5.7.31
S29/S31 = (841/840)/(961/960)	(8/7)/(31/29)2	6728/6727	2.7.29.31
S30/S32 = (900/899)/(1024/1023)	(33/29)/(16/15)2	7425/7424	2.3.5.11.29
S31/S33 = (961/960)/(1089/1088)	(17/15)/(33/31)2	16337/16335	2.3.5.11.17.31
S32/S34 = (1024/1023)/(1156/1155)	(35/31)/(17/16)2	8960/8959	2.5.7.17.31
S33/S35 = (1089/1088)/(1225/1224)	(9/8)/(35/33)2	9801/9800	2.3.5.7.11
S36/S38 = (1296/1295)/(1444/1443)	(39/35)/(19/18)2	12636/12635	2.3.5.7.13.19
S37/S39 = (1369/1368)/(1521/1520)	(10/9)/(39/37)2	13690/13689	2.3.5.13.37
S41/S43 = (1681/1680)/(1849/1848)	(11/10)/(43/41)2	18491/18490	2.5.11.41.43
S45/S47 = (2025/2024)/(2209/2208)	(12/11)/(47/45)2	24300/24299	2.3.5.11.47
S46/S48 = (2116/2115)/(2304/2303)	(49/45)/(24/23)2	25921/25920	2.3.5.7.23
S49/S51 = (2401/2400)/(2601/2600)	(13/12)/(51/49)2	31213/31212	2.3.7.13.17
S52/S54 = (2704/2703)/(2916/2915)	(55/51)/(27/26)2	37180/37179	2.3.5.11.13.17
S66/S68 = (4356/4355)/(4624/4623)	(69/65)/(34/33)2	75141/75140	2.3.5.11.17.23
S78/S80 = (6084/6083)/(6400/6399)	(81/77)/(40/39)2	123201/123200	2.3.5.7.11.13

📝Note:

While a lot of these have pages, not all of them do, although that does not mean they should not. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher prime limit, as a surprising amount of 23-limit semiparticulars shown here already have pages.

Sk²⋅S(k + 1) and S(k − 1)⋅Sk² (lopsided commas)

Significance

1. Tempering out any two consecutive square-particulars, Sk and S(k + 1), implies tempering out the two associated lopsided commas as well as the associated triangle-particular and ultraparticular, so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.
2. If a comma (such as the diaschisma, 2048/2025), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering out the square-particulars composing it (such as S16 and S17 in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise.
3. It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form.

Derivation of equivalence relation

Using the clarity of S-factorizations, we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:

Sk²⋅S(k + 1) = [k - 1, k, k + 1, k + 2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:

Sk²⋅S(k + 1) = (k/(k - 1))² / ((k + 2)/k) through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1].

S(k - 1)⋅Sk² = [k - 2, k - 1, k, k + 1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies:

S(k - 1)⋅Sk² = (k/(k - 2)) / ((k + 1)/k)² through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].

Tables

Below are two tables of 43-limit lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large, so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the olympia and the phaotic comma.

Using S-factorizations to understand the significance of S-expressions

This section deals with the forms of the infinite comma families as expressed in terms of nearby harmonics in the harmonic series and as related to square-superparticulars; note that this uses a mathematical notation of [a, b, c, ...]^[x, y, z, ...] to denote a^x * b^y * c^z * ...

If instead of working through things algebraically we look at square-particulars as describing a relationship between adjacent harmonics, we can use this to understand why certain simplifications and equivalences exist in a way that is equivalent to the sometimes harder-to-understand usual algebraic form:

If we describe Sk as [k-1, k, k+1]^[-1, 2, -1] then if we write something like Sk/S(k + 2) (semiparticulars) in this form we get:

[k-1, k, k+1, k+2, k+3]^([-1, 2, -1, 0, 0] - [0, 0, -1, 2, -1] = [-1, 2, 0, -2, 1]) from which we can clearly see that we have two (k+2)/k's making up a (k+3)/(k-1). An exercise to the reader is to go through the other forms discussed on this page to derive similar expressions. (For example, through cancellation it's easy to prove that 1/n-square-particulars (the product of n consecutive square-(super)particulars) are equal to the ratio of the two superparticular intervals on the ends.)

