The qname Algorithm

Introduction

A qname is composed of a sequence of roots. Each root has a numerical value, and the value of a qname is formed by multiplying together the values of the roots. Affixes are placed around roots or groups of roots to combine them into larger factors. qname is based on superior highly composite numbers (SHCNs). It is the successor to my earlier pname, which was inspired by Michael Thomas de Vlieger's argam.

Phonology

qname has 17 consonants and 7 vowels:

		Labial	Alveolar	Post-alveolar	Palatal	Velar	Glottal
Nasal		m	n
Stop	Voiceless	p	t			k
Stop	Voiced	b	d			g
Fricative	Voiceless	f	s	x (/ʃ/)			h
Fricative	Voiced	v	z	j (/dʒ/)
Approximant			l		y (/j/)

		Front	Central	Back
Close		i		u
Mid	Non-rhotic	e	ı (/ə/)	o
Mid	Rhotic		r (/ɚ/)
Open			a

Each letter is pronounced like its IPA value unless otherwise noted. Additionally, 'z' can be pronounced /ts/, and 'v' can be pronounced /w/.

There are two additional consonant phonemes, 'ŷ' and 'ŵ'. 'ŷ' is pronounced and written as 'j', but when preceding a hyphenation point, it and its vowel combine into a single /i/ (spelled 'i'). Similarly, 'ŵ' followed by schwa is pronounced like /u/ and spelled 'u', whereas preceding other vowels, it is pronounced and written like 'v'. Both of these vowels can glide with the previous syllable to form the diphthongs 'ai', 'oi', and 'au'. An alternative pronunciation can exist where the schwa-containing syllables do not modify the way 'ŷ' and 'ŵ' are pronounced, and they become identical to 'j' and 'v'.

Every syllable is of the form CV, with the exception of the vocalic syllables 'a' and 'un' (0 and 1), and the combined 'i' and 'u' from above. The schwa can be elided out to form consonant clusters or word-final consonants, as long as it does not conflict with another consonant's prononciation (ex. eliding /təs/ into /ts/ is forbidden as it conflicts with 'z', but eliding /kəs/ into /ks/ is allowed.) The letter 'ı' may be omitted everywhere except before the final nasal of an SHCN suffix.

Consonant Hardening

Every consonant (except 'y', 'j', and 'h') has a hardened form. These are always a cluster of two consonants. The schwa between them is always elisible.

Cons.	Hard	Cons.	Hard	Cons.	Hard
m	mp	n	ns
p	ps	t	tx	k	ks
b	bl	d	j	g	gl
f	fk	s	sk	x	xk
v	ft	z	st
ŵ	ft	l	lk	ŷ	ŷk

Basic Roots

Weak roots are one syllable. Strong roots are two syllables with the final vowel omitted, with the form CVC. If no vowel is specified for a strong root, schwa is assumed. Every basic root with value less than or equal to 12 is weak, and those that are greater than 12 are strong.

The basic roots are of the following forms:

Smalls

Small roots represent numbers from 2 to 60.

n	Root	n	Root	n	Root	n	Root	n	Root	n	Root
		11	ze	21	lex	31	nad	41	yal	51	tas
2	da	12	va	22	zad	32	dov	42	dag	52	kad
3	te	13	kal	23	ful	33	zed	43	sil	53	saf
4	ka	14	del	24	gam*	34	das	44	zag	54	daŷ
5	pe	15	fen	25	pin	35	pal	45	nob	55	pez
6	se	16	bov*	26	yaf	36	sin	46	fol	56	noŷ
7	la	17	gas	27	naŷ	37	mag	47	fox	57	god
8	yo	18	saŷ	28	lag	38	dex	48	boŷ	58	nan
9	no	19	xig	29	nen	39	min	49	lin	59	kul
10	xu	20	ven	30	faŷ	40	yob	50	pan	60	baŷ

*When preceding a hyphenation point, 'bov' and 'gam' change to 'bo' and 'ga' respectively.

Primes

A prime root represents a prime from 2 to 659 (the 120th prime). Prime roots under 60 coincide with the small roots.

