The Complete Formal FJS Description

This page is a de facto reference manual for using the FJS. It contains all the rules you will need to use the FJS, including its shorthand elements. However it will not be of much use to you unless you have read through the Crash Course first.

Everything concerning experimental elements is marked with the symbol (*).

The Radius of Tolerance

The radius of tolerance is a constant, by definition equal to 65/63.

Reduced Forms

The regular reduced form is $\text{red}(d)=d \cdot 2^{-\lfloor\log_2 d\rfloor}$

The balanced reduced form is $\text{reb}(d)=\frac{1}{\sqrt{2}} \text{red}(\sqrt{2} \cdot \text{red}(d))$

The FJS Master Algorithm

1. Input the desired prime interval and call it p.
2. Let k = 0.
3. Consider the interval of k Pythagorean fifths and call it P.
4. Is the difference between p and P less than the radius of tolerance?
5. If so: k is the fifth shift. Output. End.
6. If not: move to the next k in the sequence: (0, 1, −1, 2, −2, 3, −3, …) and repeat from step 3.

In step 4, the meaning of “difference” is the absolute value of the cent size of $\text{reb}\left( \frac{p}{3^k} \right)$

The Formal Comma

The formal comma of a prime p with a fifth shift g is given by:

$$\text{comma} (p)=\text{reb}\left( \frac{p}{3^g} \right)$$

Inverse procedure – finding the fifth shift knowing the formal comma without using the master algorithm or the radius of tolerance: The fifth shift is the exponent of 3 in the factorization of the reciprocal of the formal comma.

Converting JI Ratios into FJS Representations

1. Factorize the ratio.

2. For every prime p greater than 3 with an exponent of α_p, remember the α_p for every p and multiply the initial ratio by

   $$(\text{comma} (p))^{-\alpha _p}$$
3. The result will be Pythagorean. Transform it into FJS form.
4. Now add the α_p as accidentals – if positive, then otonal; if negative, then utonal.
5. Take the reduced form and adjust octaves as required.

Converting FJS Representations into JI Ratios

1. Convert the Pythagorean part of the FJS interval into a Pythagorean ratio.
2. For every otonal accidental p, multiply by the formal comma of p. If utonal, then divide.
3. Take the reduced form and adjust octaves as required.

FJS Interval Arithmetic

• The sum of the FJS representations of two JI ratios is the FJS representation of the product of these ratios.
• The difference of the FJS representations of two JI ratios is the FJS representation of the quotient of these ratios.
• The inversion of the FJS representation of a JI ratio is the FJS representation of the reciprocal of the ratio.
• To add two FJS intervals, add their Pythagorean parts (conventional interval classes) and merge the accidentals.
• To subtract two FJS intervals, subtract their Pythagorean parts, then swap the otonal with the utonal accidental for the second interval, then merge. Subtracting an FJS interval is the same as adding its inverse.
• To invert an FJS interval, invert its Pythagorean part and swap the otonal with the utonal accidental.
• You can also add an FJS interval to an FJS note, or subtract an FJS note from another FJS note.

Rules for Generating FJS Interval and Note Names

1. By default, the FJS uses Pythagorean tuning for traditional staff notation. So the interval C–E is 81/64, the interval C–D♭ is 256/243, the interval D♭–C♯ is 531441/524288, etc.

2. A single comma of the prime p (greater than 3) will use the appended number “p” (preferably as a superscript) to represent a deviation by that comma in its original direction. E.g. E⁵ is an interval of 81/80 below E (because the comma of 5 is 80/81); F¹¹ is an interval of 33/32 above F (because the comma of 11 is 33/32). These note names are pronounced with the number appended: “E-five”, “F-eleven”. I do not expect them to cause confusion with chord symbols, since it is often clear from context whether a single note or a chord is meant. In the case of confusion, an optional “super” prefix can be attached. Note that the original direction is preserved.

