SFINX - Xenharmonic Guitar Learning App

SFINX 
Stringed Fretted Instruments Notes Explorer

SFINX - (Stringed Fretted Instruments Notes Explorer)

NEW VERSION IS COMING!

1. Introduction

This free web-based app is designed to help you explore and experiment with stringed fretted instruments from traditional to alternative tunings and xenharmonic (micro/macro-tonal) guitars.

Key Features:

• Generate scales and chord diagrams for custom instruments and tunings
• Play and find chord progressions on an interactive virtual instrument/guitar
• Utilize a built-in tuner and fretboard calculator

Basic Concepts:

• Alternative Tuning: SFINX focuses on exploring alternative tunings that go beyond the usual 12-tone equal temperament. It allows users to design custom tunings using any possible set definition.
• Virtual Fretboard: The app provides an interactive virtual instrument/fretboard interface to visualize and experiment with different tunings, scales, and chords.
• Preset Management: Users can save and load instrument and scale presets for quick access and experimentation.

 1.1 General App Workflow 🎸

SFINX accepts any set of pitch values as input. These can be specified in multiple formats:

• Frequency ratios (e.g., 2:3)
• Fractions (e.g., 3/2)
• Decimal representations (e.g., 1.5)
• Cents (e.g., 701¢)
• Milioctaves (e.g sqrt(2) is at 500 mocts)
• Or any combination of the above.

1.1.1 Validation and Range Adjustment

• The app first validates the set by attempting to map it into the audible range (20–20 000 Hz), even if custom min/max ranges are set.
• Before defining the guitar’s physical parameters, you can assign any value from the set as the unison at a specific frequency, on a specific string and fret.
• From this reference point, the tuning system is mapped across the fretboard, taking into account:
• Number of strings and frets
• Tuning relationships between strings

1.1.2. Built‑In Tuning Generators

SFINX includes generators for:

• Equal divisions (EDO/EDx)
• Group tunings
• Random tunings for exploration

1.1.3. Default Guitar Setup

By default, SFINX builds:

• A 6‑string guitar
• A fret count that does not exceed two octaves
• A string‑to‑string tuning pattern that mimics a common gauge setup

These parameters are fully customizable, making the app equally useful for:

• Standard guitars
• Non‑conventional instruments (e.g., left‑handed 10‑string guitars, 24‑fret extended‑range setups, drop tunings)
• Microtonal and experimental designs

Finding accurate scales and chord diagrams for such setups is notoriously difficult, SFINX automates it.

1.1.4. Analytical Class System

Analytical classes in SFINX operate independently of both:

• The tuning definition
• The guitar’s physical parameters
• Any class system you create can be transported to another instrument.
• By default, the app includes:
• 12‑EDO classes (C, C♯, D, …, B)
• Common scales saved by name (e.g., Diatonic, Pentatonic)
• Classes are defined numerically and can be:
• Named (e.g., “C”, “D♯”)
• Left as indexed numbers by default

Key Advantage

This separation of tuning, instrument layout, and analytical classes means you can:

• Apply the same class system to multiple tunings
• Experiment with alternative analytical equivalences
• Work with systems that aren’t even EDO, including harmonic series subsets or irrational divisions.

2. User Interface & Controls

2.1 Main Layout Overview: Help(outdated)

Default presets include normal 12ed2 guitar and bass.
SFINX saves presets per browser, allowing you to import and export settings. Overwriting existing presets is possible, but remember to save the instrument preset after creating or modifying scales.

2.2 Instrument Setup

• Strings and Frets: Define the physical parameters of your virtual instrument. Number of strings and maximum frets per string.
• Tuning Pattern: Determine the open string note of each string relative to the previous one. This allows for various tuning configurations, including drop tunings.
  Each string you add, also adds a control on its left that sets the open string note in relation with its previous string (that is why the lowest string doesn't have one).
  The standard guitar tuning pattern is 5, 5, 5, 4, 5.
  A one step drop tuning pattern is: 7, 5, 5, 4, 5.
• Min/Max Frequency: If no reference fret is used, the lowest or highest available pitch in Hertz can be defined from this point, and the rest of the guitar is mapped accordingly. If the guitar's parameters don't match, the range is clipped to the audible spectrum, and an error is displayed for exceeding frets or strings.
• Reference Fret Mapping: Specify the exact fret and string along with the desired frequency. For instance, the open string (0 fret) of the 5th string can be set to 220 Hz.

