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:
- RoT = sqrt(531441/524288), so all the formal commas will change accordingly.
- Pythagorean commas are part of the chain and stack up to twice (^, ^^, v, vv).
- Perfect, Augmented and Diminished are deleted from the system.
- The augmented fourth (A4) is now called "tritone" (T). The diminished fifth is now a down tritone. (vT)
- 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.
- The commas are spoken as cup(^), crup(^^), cown(v), crown(vv).
- Synonyms exist here, for example: ^^M3 = vv4^3.
| 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.
| 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.
| 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
| 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 | ||
| 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.