3. A single comma of the prime p will use the appended number “p” as a subscript to represent a deviation by that comma in the opposite of its original direction. E.g. E♭₅ is an interval of 81/80 above E♭; A₁₉ is an interval of 513/512 below A. These are pronounced with the number appended, but with a “sub” prefix attached: “E-flat-sub-five”, “A-sub-nineteen”.

4. To represent combinations of movements by commas in the same orientation (either all otonal or all utonal), multiply the numbers that the original commas would contain (rather than add or concatenate them). This is because these numbers are primes. A sum of primes cannot be uniquely resolved into its summands, and neither can a contenation of primes, but due to the Fundamental Theorem of Arithmetic, a product of primes can. Therefore, from the final product, we can always uniquely deduce which primes were multiplied. Order does not matter. E.g. G♯²⁵ is two 81/80 intervals below G♯ and there is no other interpretation, since 25 = 5 × 5 and no other prime decomposition exists.

5. Usually the product allows the deduction of the prime factorization immediately at first sight. Some of the most frequent products are all recognizable: 25, 35, 49, 125, 625, 343, 55, 77, 121 are all familiar. If it is completely unclear at first sight what a number factorizes to, e.g. 119 = 7 × 17, an alternative notation can be used where all factors are listed, separated with commas (punctuation marks). The example from above could alternatively be written as G♯^5,5 and both notations are equally correct. If commas are listed out individually, it is preferred to list them in non-descending order, so D¹¹⁹ is D^7,17 rather than D^17,7. These extremes can also be combined in any way, and individual factors may be multiplied in any combination, e.g. G^11,125 is a perfectly valid representation of G¹³⁷⁵. These are pronounced as normal: “G-eleven-one-hundred-twenty-five”.

6. Products of positive and negative commas are independent. They are written as a combined subscript and superscript, e.g. F⁷₅ represents F lowered by 64/63 and then raised by 81/80 (or in the opposite order; first raised by 81/80 and then lowered by 64/63; the result is the same). Note that the products used to represent positive and negative commas will always be coprime, otherwise there is an unnecessary shift in one direction that is canceled out in the other one; *C³⁵₅ is the same note as C⁷.

7. The number 1 represents canceling any microtonal accidentals, similar to the natural sign used to cancel standard accidentals. It can be written either in a superscript or a subscript. For example, the sequence C – D – E⁵ – E¹ (or simply C – D – E⁵ – E) represents the intervals 1/1, 9/8, 5/4, 81/64 above C.

8. Intervals may have the same modifiers as notes.

Rules for Using FJS Note Names in Notation

1. To add positive comma shifts to a note, write the corresponding number that results from combining these shifts in front of it, as an accidental. For consistency, just like standard accidentals are prepended to a note on a staff but appended to its name, the same happens to microtonal accidentals. They are written in front of the note, and if there is a standard accidental as well, then in front of the accidental.

2. To add negative comma shifts to a note, write the corresponding number that results from combining these shifts in front of it, but prepended with a stroke (e.g. “/35”). Note that e.g. “7/5” is not an alteration by a 7/5 interval, but by 63/64 compounded with 81/80, which is 5103/5120.

3. If both positive and negative comma shifts appear, then the total positive shift should be written to the left of the total negative shift.

4. Pythagorean accidentals divide notes with the same diatonic pitch class (C, D, E, F, G, A, or B) into their Pythagorean pitch classes. Hence, a Pythagorean accidental applies to all notes with the same diatonic pitch class and octave in the same bar, until overwritten or canceled. Analogically, microtonal accidentals divide notes with the same Pythagorean pitch class into their absolute JI pitches. Hence, a microtonal accidental applies to all notes with the same Pythagorean pitch class and octave in the same bar, until overwritten or canceled. This means that, if the first E in a bar is written with a +5 accidental, then all other E in that bar in the same octave will also have the +5 accidental implied (until it is canceled with a 1), however, an E♭, just as much as a D or an F, will not have that accidental implied, even if it is in the same bar.

5. A “1” accidental cancels any microtonal accidentals for the Pythagorean pitch class it is applied to.

Rules for Using FJS Key Signatures (*)

1. FJS key signatures work exactly like Pythagorean key signatures, are independent of them, and are written to the right of Pythagorean key signatures if they exist in the same score.