2.3 Visualization Tools

• Interval Rulers: Measure intervals and string lengths for in-depth analysis.
• The pink ruler marks always the "equave".(the last number in the interval list, or the interval used for the division in equal-division systems, outdated)
• The fixed green 12ed2 ruler is useful to measure other tunings, since most musicians are familiar with the role of those tempered 12, is easy to relate new tunings using the ruler for quick comparison. (it can be moved)
• Calculator ruler: Displays a ruler (orange) with the relative size of the string when hovering a fret.
• Size: Sets the nut to bridge length for calculations. In units.

For deeper interval analysis and measurement beyond the basic tools provided, explore my other app, 'Interval Matrix.'. Unlike SFINX, which focuses on a fixed interval matrix inherent to the guitar, this app offers a more flexible and comprehensive approach to interval exploration. (See Fretboard-Matrix Equivalence)

• Highlights: Select specific notes on the fretboard to focus on particular scales or chords. The number of notes is determined by the classes.
  Here is where you set the scale to be displayed on the fretboard.
  Each marked note, adds a control for arpeggiate a chord.
• Chord Pattern: Define the notes for each chord position.
  Each string added, adds a control for selecting a note for the chord-row.
• Colors: Customize the appearance of notes based on:
• Single: Plain color.
• Class: Notes get a unique color by its class.
• Row: Colorize by chord row, controlled by the chord pattern.
• Chroma: Notes get a color by their musical chroma(see section 3), this uses a custom sRGB hue wheel, and the root color is red.

2.4 Playback & Sound

• Arpeggiator: Play notes sequentially in different patterns (up, down, converge, diverge).
• Speed: Arpeggio note separation speed in seconds.
• Synth Sound: Choose from available sounds for the virtual guitar.
• Note Length: Control the duration of played notes.
• Volume: Adjust the overall volume of the virtual guitar.

Tuning System(outdated)

Decimal Ratio List: The program accepts intervals in decimal ratio, comma-separated. The last ratio represents the octave equivalent.
Equal Division: Divide the octave into a specified number of equal intervals.

2.5 Display Adjustments

• Width, Length, Zoom: Adjust the visual representation of the fretboard without affecting the tuning or sound.

3. Core Concepts: Basics of Music and Tuning

(the "minimum universal theory")

To understand the complexity and musical possibilities of any system, SFINX focuses on a small set of universal principles. These are the foundations from which any tuning, traditional or experimental, can be explored.

3.1 Tuning Systems – App Philosophy

Just as music resists a single absolute definition, so too do tuning systems. Across music theory, numerous philosophical schools offer different views on what a tuning system is and how it should be analyzed. SFINX is built on a generalized framework designed to accommodate them all. Here, the core axiom is simple: a tuning system is any non‑empty set of pitches. How that set is analyzed, whether by octave, chroma, geometric repetition, or custom equivalence, is personal and flexible.

Understanding Properties of sound perception vs Music Manipulation

The main focus is on distinguishing the types of equivalences that are often confused in music (perceptual, analytical, geometrical).

3.2 Octave

In music, an octave is the interval between a reference pitch \( P \) and another pitch with twice its frequency, corresponding to the second harmonic. More generally, octaves from a reference pitch \( P \) are defined by all frequencies of the form \( P \times 2^n \), where \( n \) is an integer. Related intervals include the double octave (fourth harmonic) and the sub-octave (second subharmonic), among others.

Octaves are perceptually identified as similar, as exemplified by all C notes on a piano. This perceptual equivalence is rooted in the harmonic structure of timbre: in sound sources with harmonic spectra (where overtones align closely with the harmonic series), multiplying or dividing frequency by 2 results in spectral redundancy. No new overtone content is introduced, and our auditory system, evolved in a world saturated with such sounds, detects this redundancy as similarity.

