User:Eufalesio/Punctional Just System

This article features a notation system that is not meant to be taken too seriously, similar to Breeds' Dwarf system for MOS scales, as part of its design is obviously jocular in nature, but I think that's part of the charm.

Preamble

The Punctional Just System (PJS) is a logical system for ∞-limit just intonation, based on the Functional Just system. One of the differences with the Functional Just System is that its radius of tolerance (RoT) is fixed to be half of the pythagorean comma. This creates a 53-form scale which is almost an exact copy of 53edo. The another difference is that it incorporates an entire system of nomenclature that is filled to the brim with puns, including the acronym of the system.

So, put on your pajamas (PJS) and sit back... because things will only get better from here.

Bases

Unlike standard FJS which has RoT = 65/63 and thus creates an almost 12-form scale, PJS requires a 53-form scale for it to shine. To the diatonic interval names P1, m2, M2, m3, M3, P4, pythagorean commas are added, up to twice up or down. That's MFJS (Mercator Functional Just System), the boring brother of PJS.

MFJS

In MFJS:

1. RoT = sqrt(531441/524288), so all the formal commas will change accordingly.
2. Pythagorean commas are part of the chain and stack up to twice (^, ^^, v, vv).
3. Perfect, Augmented and Diminished are deleted from the system.
4. The augmented fourth (A4) is now called "tritone" (T). The diminished fifth is now a down tritone. (vT)
5. Intervals beyond ±26 fifths will have Mercator's commas added or subtracted to their notation. The mercator comma is treated as a formal comma, and will be written ^3 _3 respectively.
6. The commas are spoken as cup(^), crup(^^), cown(v), crown(vv).
7. Synonyms exist here, for example: ^^M3 = vv4^3.

MFJS interval regions

Fifth shift region	Notation	Fifth shift	Region span	Formal comma
0	1	0	0.000c to 11.730c
1	^1	12	11.730c to 35.190c
2	^^1	24	35.190c to 55.035c
3	vm2	-17	55.035c to 78.495c
4	m2	-5	78.495c to 101.955c	67	16281/16384
5	^m2	7	101.955c to 125.415c	17	2176/2187
6	^^m2	19	125.415c to 148.875c
7	vvM2	-22	148.875c to 168.720c
8	vM2	-10	168.720c to 192.180c	71	4192479/4194304
9	M2	2	192.180c to 215.640c	9
10	^M2	14	215.640c to 239.100c	73	4784128/4782969
11	^^M2	26	239.100c to 258.945c	37	2542620639232/2541865828329
12	vm3	-15	258.945c to 282.405c
13	m3	-3	282.405c to 305.865c	19	513/512
14	^m3	9	305.865c to 329.325c
15	^^m3	21	329.325c to 349.170c
16	vvM3	-20	349.170c to 372.630c	79	275455967679/274877906944
17	vM3	-8	372.630c to 396.090c	5	32805/32768
18	M3	4	396.090c to 419.550c
19	^M3	16	419.550c to 443.010c	41	42991616/43046721
20	vv4	-25	443.010c to 462.855c	83	70324954583769/70368744177664
21	v4	-13	462.855c to 486.315c
22	4	-1	486.315c to 509.775c
23	^4	11	509.775c to 533.235c	43	176128/177147
24	^^4	23	533.235c to 553.080c	11	94489280512/94143178827
25	vvT	-18	553.080c to 576.540c	89	34480423521/34359738368
26	vT	-6	576.540c to 600.000c
27	T	6	600.000c to 623.460c
28	^T	18	623.460c to 646.920c	23	385875968/387420489
29	vv5	-23	646.920c to 666.765c	47	4424729404869/4398046511104
30	v5	-11	666.765c to 690.225c
31	5	1	690.225c to 713.685c
32	^5	13	713.685c to 737.145c
33	^^5	25	737.145c to 756.990c
34	vm6	-16	756.990c to 780.450c
35	m6	-4	780.450c to 803.910c
36	^m6	8	803.910c to 827.370c
37	^^m6	20	827.370c to 850.830c	13	3489660928/3486784401
38	vvM6	-21	850.830c to 870.675c
39	vM6	-9	870.675c to 894.135c	53	1043199/1048576
40	M6	3	894.135c to 917.595c
41	^M6	15	917.595c to 941.055c
42	vvm7	-26	941.055c to 960.900c
43	vm7	-14	960.900c to 984.360c	7	33480783/33554432
44	m7	-2	984.360c to 1007.820c
45	^m7	10	1007.820c to 1031.280c	29	59392/59049
46	^^m7	22	1031.280c to 1051.125c
47	vvM7	-19	1051.125c to 1074.585c	59	68573426553/68719476736
48	vM7	-7	1074.585c to 1098.045c
49	M7	5	1098.045c to 1121.505c	61	244/243
50	^M7	17	1121.505c to 1144.965c
51	vv8	-24	1144.965c to 1164.810c	31	8755315630911/8796093022208
52	v8	-12	1164.810c to 1188.270c
53	8	0	1188.270c to 1200.000c