2. By default, an FJS key signature can only apply microtonal accidentals to those pitches which are diatonic in the key that a Pythagorean key signature has already supplied. For example, if a piece is in A major, and has the Pythagorean key signature ♯FCG, then an FJS key signature can usually only apply microtonal accidentals to the notes A, B, C♯, D, E, F♯, and G♯. Because FJS key signatures are nonstandard, there is no particular order in which these accidentals must appear. However, due to the Pythagorean nature of the FJS, it is found that these key signatures will often exhibit Pythagorean patterns, and so they will appear as fragments of the standard FCGDAEB or BEADGCF order.

3. An FJS key signature can supply microtonal accidentals to non-diatonic pitches by selecting an appropriate space or line on the staff for the diatonic pitch class, writing the FJS accidental, and then appending a Pythagorean accidental in parentheses to indicate the Pythagorean pitch class. The parentheses are required to indicate that the Pythagorean accidental itself is not part of the key signature, instead the meaning is as follows: if the Pythagorean accidental is used, then the microtonal accidental that follows will also be applied. For instance, an FJS key signature for C harmonic major could be B⁵ E⁵ A⁵ A₅(♭), indicating that all B, E, and A should be altered by a +5 accidental, and all A♭ should be altered by a −5 accidental. In this case, A is not flat by default; but if it is, then it is also altered by −5.

4. As with standard key signatures, FJS key signatures do not “mix” with proper FJS accidentals. Rather, standalone FJS accidentals overwrite those that would be due to an FJS key signature. An E₇ indicated by writing E with a −7 accidental before it is still E₇, even if there is an FJS key signature that says E⁵ instead. (It does not, for instance, become E⁵₇.)

5. A Pythagorean key signature change does not automatically also trigger an FJS key signature change. A piece which begins with the key signatures ♯FCG and F⁵ C⁵ G⁵ will not have its FJS key signature canceled when the piece changes key to A♭ major (♭BEAD). If no FJS key signature change has happened, then it is still assumed that in this key of A♭ major, if an F♯ appears, then alter it by +5. This may seem strange, since an F♯ is never expected to appear in A♭ major. However, it actually makes lots of sense because it helps avoid the need to restate the FJS key signature every time that a simple modulation occurs (e.g. from ♯FCG to ♯FC). In such cases, it may very probably be desired that the F♯ and C♯ (and G♯, which may appear in D major as part of a II chord) remain altered by +5.

6. An FJS key signature change will cancel all alterations due to the previous FJS key signature. If the Pythagorean key signature has changed as well, which it always does before the FJS key signature, then of course a new key signature is always defined with respect to the new set of seven diatonic pitches. For example, ♯FCG F⁵ C⁵ G⁵ is quintal A major. Changing the Pythagorean key signature to ♭BEAD causes ♭BEAD F⁵(♯) C⁵(♯) G⁵(♯). However, if the FJS key signature (written as F⁵ C⁵ G⁵) is restated immediately after the Pythagorean switch to ♭BEAD, i.e. if we have the combination ♭BEAD F⁵ C⁵ G⁵, this has a different effect because the F, C, and G that are diatonic to A♭ major are now different: F♮, C♮, G♮ instead of F♯, C♯, G♯.

Rules for Using FJS Transposition Lines (*)

1. There exist FJS transposition lines, which work analogically to octave lines, but apply an FJS accidental instead of shifting by an octave.

2. An FJS transposition line can be written either as alta or bassa, with no difference in meaning.

3. An FJS transposition line begins with a compound FJS accidental, which appears just as it would if it was written on a staff. For example, a transposition line of +5 will begin with “5”, a transposition line of +7−5 will begin with “7/5”, etc. As with octave lines, “loco” following such a line is optional.

4. All notes under the transposition line have its FJS pitch shift added to its pitch shifts as applied by FJS accidentals and key signatures.

The Pythagorean Comma (*)

This notation element is nonstandard and it cannot be used within the calculator.