Crucially, the octave does more than suggest similarity; it introduces a cycle. This perceptual equivalence transforms the linear, undifferentiated pitch continuum (spanning roughly 20 Hz to 10-20 kHz, with individual resolution limits around 5-25 cents) into cyclical chromas. Within this framework, pitch difference becomes inferential and relational, allowing tones separated by octaves to be treated as functionally equivalent.

This octave equivalence allows for the transposition of chords and voicings while preserving their musical function. It facilitates transcription across ranges, such as adapting piano music (with ~80 notes) to guitar (with fewer than 30), and supports the substitution of notes with their octave duplicates without significant loss of tonal meaning or structural role.

For a deeper exploration of the octave’s ontological status, see Tonal Constancy.

3.3 Chroma

Chroma denotes the relative position of a pitch within a perceptual cycle. For harmonic timbres, this cycle is typically the octave, meaning the chroma of a pitch remains consistent across its octave equivalents. For example, the 3rd, 6th, 12th, 24th, and 48th harmonics all share the same chroma in octave space; these correspond to the interval of a fifth.

Mathematically, this can be understood by observing that the base-2 logarithms of these harmonic numbers share the same fractional part. In musical contexts, especially within just intonation, logarithmic representations are often avoided in favor of more direct ratio-based reasoning.

A formal expression of chroma is:
\[
\text{chroma}(x) = 2^{\log_2(x) \bmod 1}
\]
Alternatively, using a multiplicative modulo operation:

\[
\Xi(x) = x \bmod 1{:}2
\]
This formulation highlights that chroma is invariant under space scaling, multiplying or dividing a pitch by powers of, in this case, 2 (i.e., \(2^n\), where \(n \in \mathbb{Z}\)) does not alter its chroma. For instance, relative to a reference pitch of 1:1, the chroma of 3, 6, 12, 24, etc. is 1.5, corresponding to the frequency ratio 2:3, a fifth.

Example: Harmonic Instruments

On traditional harmonic instruments like the guitar or piano, a frequency multiple of 5 from the fundamental, i.e., the 5th harmonic (and its octave multiples: 10th, 20th, etc.), corresponds to a major third. In octave space, this maps to the ratio 4:5, meaning its chroma is that of a major third. These harmonics align with the base-2 logarithmic structure of pitch perception, even if the tuning system itself doesn’t explicitly include octaves.

Non-Octave Systems and Timbre Manipulation

Even in systems that omit octaves, such as those based on irrational cycles like π (e.g., a 1:3.14 repetition) our auditory system still tends to perceive pitch relationships through a base-2 logarithmic lens, due to biological conditioning and harmonic familiarity.

However, if the timbre is manipulated such that the perceptual equivalence is based on a different cycle, say, 1:π then the chroma of a pitch becomes relative to that new cycle. In such a system, a frequency multiple of 5 (and all values of \(5 \times 3.14 \times n\)) might no longer be perceived as a major third. Instead, its chroma would be defined by:

\[
\text{chroma}_\pi(x) = \pi^{\log_\pi(x) \bmod 1}
\]
This represents a base-π chroma, and reflects how timbre and spectral structure can reshape the perceptual geometry of pitch space.

Key Insight

Chroma is not an intrinsic property of an isolated pitch. It emerges from the relationship between a pitch and a reference point, shaped by both spectral content and perceptual context. A pitch \( P \) alone does not possess chroma; only when considered in relation to a reference pitch \( A \), do \( P \) and all its perceptual equivalents (e.g., \( P \times 2^n \) or \( P \times \pi^n \)) share a common chroma.

Note: Timbre manipulation and psychoacoustic thresholds continue to challenge and expand our understanding of chroma. As sound synthesis evolves, even the logarithmic perception of pitch may take on new forms.

Next: separating two different invariance concepts that often get conflated because of how Xedo “cheats” by aligning them.

3.4 Geometrical Repetition

(structural / physical invariance, if any)

Definition: 

The smallest positive interval \(R_g\) such that the set is invariant under translation by \(R_g\) in its raw coordinate space (cents, ratios, etc.).
Key point: This is about the actual step size that tiles the set without changing its internal relationships.