I think 89-limit is enough. I myself wouldn't care beyond the 31- or 29- limit, but that's besides the point.

Punny names

PJS takes the base from MFJS, and puts in some spice to really drive her point home. By incorporating puns into the system, each interval becomes unique and recognizable, even if whimsically nonsensical. These are punny names. It even works in more languages! This is done by changing the first consonants of names, preserving rhymes but creating still recognizable interval and pitch names. It also subtly changes MFJS so that crown/cown/cup/crup and over/under are substituted with (case sensitive) L/z/p/s and h/k. These don't understand for anything; they aren't acronyms. Just consonants.

PJS names in English

Symbol	L/R (-2 commas)	D/Z (-1 comma)	–	P (+1 comma)	S (+2 commas)
Flat	rat	dat	flat	pat	sat
Natural	lu/loo (lateral)	du/due (datural)	natural	pu/pew (patural)	su/due (satural)
Sharp	larp	darp	sharp	parp	sarp
Minor	liner	diner	minor	piner	siner
Major	rager	dajor	major	pager	sager

Some notes:

• Using the -atural names for pitches is optional, so vvE can be called E lateral or E loo. Technically, the names should be "lat", "latural", "linor", "dinor", "pinor", "sinor", "lajor", "pajor", "sajor", but the system wouldn't be as punny without some respelling. Finally, some good feature of English's horrendous orthography.
• The choice of L/R, D/Z, P, S depends on the language, optimized for maximum punniness without sacrificing legibility.

When Mercator commas are added, they modify the name of the nominal in... interesting ways.

Mercator extra fifth shift	unison	second	third	fourth	tritone	fifth	sixth	seventh	octave
-159	u3gison	3necond	3nerd	3north	try3known	3nifth	3nixth	3neventh	on3tave
-106	u2gison	2necond	2nerd	2north	try2known	2nifth	2nixth	2neventh	on2tave
-53	ugison	necond	nerd	north	tryknown	nifth	nixth	neventh	ontave
+53	umison	mecond	myrrh'd	morth	trymoan	mifth	mixth	meventh	optave
+106	u2mison	2mecond	2myrrh'd	2morth	try2moan	2mifth	2mixth	2meventh	op2tave
+159	u3mison	3mecond	3myrrh'd	3morth	try3moan	3mifth	3mixth	3meventh	op3tave

Some notes:

The numbers are pronounced as they are inside the word, so "u2gison" is pronounced /juˈtuːgisən/. After all, writing it as utwogison just doesn't work. At least for me, it doesn't. So keep the numbers, it makes the system a bit more bizarre.

Feast your brains upon the marvel of PJS interval and pitch names.