π(n)	n	Root	π(n)	n	Root	π(n)	n	Root	π(n)	n	Root	π(n)	n	Root	π(n)	n	Root
1	2	da	21	73	sag	41	179	nib	61	283	pom	81	419	tal	101	547	men
2	3	te	22	79	tid	42	181	mob	62	293	sam	82	421	pub	102	557	nol
3	5	pe	23	83	van	43	191	teg	63	307	yub	83	431	biz	103	563	lol
4	7	la	24	89	kom	44	193	lus	64	311	gad	84	433	pon	104	569	luz
5	11	ze	25	97	man	45	197	lod	65	313	teb	85	439	sad	105	571	dub
6	13	kal	26	101	fel	46	199	pad	66	317	dis	86	443	lad	106	577	sig
7	17	gas	27	103	kob	47	211	siv	67	331	hol	87	449	fan	107	587	nil
8	19	xig	28	107	nix	48	223	kam	68	337	bel	88	457	laŷ	108	593	has
9	23	ful	29	109	kub	49	227	yen	69	347	tul	89	461	kid	109	599	maŷ
10	29	nen	30	113	zin	50	229	san	70	349	yeb	90	463	tol	110	601	dam
11	31	nad	31	127	gal	51	233	sib	71	353	lud	91	467	pod	111	607	lon
12	37	mag	32	131	yel	52	239	tel	72	359	haf	92	479	yul	112	613	kog
13	41	yal	33	137	las	53	241	yod	73	367	tan	93	487	neb	113	617	yan
14	43	sil	34	139	sel	54	251	xen	74	373	vol	94	491	pud	114	619	fov
15	47	fox	35	149	bol	55	257	xiz	75	379	len	95	499	mal	115	631	mov
16	53	saf	36	151	kib	56	263	bal	76	383	som	96	503	kum	116	641	liv
17	59	kul	37	157	lub	57	269	lan	77	389	lid	97	509	bag	117	643	ten
18	61	log	38	163	sol	58	271	xil	78	397	pul	98	521	kaf	118	647	gon
19	67	kaŷ	39	167	yed	59	277	pas	79	401	vaŵ	99	523	vaŷ	119	653	lig
20	71	kas	40	173	zil	60	281	nid	80	409	hag	100	541	fal	120	659	xin

SHCNs

The first nine SHCNs, up to 55440, have basic root forms. SHCNs under 60 coincide with small roots.

n	Root	n	Root
2	da	360	kol
6	se	2520	saŵ
12	va	5040	pol
60	baŷ	55440	den
120	han

Specials

The root for 0 is 'a' and the root for 1 is 'un'. These roots are exclusively used in the words for 0 and 1.

Affixes

Compound roots, roots that are composed of multiple other roots, can be either single or multiple. A single root is defined to be either a single basic root or any compound root which does not have an exponent at the outermost level. Numerically, single roots represent numbers less than or equal to 60, primes, primorials, and SHCNs, but not their powers. Compound roots which are not single roots (they are products of powers of single roots) are multiple roots. A constituent root is a single root, or a single root with an exponent prefix.

A compound root is considered strong if it ends in a strong basic root, prime '-k' or '-t', or an SHCN suffix. A compound root is considered weak if it ends in a weak basic root or the primorial '-vi', '-bi', or '-po'.

A hyphenation point is a point in the word where a hyphen can be inserted. Hyphenation points occur at every point that is directly between the end of a numerical root or suffix and the beginning of another numerical root or 'su-'. There are also hyphenation points at the beginning and end of the word.

Primes

There are two separate affixes to produce primes larger than 659. The first is '-k'. This is appended to any single root with value n, and it produces the nth prime. The second prime suffix is the circumfix 'su-t'. When surrounding a multiple root with value n, it produces the nth prime. In the root is strong and ends with 'd', 'k', 'l', 'v', or 'ŵ', the vowel 'o' is inserted between it and the prime suffix. Any other strong root has an 'a' inserted between it and the suffix.

If there are several consecutive instances of the same prime suffix, the 'k' or 't' will change to 'y' in an alternating manner such that the outermost suffix does not change (for instance, in a run of four '-k' suffixes, the first and third change to 'y'.)

Primorials

There are three affixes that produce primorials. The first, '-vi', appends to a prime root n, inserting a schwa if the prime is strong, and produces the primorial with highest factor n. The other two, '-bi' and 'su-po' act like prime '-k' and 'su-t'. '-bi' attaches to a single root n, and produces the primorial whose highest prime factor is the nth prime. 'su-po' produces the same by surrounding a multiple root. All three suffixes insert schwa if they attach to a strong root.