Example:

• In 12edo, \(R_g = 100\) cents, because every pitch is an integer multiple of 100 cents and transposing by 100 cents preserves all interval relationships.
• In Pythagorean 12-tone, \(R_g = 1200\) cents (the octave), because the steps are not equal , transposing a triad by a single step changes the exact interval sizes, so only a full octave preserves the internal structure.

Details in appendix.

3.5 Pitch Class

In modern music theory, a pitch class (or simply class) is an abstract concept analogous to an equivalence class in algebra. It denotes a set of pitches related by a specific, defined equivalence relation. This relation is not inherently perceptual and can be arbitrarily defined according to the needs of a particular musical system or analytical framework.

In octave-based systems such as 12‑tone equal temperament (12‑TET), the most common equivalence relation is octave equivalence, which causes pitch class and chroma to coincide. However, this is only one possible choice, other equivalence relations can be imposed for analytical or compositional purposes.

A pitch class functions as an assigned label or identifier for a pitch within a defined system. This assignment is absolute for that pitch in the given framework, but it does not inherently:

• Convey the pitch’s intervallic relationship to other pitches
• Indicate its harmonic or tonal function

In short, the pitch class names the pitch in the context of the chosen equivalence, but the relationships between classes, and their musical meaning, are determined separately.

3.6 Analytical Equivalence

3.6.1 Definition:

An analytical equivalence is established by choosing a fixed number of pitch classes and defining a modular space within which musical elements, such as scales, chords, and progressions, are analyzed and manipulated.
The number of classes often corresponds to a musically significant interval that serves as the boundary of the analytical cycle, but this is not required. In algebraic terms, these are equivalence classes: once defined, each pitch is assigned to a class, often given symbolic names (e.g., Do, Re, Mi or letter names with sharps/flats).

3.6.2 Key Characteristics

• Independent of geometry or perception: Analytical equivalence does not depend on the tuning’s inherent geometrical structure or repetition or perceptual chroma.
• Flexible perspective: You can impose any class system that is useful for analysis, composition, or experimentation.
• Naming system: Once the equivalence is chosen, pitches are grouped and labeled according to that system, regardless of their actual frequency ratios.

3.6.3 Why It’s Useful

Analytical equivalence allows you to assign a meaningful or informative class system to tuning systems that might otherwise resist simple description, such as irrational divisions of irrational intervals.

Example: 11.5 equal divisions of π:

• Question: How many notes does it have?
• Answer: That depends on the analytical equivalence you choose. You could define it to have 10, 15, or any other number of classes for convenience.

3.6.4 Case Study: Equalized Bohlen–Pierce

Often labeled 13ed3:

• “13” here is not the number of classes, it’s a title indicating the step size (≈146 cents) and its connection to the third harmonic (3/1 tritave).
• This label is informative about origin and step size, but not directly about how it maps to harmonic instruments like piano or guitar.
• This system has a near-octave of ≈1180 cents:
• Melodically, it can pass as an octave.
• Harmonically, it often sounds “out of tune” relative to octave-based harmony.
• For chord-building intuition, it can be analytically treated as having 8 classes, since it behaves locally like 8-EDO.

3.6.5 Beyond EDO

Analytical equivalence works for:

• Non-EDO systems (e.g., harmonic series subsets)
• Experimental tunings with no obvious periodicity
• Custom cycles chosen for compositional effect

3.6.6 Extra Example

In a 100-cent step scale (true geometrical repetition = 100 cents), you could define only 2 analytical classes:

• Class A
• Class B

Playing only the A’s yields the whole-tone scale, but with the class system acting as a shortcut.

3.6.7 Core Insight

Analytical equivalence is a lens, not a property. It’s the act of deciding what counts as “the same” for the purposes of analysis, naming, and manipulation, free from the constraints of the tuning’s physical structure or perceptual cycles.

3.7 Summary

3.7.1. Octave (or chosen perceptual cycle)

• The perceptual repetition unit , often 2:1, but could be any, π‑periodic, its always timbre‑based. (subjective thresholds).
• Turns the infinite pitch continuum into a cyclic space.

