The Helmholtz-Ellis JI Pitch Notation (HEJI) | 2020 | LEGEND revised by Marc Sabat and Thomas Nicholson | PLAINSOUND MUSIC EDITION | www.plainsound.org in collaboration with Wolfgang von Schweinitz, Catherine Lamb, and M.O. Abbott, building upon the original HEJI notation devised by Marc Sabat and Wolfgang von Schweinitz in the early 2000s PYTHAGOREAN JUST INTONATION | generated by multiplying / dividing an arbitrary reference frequency by PRIMES 2 and 3 only notate a series of perfect fifths above / below a reference — 3/2 ≈ ±702:0 cents (ca. 2c wider than tempered) ::: ::: E e n v V each new accidental therefore represents 7 fifths, altering by one apotome — 2187/2048 ≈ ±113:7 cents Frequency ratios including higher prime numbers (5–47) may be precisely written by adding the following distinct accidental symbols. Custom indications for higher primes or various enharmonic substitutions may be invented as needed by simply defining further symbols representing the relevant ratio alterations. PTOLEMAIC JUST INTONATION | PRIMES up to 5 includes the consonant just major third — 5/4 ≈ ±386:3 cents (ca. 14c narrower than tempered) D d m u U F f o w W alteration by one syntonic comma — 81/80 ≈ ±21:5 cents C c l t T G g p x X alteration by two syntonic commas — 81/80 · 81/80 ≈ ±43:0 cents ~u = e |f = v alteration by one schisma to notate an exact enharmonic substitution — 32805/32768 ≈ ±2:0 cents SEPTIMAL JI | PRIME 7 includes the consonant natural seventh — 7/4 ≈ ±968:8 cents (ca. 31c narrower than tempered) < > alteration by one septimal comma — 64/63 ≈ ±27:3 cents (Giuseppe Tartini) , . alteration by two septimal commas — 64/63 · 64/63 ≈ ±54:5 cents UNDECIMAL | PRIME 11 includes the undecimal semi-augmented fourth — 11/8 ≈ ±551:3 cents (ca. 51c wider than tempered) 5 4 alteration by one undecimal quartertone — 33/32 ≈ ±53:3 cents (Richard H. Stein) TRIDECIMAL | PRIME 13 includes the tridecimal neutral sixth — 13/8 ≈ ±840:5 cents (ca. 59c narrower than a tempered major sixth) 0 9 alteration by one tridecimal thirdtone — 27/26 ≈ ±65:3 cents (Gérard Grisey) HIGHER PRIMES 17 – 47 : ; alteration by one 17-limit schisma — 2187/2176 ≈ ±8:7 cents \ / alteration by one 19-limit schisma — 513/512 ≈ ±3:4 cents 6 3 alteration by one 23-limit comma — 736/729 ≈ ±16:5 cents (James Tenney / John Cage) 7 2 alteration by one 29-limit sixthtone — 261/256 ≈ ±33:5 cents 1 8 alteration by one 31-limit quartertone — 32/31 ≈ ±55:0 cents (Alinaghi Vaziri) à á alteration by one 37-limit quartertone — 37/36 ≈ ±47:4 cents (Ivan Wyschnegradsky) - + alteration by one 41-limit comma — 82/81 ≈ ±21:2 cents (Ben Johnston) è é alteration by one 43-limit comma — 129/128 ≈ ±13:5 cents ì í alteration by one 47-limit quartertone — 752/729 ≈ ±53:8 cents CENTS HEJI accidentals may be combined with an indication of their deviation in cents from equal temperament as read on a tuning meter; A 440 Hz is usually defined to be ±0 cents. ± − n If this deviation exceeds 50 cents, the nearest tempered pitch-class may be added: e.g. A0 ( 65 cents from An) could include the annotation Ae+35 placed above or below its accidental. TEMPERED NOTES | may be combined with cents deviations to notate free microtonal pitches ::: A a i j r z Z ::: indicate the respective equal tempered quartertone; show which pitch is assigned a deviation of 0c The Helmholtz-Ellis JI Pitch Notation (HEJI) | 2020 Harmonic / Subharmonic series 1–49 notated by modifications of Pythagorean notes with dedicated microtonal accidental symbols for primes 5 through 47 revised by Marc Sabat & Thomas Nicholson in collaboration with Wolfgang von Schweinitz, Catherine Lamb and M.O. Abbott Ratios represent the amount of modification of the Pythagorean notes by each additional symbol, building upon the original HEJI devised by Marc Sabat and Wolfgang von Schweinitz cents indications are deviations that would be shown on a tuning meter with A = 0 cents Standard otonal notation above A +51 n +4 +51 +28 +2 +2 +2 +2 +5 +4 n n u 4 u n < n :v n / u < 4 3v n n -14 -31 -14 n F+401v -3<1 -1u2 -2 -14 -29 pnartial 5 7 11 13 17 19 23 n n Ö Ö Ö Ö Ö Ö Ö interval M3 m7 P4 M6 aug8 m3 aug4 alteration (81:80) (64:63) (32:33) (27:26) (2187:2176) (512:513) (729:736) E+38 +51 +28 E-34 +2 +29 +12 +5 +4 +51 +6 +30 n , < 2 u 1 n 4 :v <m n á / 0v u +v < é 4 u 3v ív t 0v v -27 F+41 -31 -12 A +45 A -47 -45 -2 C+42 -14 -29 -10 e v 29Ö 31Ö 37Ö 41Ö 43Ö 47Ö m7 P8 M2 M3 P4 aug4 (256:261) (32:31) (36:37) (81:82) (128:129) (729:752) Standard utonal notation below nE +2 +2 +16 +2 G -39 n n +16 +33 +2 v +33 +14 +2 +4 +16 +31 n n o 5 n o n 9 >v o n ;e n n >v -2 -49 -3 -2 \v o > -459 -62e6 n 5 7 n 11 13 17 19 23 Ü Ü Ü Ü Ü Ü Ü M3 m7 P4 M6 aug8 m3 aug4 (80:81) (63:64) (33:32) (26:27) (2176:2187) (513:512) (736:729) +29 Gv-39 +33 +14 F-43 +2 +47 +4 Cv-41 +16 +31 +12 g 9 -4n >v 7v o -28 8 n -551 ;-e3 >o -2n -49à \v 9 o -2-7 > -1è0 -459 f 6-2e6 A+ì3e6 n 29 31 37 41 43 47 A.