Punny intervals and pitches

Simplified name	Punny interval name	Punny pitch name (from D)
1	unison	D natural
p1	punyson	D pew
s1	sunison	D sue
zm2	diner second	E dat
m2	minor second	E flat
pm2	piner second	E pat
sm2	siner second	E sat
LM2	rager second	E larp
zM2	dajor second	E darp
M2	major second	E natural
pM2	pager second	E pew
sM2	sager second	E sue
zm3	diner third	F due
m3	minor third	F natural
pm3	piner third	F pew
sm3	siner third	F sue
LM3	rager third	F larp
zM3	dajor third	F darp
M3	major third	F sharp
pM3	pager third	F parp
L4	lourth	G loo
z4	dourth	G due
4	fourth	G natural
p4	pourth	G pew
s4	sourth	G sue
LT	lye tone	G larp
zT	dry tone	G darp
T	tritone	G sharp
pT	pry tone	G parp
L5	lifth	A loo
z5	difth	A due
5	fifth	A natural
p5	pifth	A pew
s5	sifth	A sue
zm6	diner sixth	B dat
m6	minor sixth	B flat
pm6	piner sixth	B pat
sm6	siner sixth	B sat
LM6	rager sixth	B loo
zM6	dajor sixth	B due
M6	major sixth	B natural
pM6	pager sixth	B pew
Lm7	liner seventh	C loo
zm7	diner seventh	C due
m7	minor seventh	C natural
pm7	piner seventh	C pew
sm7	siner seventh	C sue
LM7	rager seventh	C larp
zM7	dajor seventh	C darp
M7	major seventh	C sharp
pM7	pager seventh	C parp
L8	loctave	D loo
z8	doctave	D due
8	octave	D natural

There's also "loonyson", "dunyson", "liner second", "liner third", "sager third", "sigh tone", "liner sixth", "sager sixth", "sager seventh", "poctave" and "soctave", but they are beyond the range or not octave reduced, so these are outsider regions.

Formal commas in English have their first consonant changed to h if it's otonal, and k if it's utonal. But what about other languages? Well, the consonants change the same, but to other ones. Feast your eyes.

PJS in other languages

Punny PJS names in German

	L/R	D/Z	–	P (pf)	S (sch)	+ Mercator	- Mercator	Otonal comma	Utonal comma
Flat	Les	Zes	-es	Pfes	Sches	be-	ne-	h-	k-
Natural	Lu	Zu	Natural	Pfu	Schuh
Sharp	Lis	Zis	-is	Pfis	Schis
Minor	Leine	Zeine	Kleine	Pfleine	Schleine
Major	Roß	Droß	Groß	Pfroß	Schroß
Unison	Rime	Drime	Prime	Pfrime	Schrime	Pribe	Prine
Second	Sekonde					Bekonde	Nekonde
Third	Terz					Berz	Nerz
Fourth	Larte	Zwarte	Quarte	Pfarte	Schwarte	Barte	Narte
Tritone	Ritonus	Zitonus	Tritonus	Pfritonus	Schritonus	Tritobus	Tritogus
Fifth	Linte	Zwinte	Quinte	Pfinte	Schwinte	Binte	Ninte
Sixth	Lixte	Zixte	Sixte	Pfixte	Schixte	Bixte	Nixte
Seventh	Leptime	Zeptime	Septime	Pfeptime	Scheptime	Beptime	Neptime
Octave	Loktave	Zoktave	Oktave	Pfoktave	Schoktave	Obtave	Ontave

Punny PJS names in Spanish

	L/R	D/Z	–	P	S (j)	+ Mercator	- Mercator	Otonal comma	Utonal comma
Flat	lemol	demol	bemol	pemol	jemol	me-	ñe-	b-	f-
Natural	lu (latural)	du (datural)	natural	pu (patural)	ju (satural)
Sharp	lostenido	dostenido	sostenido	postenido	jostenido
Minor	lenor	denor	menor	penor	jenor
Major	layor	dayor	mayor	payor	jayor
Unison	lunísono	dunísono	unísono	punísono	junísono	umísono	uñísono
Second	legunda	degunda	segunda	pegunda	jegunda	megunda	ñegunda
Third	lercera	dercera	tercera	percera	jercera	mercera	ñercera
Fourth	luarta	duarta	cuarta	puarta	juarta	muarta	ñuarta
Tritone	litrono	dritono	tritono	pritono	jitrono	tritomo	tritoño
Fifth	linta	dinta	quinta	pinta	jinta	minta	ñinta
Sixth	lexta	dexta	sexta	pexta	jexta	mexta	ñexta
Seventh	léptima	déptima	séptima	péptima	jéptima	méptima	ñéptima
Octave	loctava	doctava	octava	poctava	soctava	optaba	ontava

Italian/French are basically the same as Spanish, only that they use Z for dièse/diesis (zièse/ziesis), and S, not j. (sage-heure/saggiore). The ñ- is also kept, but spelt gn- as their orthography commands. Apart from that, In French there are more homophones!