Sk = [k-1, k, k+1]^[-1, 2, -1]

Sk * S(k+1) = [k-1, k, k+1, k+2]^[-1, 1, 1, -1]
 = [k-1, k, k+1(, k+2)]^[-1, 2, -1(, 0)] * [(k-1,) k, k+1, k+2]^[(0,) -1, 2, -1]

S(k-1) * Sk * S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 1, 0, 1, -1]
 = ( (k-1)/(k-2) )( k/(k-1) ) * ( k/(k-1) )/( (k+1)/k ) * ( (k+1)/k )/( (k+2)/(k+1) )
 = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) = ( (k-1)(k+1) )/( (k-2)(k+2) )

k-2  k-1   k   k+1   k+2
-1    2   -1    0    0
 0   -1    2   -1    0
 0    0   -1    2   -1
========================
-1    1    0    1   -1

Sk / S(k+1) = [k-1, k, k+1, k+2]^[-1, 3, -3, 1]
 = [k-1, k, k+1]^[-1, 2, -1] * [k, k+1, k+2]^[1, -2, 1]
 = (k+2)/(k-1) * ( k/(k+1) )^3 = (k+2)/(k-1) / ((k+1)/k)^3

S(k-1) / S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 2, 0, -2, 1]
 = [k-2, k-1, k]^[-1, 2, -1] * [k, k+1, k+2]^[ 1, -2,  1]
 = [k-2, k-1, k]^[-1, 2, -1] / [k, k+1, k+2]^[-1,  2, -1]
 = (k+2)/(k-2) * ((k-1)/(k+1))^2 = (k+2)/(k-2) / ((k+1)/(k-1))^2

k-2  k-1   k   k+1   k+2
-1    2   -1    0    0
 0    0    1   -2    1
========================
-1    2    0   -2    1

This technique will be called "S-factorizations", as it is uses a certain format for expressing factorization (analogous to monzos) that is uniquely suited for interpreting the relationships described by S-expressions.

Note that the redundancy in these factorizations (in the sense that there are generators that are not linearly independent of the others) is a property that reflects the reality of equivalent S-expressions.

The generalisation of this method using commutative group theory is discussed in S-expression/Advanced_results#Abstraction, though the ideas are very simple for anyone with simple mathematical training willing to learn the very basics needed.

Using S-factorizations to show a useful equivalence/redundancy of S-expressions

Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:

The general S-expression equivalence

Consider:

Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:
S(2k-1) * S(2k) * S(2k) * S(2k+1)
 = [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(
   [-1,    2,   -1]
       + [-2,    4,   -2]
             + [-1,    2,   -1]
 = [-1,    0,    2,    0,   -1] )

From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex k:

[math]\displaystyle{ \large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1) }[/math]

...where we use the notation Sk^p to mean (Sk)^p rather than S(k^p) for convenience in the practical analysis of regular temperaments using S-expressions.

For tuning theory only integer k > 1 is of relevance. Technically, rational k other than 1 correspond to rational commas too; the most relevant case for tuning theory is that half-integer k work as an alternative notation for odd-particulars, though for intuitively understanding the notation, the method described in #Abstraction may be recommendable as having (in a mathematical sense) exact analogues for every infinite family of commas defined in terms of an analogue of an S-expression, for which the most musically fruitful example is Ok = (k / (k - 2))/((k + 2) / k) for odd k as relevant to no-twos subgroup temperaments.

Equivalent S-expressions

Significance and meaning

All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/n-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/n-square-particular expressions become very common for higher n, but are still quite rare for small n.

A useful general rule

While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:

$$ \large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1) $$

This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from #Examples. See S-expression/Advanced results for mathematical details.

Examples

Here is an incomplete list of examples.

Comma	S-expressions
28/27	S7⋅S8, S4/S6
36/35	S6, S8⋅S9
64/63	S8, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
81/80	S9, S6/S8
176/175	S8/S10, S22⋅S23⋅S24
243/242	S9/S11, S15/(S22/S24 = S55)
325/324	S25⋅S26, S10/S12
540/539	S12/S14, (S9⋅S10)/S7, (S6/S7)/(S8/S10)
676/675	S26, S13/S15
1225/1224	S35, S49⋅S50
3025/3024	S55, S22/S24, (S25/S27)⋅S99
2601/2600	S51, S17/(S25⋅S26)
9801/9800	S99, S33/S35
25921/25920	S161, S46/S48
123201/123200	S351, S78/S80

📝Note:

Examples that can easily (with one or two algebraic rewriting steps) be shown to result from the aforementioned useful general rule are not included.

📝Note:

Where a comma written in the form a/b is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of 3025/3024 as there are many S-expressions for it so restating them each time it appears seems inconvenient.

💡Tip:

Feel free to expand with any equivalences you find that you think are valuable.

A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.