3.7.2. Chroma

• The position within that cycle , the “color” of a pitch, invariant under the cycle’s repetition.
• In octave space, all C’s share the same chroma; in other cycles, the mapping changes.

3.7.3. Geometrical Repetition (if any)

• The structural repetition of the tuning itself, the smallest shift that leaves the set’s raw geometry unchanged.
• In 12‑EDO: 100 ¢; in Pythagorean 12‑tone: 1200 ¢; in other tunings: could be none.

3.7.4. Analytical Equivalence (Classes)

• The imposed modular space for analysis — how you decide “these pitches count as the same” for naming, chord building, and scale logic.
• Independent of the tuning’s geometry or perceptual cycles.
• Lets you treat wildly different tunings with the same symbolic framework.

With just those four, a user can:

• Load any pitch set into the app.
• Map it to any instrument geometry.
• Apply any analytical lens (octave‑based, tritave‑based, custom modulus).
• Start building scales, chords, and progressions without needing to “speak” a particular school’s theory.

Everything else, cadences, modes, voice‑leading, genre styles, is layered on top of these fundamentals.
it’s theory‑agnostic but still complete enough to navigate the app and make music right away.

3.8 Extra

Analytical equivalence is essentially naming and organizing abstract ratios under a chosen “modulus”, whether that’s the octave, 900 cents, a tritave, or something else entirely. It’s the act of saying “these two pitches are the same for the purposes of my analysis” even if, physically, they’re not the same frequency ratio.

Jazz example:

• You impose a 900 cent cycle instead of the octave.
• You treat pitches separated by 900 cents as the same “class”, just like calling them the same letter name in a custom alphabet.
• This lets you build chord progressions and voice-leading patterns that have their own internal logic, independent of the usual octave-based chroma.

Musicians have been doing this forever, often without formalizing it.

• Like previous example, in jazz, it’s common to treat certain intervals as “equivalent” for reharmonization tricks.
• In experimental tuning, analytical equivalence is the key to making sense of irrational divisions or non-octave systems, you pick the modulus that makes the relationships meaningful for your purpose.

The framework then separates:

• Geometrical repetition (if any), what the set actually repeats at in raw space.
• Analytical repetition, what you choose to treat as equivalent for naming, analysis, and composition.

That separation is what lets you say:

    > “12edo is analytically octave-based, but geometrically it repeats every 100 cents, and I can choose to analyze it under any modulus I want.”

4. Practical Features

4.1. Instant Fretboard Accuracy

• Auto‑calculates exact fret positions for any tuning system, including JI, EDO, non‑octave, and irrational divisions.
• Eliminates manual math errors and saves hours of setup time.

4.2. Microtonal “Test Drive” Before You Commit

• Simulate any microtonal fretboard before physically modifying or buying a guitar.
• Compare multiple tunings side‑by‑side to see which feels and sounds right.

4.3. Interactive Learning & Composition Tool

• Clickable/Playable fretboard — hear notes, chords, and scales instantly.
• Built‑in arpeggiators and backing track generators for real‑time practice and composition.

4.4. Scale & Chord Visualization

• Displays accurate diagrams for any tuning, not just standard 12‑EDO.
• Works with custom analytical class systems, so you can label notes however you want (letters, numbers, symbols).

4.5. Cross‑Instrument Portability

• Analytical classes and scale definitions can be applied to any instrument layout, not just the guitar you started with.
• Great for multi‑instrumentalists or luthiers designing new instruments.

4.6. Flexible Physical Parameters

• Supports non‑standard guitars: extended‑range, left‑handed, drop tunings, altered string counts.
• Adjustable fret counts, string intervals, and scale lengths.

4.7. Creative Experimentation

• Try irrational cycles (e.g., π‑periodic systems) or non‑harmonic timbre‑based equivalences.
• Explore “what if” scenarios for chord shapes, voicings, and modulations in exotic tunings.

4.8. Educational Value

• Perfect for teaching tuning theory, interval relationships, and fretboard geometry visually.
• Bridges the gap between abstract theory (interval matrices, chroma, equivalence) and hands‑on playing.