Examples (Just)

Examples	PJS	Punny English	Punny German	Punny Spanish	Punny French	Punny Italian
5/4	zM3h5	hive dajor third	Hünf Droß Terz	tercera dayor binco	tierce dajeure binq	tierza daggiore binco
7/4	zm7h7	heaven diner seventh	Hieben Leine Septime	séptima denor biete	septième dîneur bête	settima dinore bette
11/8	s4h11	heleven sorth	Helf Schwarte	juarta bonce	sarte ronce	suarta bundici
13/8	sm6h13	hirteen siner sixth	Heizehn Schleine Sexte	sexta menor brece	sixte sineure reize	sesta sinore bredici
7/5	zTh5k7	kive heaven drytone	Künf Hieben Dritonus	dritono finco biete	driton finq bête	dritono finco bette
13/10	sM3_5^13	kive hirteen sager third	Künf Heizehn Schroß Terz	tercera jayor finco brece	tierce sage heure finq braise	terza saggiore finco bredici
	L4^3_5^13	kive hirteen lunorth	Künf Heizehn Wunarte	luñuarta finco brece	lugnarte finq braise	lugnuarta finco bredici
81/80	p1^5	hive punison	Hünf Punison	punísono brinco	puni son binq	punisono binco
29/16	pm7^29	hentynine piner seventh	Hiebenundzwanzig Pleine Septime	séptima penor beintibueve	septième pineure bingt-beuf	settima pinore bentibove
8192/8019	s1_3^11	heleven sugison	Helf Schrine	suñísono bonce	sugnison bonze	sugnisono bundici
140/99	T_3^5^7_11	hive heaven keleven try known	Hünf Hieben Kelf Trinogus	tritomo rinco biete fonce	tritome binq bête fonze	tritomo rinco bette fundici
Vavoom comma	z1^3*2^5*17	seventeenfold hive du2mison	Siebzehnfach Hünf Dri2be	du2mísono binco por diecisiete	du2mison binq fois dixsept	du2misono binco por diciasette

Be it said that the only languages I am truly fluent in are English and Spanish, knowing more or less some French. I'm open to suggestions so that the consonants chosen in this system don't produce that nonsensical of names.

Heavens, that's a quite system. I give it a 7/8.

PUS

It may not be used strictly for just intervals, and instead be used to name edos that decently approximate chains of fifths notation well enough, using the main L/p/d/s names for 41L 12s, and "mercator" accidentals which could stand in simply for fractions of pyth comma or edosteps. That way, we can avoid talking about hives and heavens. Such a system is called the horribly abreviated Punctional Unjust System (PUS).

Examples

Edo classes	Edo	5/4	7/4	11/8	13/8	19/16	29/16
Meantonoids (L/d/p/s alters by meantone dieses)	12	major third	minor seventh	tritone	minor sixth	minor third	minor seventh
	19		diner seventh	drytone	major sixth
	31			pourth	dajor sixth
	43			sourth			piner seventh
	50			pourth		minor nerd
Superpythoids (Alters by HUGE pyth commas)	22	major nerd	minor seventh	morth	minor mixth
	27				minor 2mixth
	34			pourth	piner sixth
	49	major 2nerd		mourth	minor 2mixth
	80		minor neventh	pourth	piner sixth
	87		minor 2neventh	pumorth
Schismoids (L/d/p/s alters by pythagorean commas)	41	dajor third	diner seventh	sourth	siner sixth	minor third
	53
	94
	130		diner neventh	stew mourth	siner mixth
	159			smourth	siner sixth
	217	dajor myrrh'd	diner seventh	sourth	siner nixth	minor myrrh'd	minor meventh
	270						minor 2meventh
	311

*PUS works worse with fifths whose pyth comma is beyond 20-28c. Note that for meantone and other systems where the pythagorean comma is negative, for it to make sense, the L/p/d/s consonants refer to meantone dieses, and fractions of those dieses. For superpyth, since the comma maps to something huge, n/m fractions must also be used. No mavila, blackwood, whitewood, or any weird fifth tuning with this system.