It suffices to show every superparticular number including 2/1 has an expression using square-particulars:

$$ \begin{align} & 2/1 = S_2 \cdot S_2 \cdot S_3\ ,\\ & 3/2 = S_2 \cdot S_3\ ,\\ & 4/3 = S_2\ ,\\ & \frac{a/(a - 1)}{(b + 1)/b} = \prod_{k=a}^b \left( S_k = \frac{k/(k - 1)}{(k + 1)/k} \right) \\ & \ \ \ = \frac{a/(a - 1)}{(a + 1)/a} \cdot \frac{(a + 1)/a}{(a + 2)/(a + 1)} \cdot \frac{(a + 2)/(a + 1)}{(a + 3)/(a + 2)} \cdot\ \ldots \cdot \frac{b/(b - 1)}{(b + 1)/b} = \frac{a/(a - 1)}{(b + 1)/b} \\ & \implies \frac{a/(a - 1)}{(b + 1)/b} = S_a \cdot S_{a + 1} \cdot S_{a + 2} \cdot\ \ldots \cdot S_b \\ & \implies \frac{S_2 \cdot S_2 \cdot S_3}{\prod_{a = 2}^k S_a} = 2 \cdot \left( \frac{2/(2 - 1)}{(k + 1)/k} \right)^{-1} = 2 \cdot \left( \frac{(k + 1)/k}{2} \right) = (k + 1)/k \end{align} $$

From here it should not be hard to see how to make any positive rational number. For 11/6, for example, we can do (11/10)(10/9)(9/8)…(2/1) = 11 and then divide that by (6/5)(5/4)(4/3)(3/2)(2/1), meaning 11/6 = (11/10)(10/9)(9/8)(8/7)(7/6) because of the cancellations, then each of those superparticulars we replace with the corresponding S-expression to get the final S-expression. This final S-expression is likely to be far from the most efficient or interesting expression; the redundancy in S-expressions is a strength and feature, as it tells us that there are more than the trivial connections between commas and intervals and that S-expressions can be wielded as a mathematical tool/language to investigate and identify them.

Glossary

Superparticular

The interval/comma between two consecutive harmonics. See superparticular.

These are of the form ⁠k + 1/k⁠.

Square-particular

A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.

These are of the form ⁠k²/k² − 1⁠ = Sk.

Triangle-particular

A superparticular interval/comma whose numerator is a triangular number. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.

These are of the form ⁠k² + k/k² + k − 2⁠. (This always simplifies to a superparticular.)

1/n-square-particular

A comma which is the product of n consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.

These are of the form Sa⋅S(a + 1)⋅…⋅Sb = ⁠⁠a/a − 1⁠/⁠b + 1/b⁠⁠ = ⁠ab/(a − 1)(b + 1)⁠.

Replacing/substituting a with k and b with k + n - 1 gives us an equivalent expression that includes the number of square-particulars n:

Sk⋅S(k + 1)⋅…⋅S(k + n − 1) = ⁠⁠k/k − 1⁠/⁠k + n/k + n − 1⁠⁠ = ⁠k(k + n − 1)/(k − 1)(k + n⁠

For b = a + 1 these can also be called triangle-particulars, in which case they are always superparticular.

These have implications for whether consistency in the (n + k) = (b + 1)-odd-limit is potentially possible in a given temperament; see the section on 1/n-square-particulars.

Odd-particular

An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.

These are of the form ⁠2k + 1/2k − 1⁠.

Throdd-particular

An interval/comma between two harmonics 3 apart which is not superparticular.

These are of the form ⁠3k + 1/3k − 2⁠ or ⁠3k + 2/3k − 1⁠.

Quodd-particular

An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.

These are of the form ⁠4k + 1/4k − 3⁠ or ⁠4k + 3/4k − 1⁠.

n-odd-particular

An interval/comma between two coprime harmonics n apart (also called as delta-n ratio). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.

If n is a prime, an n-odd-particular interval is between two harmonics n apart which is not superparticular. For example, 5-odd-particular intervals are of the form ⁠5k + 1/5k − 4⁠, ⁠5k + 2/5k − 3⁠, ⁠5k + 3/5k − 2⁠, or ⁠5k + 4/5k − 1⁠.

If n is a composite, an n-odd-particular interval is between two harmonics n apart which is neither superparticular nor of m-odd-particular intervals where m is any other divisor of n. For example, 6-odd-particular intervals are of the form ⁠6k + 1/6k − 5⁠ or ⁠6k + 5/6k − 1⁠.

Ultraparticular

An interval/comma which is the ratio of two consecutive square-particulars.

These are of the form ⁠Sk/S(k + 1)⁠.

Semiparticular

A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:

These are of the form ⁠Sk/S(k + 2)⁠.

S-expression

An expression using the Sk shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.

S-factorization

An expression that takes a list of consecutive integer harmonics including the kth harmonic and raises them to integer powers, similar to a smonzo but uniquely suited to analysing S-expressions.

For example: Sk = [k − 1, k, k + 1]^{[−1, 2, −1]} because Sk = (k − 1)⁻¹k²(k + 1)⁻¹.

S-comma

Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).

Indirect S-comma

Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions; for example, the schisma admits at least three such representations.

See further

• Advanced results – for the harder-to-reach algebra