4.9 Shareable, Interactive Setups

• Because SFINX runs in the browser, you can configure an instrument, set the tuning, define analytical classes, load scales, and even create chord progressions, and then generate a unique shareable URL.
• Anyone with the link sees exactly what you set up: the same fretboard layout, same tuning, same class system, same progression.
• Perfect for:
• Sending musical ideas to bandmates
• Sharing teaching examples with students
• Posting microtonal experiments in forums or social media
• Recipients don’t need to install anything, they just open the link and can interact with the fretboard immediately.

5. Analytical Tools & Visualization

5.1 Color‑Coded Octaves and Chroma Analysis

The infinite, continuous, and cyclical nature of musical chroma parallels the visual spectrum. When color‑coding is applied to musical notation, it becomes a powerful tool for visualizing pitch relationships, whether in complex microtonal contexts or conventional 12‑tone systems.

Perceptual Questions

Color‑coding raises intriguing perceptual issues:

• Are there enough distinct colors in the visual spectrum to represent subtle pitch distinctions?
• Where are the perceptual boundaries between adjacent note/color pairings?
• As Newton observed: “the just confines of the colours are hard to be assigned, because they pass into one another by insensible gradation.”

This is strikingly similar to a core question in music theory: When does one pitch function shift to the next?
Our ears often resolve such ambiguity by prioritizing contextual relationships over exact interval boundaries.

5.1.1 Color in Notation Systems

Many notation systems, from traditional staff notation to MIDI rolls, have experimented with color to represent octave equivalence and pitch classes.
Modern software makes it easy to integrate color‑coding into both microtonal and standard workflows, aiding learning and analysis.

Example (12‑EDO):

In a color‑coded MIDI roll, 12 evenly spaced hues from the sRGB wheel are assigned to the 12 pitch classes, with an arbitrary origin (e.g., red = C).

• A nine‑note chord might display only three colors, instantly revealing it’s a major triad (R, G, B) without manual interval calculation.

5.1.2 The Spiral Harp: A Case for Color Coding

The Spiral Harp is a virtual instrument that generates pitches from the lengths of spiral polygonal chain segments.

• It can produce over 1,000 distinct non periodic pitches in the audible range.
• Traditional labeling is impractical, and enumerating all interval ratios is infeasible.

Solution: Color‑coding by octave equivalence.

• Strings of the same color share the same chroma and produce consonant sonorities.
• An alternative sRGB hue wheel is used, revealing not only octave equivalence but also intervallic relationships.

5.1.3 Color and Interval Relationships

Complementary colors (red–cyan, orange–blue, yellow–violet, green–magenta) correspond to tritone relationships.

• In music theory, the tritone is the geometric mean of the octave:

\[
\text{Tritone ratio} = \sqrt{2}
\]

• In color perception research, complementary “color attractors” in the spectrum often appear at wavelength ratios approximating \(\sqrt{2}\), a striking cross‑domain coincidence.
• Unlike perfect fifths and fourths, which invert into each other, the tritone is symmetrical under inversion, reinforcing its ambiguous, “achromatic” quality.

5.1.4 In the App

SFINX supports different independent coloring modes:

1. By Analytical Class, colors follow your chosen modular analysis (e.g., 12‑EDO letters, custom class systems).
2. By Perceptual Chroma, colors follow the pitch’s position within the perceptual cycle (octave or other).
3. Chord-Row.
4. Plain.

This approach means you can:

• See how analytical equivalence maps onto perceptual space.
• Instantly spot consonances, dissonances, and structural patterns, even in dense microtonal textures.

5.2 Interval & Chroma Matrices

5.2.1. Overview

In music theory, interval matrices are analytical tools for exploring the relationships between pitches in a tuning system or scale.

• Some tunings are octave‑periodic.
• Others use alternative periods or have no periodicity, potentially generating an infinite number of chromas.

To fully understand such systems, it’s often necessary to calculate pitches beyond the minimal generating set, revealing scale extensions and emergent musical possibilities.

5.2.2. Interval Matrix (Definition)

Prime numbers up to 19. Delta = -1, octave-space.
\(\mathbf{Ä}_{1:2}^{-1}(P_{19})\)

An interval matrix is a table showing the intervals between all pairs of pitches in a given tuning or scale.