The Pythagorean comma can be allocated to the factor of 3. Hence e.g. 15/14 (a type of chromatic semitone) can be changed to a diatonic semitone and hence can be used to conveniently write a trill. +3 raises by a Pythagorean comma (adds twelve factors of 3, so otonal) and −3 lowers. So B♯ is the same as C³, and D♭♭₇ is the same as C₂₁.

Using the FJS for Notation Beyond JI (*)

Any number, not just primes, can be the input of the FJS master algorithm. In fact, although the FJS was not constructed for this purpose, the entire algorithmic procedure can be carried out for many irrational numbers. Any positive real number works, but a few (in particular, integer roots of rational numbers) will obliterate the FJS’s pristine and fragile bijection, leaving a mere surjection behind (unambiguous reading, but no longer unambiguous writing).

This means the FJS can be used to notate e.g. the 833 cents scale, a scale based on φ (the golden ratio). In the FJS, φ/1 is a m6^φ.

Moreover, expanding the FJS to notate powers of φ in addition to the rational numbers preserves all its amazing properties.

Among others, the following famous irrational constants work, too:

• π; m6^π (τ, the “true circle constant”, has the same representation, since it’s just 2π, so an octave of π)
• e; P4^e
• γ; M2^γ (Euler-Mascheroni constant)

All transcendental numbers work.

Some of these make quite a lot of sense musically. φ is the interval obtained when looking for intervals whose summation and difference tones are equidistant (which is a direct consequence of the appearance of φ in the Fibonacci sequence). You may also have heard that φ is the most irrational number, which makes music involving it the polar opposite of just intonation in a way; this is one way where the golden ratio is not the standard of absolute beauty, but its exact opposite! e, due to its most famous limit definition, shows up as the limit in the following sequence: octave, two 3/2 fifths, three 4/3 fourths, four 5/4 major thirds, five 6/5 minor thirds, et caetera. (Completely unrelated are the uses of these constants as logarithmic ratios, e.g. in golden meantone and in Lucy Tuning.)

There has been no attempt so far to represent negative or complex numbers in the FJS. What could that even mean?

More Experimental Features (*)

From this point onwards, the FJS is open source! Make new shorthand features if you feel they are needed. You can make entire FJS dialects with nonstandard radii of tolerance (the calculator has a function for that), having all positive accidentals be upward and all negatives be downward, or even using a nonstandard master algorithm (e.g. instead of finding the simplest Pythagorean approximation within a radius of tolerance, maybe finding the nearest approximation within a Pythagorean simplicity of tolerance).

I’m not a stubborn original creator who won’t let anyone modify their work if the changes benefit the work. The FJS used to have so many bad features, some of which have gone through last-minute changes or removals. If you think you’ve made a nice new feature, share it with me by e-mail and I’ll be happy to add it here.

Examples

The following two notes are G♭⁷₅ and F♯⁵₇ respectively.

The following four bars read: C♯⁵ C♯⁵; C♯⁵ C♯; C♯⁵ C; C♯⁵ C⁵.

The following is the just major scale in A; A, B, C♯⁵, D, E, F♯⁵, G♯⁵, A.

The following is the harmonic series fragment 8–16 in F; F, G, A⁵, B♭¹¹, C, D♭¹³, E♭⁷, E⁵, F. (The key signature is not practical, but possible.)

The following chord progression contains two 10:12:14:17 diminished seventh chords and two 4:5:6 major triads, on A, D, G, and C, in that order.

This chord progression demonstrates the usefulness of the rule that a Pythagorean key change doesn’t reset an FJS key. It reads: A major, E major, A major, A dominant 7^th, D major, G major, D major/A, E major/A, A dominant 7^th, D major; where all major triads are 4:5:6 and all dominant seventh chords are 4:5:6:7. The FJS key stays the same after the modulation from A to D, which is useful because the G♯ in D is expected to be +5.

The following example uses the eight notes C, D, E⁵, F, G, A⁵, A, and B⁵ without a key signature.

The following example denotes the constants φ, π (or τ), e, and γ, in the key of A.