-3v6 Ü Ü Ü Ü Ü Ü m7 P8 M2 M3 P4 aug4 (261:256) (31:32) (37:36) (82:81) (129:128) (752:729) cc 2020 Plainsound Music Edition The Helmholtz-Ellis JI Pitch Notation (HEJI) | 2020 | LEGEND revised by Marc Sabat and Thomas Nicholson | PLAINSOUND MUSIC EDITION | www.plainsound.org in collaboration with Wolfgang von Schweinitz, Catherine Lamb, and M.O. Abbott, building upon the original HEJI notation devised by Marc Sabat and Wolfgang von Schweinitz in the early 2000s PYTHAGOREAN JUST INTONATION | generated by multiplying / dividing an arbitrary reference frequency by PRIMES 2 and 3 only notate a series of perfect fifths above / below a reference 3/2 ≈ ±702:0 cents (i.e. 2c wider than tempered) ::: ::: E e n v V each new accidental represents 7 fifths, altering by one apotome 2187/2048 ≈ ±113:7 cents Frequency ratios including higher prime numbers (5–47) may be notated by adding the following distinct accidental symbols. Custom indications for higher primes or various enharmonic substitutions may be invented as needed by simply defining further symbols representing the relevant ratio alterations. includes the consonant just major third PTOLEMAIC JUST INTONATION | PRIMES up to 5 5/4 ≈ ±386:3 cents (ca. 14c narrower than tempered) alteration by one syntonic comma D d m u U F f o w W 81/80 ≈ ±21:5 cents alteration by two syntonic commas C c l t T G g p x X 81/80 · 81/80 ≈ ±43:0 cents alteration by one schisma to notate an exact enharmonic substitution = = ~u e |f v 32805/32768 ≈ ±2:0 cents includes the consonant natural seventh SEPTIMAL JI | PRIME 7 7/4 ≈ ±968:8 cents (ca. 31c narrower than tempered) alteration by one septimal comma (Giuseppe Tartini) < > 64/63 ≈ ±27:3 cents alteration by two septimal commas , . 64/63 · 64/63 ≈ ±54:5 cents includes the undecimal semi-augmented fourth UNDECIMAL | PRIME 11 11/8 ≈ ±551:3 cents (ca. 51c wider than tempered) alteration by one undecimal quartertone (Richard H. Stein) 5 4 33/32 ≈ ±53:3 cents includes the tridecimal neutral sixth TRIDECIMAL | PRIME 13 13/8 ≈ ±840:5 cents (ca. 59c narrower than a tempered major sixth) alteration by one tridecimal thirdtone (Gérard Grisey) 0 9 27/26 ≈ ±65:3 cents PRIMES 17 THROUGH 47 alteration by one 17-limit schisma : ; 2187/2176 ≈ ±8:7 cents alteration by one 19-limit schisma \ / 513/512 ≈ ±3:4 cents alteration by one 23-limit comma (James Tenney / John Cage) 6 3 736/729 ≈ ±16:5 cents alteration by one 29-limit sixthtone 7 2 261/256 ≈ ±33:5 cents alteration by one 31-limit quartertone (Alinaghi Vaziri) 1 8 32/31 ≈ ±55:0 cents alteration by one 37-limit quartertone (Ivan Wyschnegradsky) à á 37/36 ≈ ±47:4 cents alteration by one 41-limit comma (Ben Johnston) - + 82/81 ≈ ±21:2 cents alteration by one 43-limit comma è é 129/128 ≈ ±13:5 cents alteration by one 47-limit quartertone ì í 752/729 ≈ ±53:8 cents CENTS HEJI accidentals may be combined with an indication of their deviation in cents from equal temperament as read on a tuning meter; A 440 Hz is usually defined to be ±0 cents. If this deviation exceeds ±50 cents, the nearest tempered pitch-class n − may be added: e.g. A0 ( 65 cents from An) could include the annotation Ae+35 placed above or below its accidental. TEMPERED NOTES | may be combined with cents deviations to notate free microtonal pitches indicate the respective equal tempered quartertone; ::: ::: A a i j r z Z show which pitch is assigned a deviation of 0c The Helmholtz-Ellis JI Pitch Notation (HEJI) | 2020 Harmonic / Subharmonic series 1–49 notated by modifications of Pythagorean notes with dedicated microtonal accidental symbols for primes 5 through 47 revised by Marc Sabat & Thomas Nicholson in collaboration with Wolfgang von Schweinitz, Catherine Lamb and M.O.