If the suffix '-vi' succeeds a strong root ending in 'v', the 'v' and '-vi' are combined into '-fi'. The point before the primorial suffix acts like a hyphenation point in that 'ŷ' and 'ŵ' are pronounced like 'i' and 'u' respectively, but otherwise cannot by hyphenated.

SHCNs

Given an SHCN, one can find its highest prime factor n. Then, there is a smallest SHCN whose highest prime factor is n. Denote this by the reduced SHCN. Then, the original SHCN has more prime factors than the reduced SHCN. Define the ancillary to be the difference in these values. If the ancillary is 0, then append the suffix '-n' to the single root n. Some highest prime factors can have ancillaries going up to 3: the ones below 659 are listed in the table below.

		Ancillary
π(n)	n	0	1	2	3
6	13	kalın	kaldın	kalsın	kalfın
17	59	kulın	kuldın	kulzın
34	139	selın	seltın	selfın
36	151	kibın	kipxın	kiblın
92	479	yulın	yulmın	yulkın

Larger known highest prime factors with ancillary 2 are n = 1709, 1907, 2753, 2939, 6301, 7229, 49433, 1117370759, with π(n) = 267, 292, 402, 424, 820, 924, 5080, 56496498.

If n is not listed in this table, it either has highest ancillary value 1 or is greater than 659. If the ancillary is 1, replace the consonant before the SHCN suffix with its hardened version. If the ancillary is 2, the single root n ends in the prime suffix '-k' or '-t'; replace them with 'p'. If the ancillary is 3, harden the 'p' into 'ps'. It is an open problem whether an ancillary of 3 ever occurs outside of n=13.

If the SHCN precedes a labial consonant, the final 'n' should be replaced by 'm'. Before the final nasal, the letter 'ı' should always be written. This vowel cannot be elided.

Exponents

Any single root can have an exponent prefix, denoting a power of the value. If the exponent is less than 12, the prefix is derived from the small root of the number, by replacing the final vowel with 'i', or 'e' if the exponent is 12 ('va' becomes 've'). Additionally, the prefix for an exponent of 8 is 'ji' instead of 'yi'. If the exponent is a primorial, which ends in '-vi', '-bi', or '-po', then that suffix is replaced by '-vo', '-ba', or '-pa' respectively. If the exponent is otherwise a strong simple root, the prefix is the simple root with the vowel 'r' at the end. Otherwise, the exponent is a multiple root, in which case the circumfix 'su-pr' is applied, and the entire unit is prepended to the base of the power.

Contractions of 'su-'

In certain situations, the prefix 'su-' can be removed. When 'su-' succeeds one of prime suffixes '-k' or '-t', optionally separated by a SHCN nasal suffix or a consonant added by hardening, which is ignored for the purpose of the following contractions, and the root preceding the prime suffix is strong and ends in 'p', 't', or 'z', the vowel preceding the prime suffix changes to 'u'. Otherwise, if the root is strong, the vowel preceding the prime suffix changes to 'e'. Finally, if the root ends in the primorial '-vi', '-bi', or '-po', the suffix is replaced by '-ji', '-gi', or '-ki' respectively. The '-fi' contraction is not applied, so '-fi' is replaced with '-vji'. In all of these cases, 'su-' is removed.

If the root preceding the prime suffix ends in a weak basic root, and the prime suffix is '-t', then 'su-' cannot be removed, but '-t' and 'su-' can combine into '-zu-'. The hyphenation point formerly between the '-t' and 'su-' is placed after the 'zu'.

If, after all other contractions, there are two 'su-' next to each other, the second one should be replaced with 'gu-'. This continues in an alternating manner, so for instance 'sususususu-' (five repetitions) becomes 'sugusugusu-'.

Decomposition

A valid word is any word that can be parsed into a number by the rules given above.

A subcanonical word is a valid word with the additional properties that the values of the constituent roots are in descending order, every constituent root of any subword that can be expressed with a basic root is expressed with a basic root, there are no compound prime roots with index 120 or less, and there are no exponents on perfect power roots (4, 8, 9, 16, 25, 32, 36, 49).

A quasicanonical word is a subcanonical word where every subword is expressed with the fewest constituent roots.

A canonical word is, given its value, the unique quasicanonical word with that value where, for every subword, the first constituent root's value is as high as possible. If there are multiple quasicanonical words with this property, the second constituent root's value must be as high as possible, and so on.