• Particularly useful for non‑equal temperaments or scales with non‑uniform step sizes.
• In equal temperaments, the matrix contains redundant patterns, making it less informative.
• Example: The diatonic scale’s interval matrix reveals the characteristic structures of its modes (Ionian, Dorian, Phrygian, etc.).

(Image.5.2.2), sRGB color‑coded interval matrix of the 3‑limit diatonic scale
Group presentation: ⟨2, 3 | 3⁷ ≡ 1⟩
Each row = a cyclic permutation of the scale.
Displayed logarithmically with a 12‑EDO ruler for reference.

5.2.3. Chroma Matrix

A chroma matrix is an extended interval matrix where the octave is fixed as the period.

• Color‑coding is applied based on octave equivalence by default, and an arbitrary reference pitch.
• One color in the matrix → the system contains only octave duplications.
• Multiple colors emerging as pitches are added → the system has infinite chroma.

5.2.4. Analytical Value

• In octave‑based tunings (even with unequal divisions), chroma matrices have limited analytical value, adding pitches beyond the octave does not produce new chromas.
• In non‑octave tunings, chroma matrices are far more revealing.

5.2.5. Example: 13‑ED3 (Bohlen–Pierce Equal Temperament)

• Period = tritave (3:1), equally divided into 13 steps.
• Interval matrix rows are identical due to equal division and arbitrary choice of equivalence class.
• Local interval relationships are the same from any pitch, so the global structure is not captured.
• Chroma analysis (folding the set into the octave) is more informative — here, the chroma matrix acts as an external interval matrix, using the octave as the modulus.

5.2.6. Key Insight

• Interval matrix → internal relationships within the system’s own period/equivalence.
• Chroma matrix → those same relationships, but folded into an external reference period (often the octave).
• Color‑coding makes finite vs. infinite chroma sets visually obvious.

5.2.7 Fretboard-Matrix Equivalence

The “interval matrix” and the “fretboard” are isomorphic representations of the same underlying mathematical object, the only differences are:

• Domain:
• Interval matrix → frequency (ratio) domain, fixed period, full permutation of the set.
• Guitar → string length domain, fixed physical constraints, partial permutation (limited rows, fixed string‑to‑string interval).
• Permutation scheme:
• Interval matrix → usually permutes by 1 step through the generating set, producing an \(n \times n\) grid.
• Guitar → permutes by the tuning interval between strings (e.g., perfect fourths), producing a subset of the full matrix.
• Period choice:
• Interval matrix → period is whatever modulus you choose for the analysis (octave, tritave, external element).
• Guitar → period is implicit in the physical layout (scale length, fret spacing), but you can still “fold” it analytically into any modulus.

Why they’re the same object

Both are visualizations of multiplicative set geometry:

• The rows are cyclic permutations of the generating set.
• The columns are positions within the chosen period.
• The coloring (by chroma, class, or other equivalence) reveals structural symmetries and repetitions.

The guitar is just a physically parameterized, incomplete interval matrix, a “cropped” and “tilted” slice of the full mathematical grid, optimized for human hands.
The interval matrix is the abstract, complete guitar, unconstrained by ergonomics, range, or tuning tradition.

> Any fretted string instrument layout is a partial, parametrized realization of an interval matrix in the string‑length domain. Conversely, any interval matrix can be interpreted as the complete, unconstrained fretboard of a generalized string instrument.

6. Special Topics

6.1 The Impact of Non‑Octave Tunings on Music

6.1.1 Basics

• Chroma content; the set of distinct pitch positions within the chosen perceptual cycle, is a key factor in understanding any tuning system.
• If the tuning’s period is the octave (or a power of the octave), the chroma set is finite.
• If the period is not an integer multiple of the octave, the chroma set is infinite.
• Infinite chroma sets are theoretically rich but often harder to use in conventional music‑making, especially in collaborative contexts.
• Octave‑based systems make it easy for musicians to match pitch classes across registers; non‑octave systems require much deeper familiarity to coordinate.

6.1.2 Details

Generating Sets (in the context of tuning)

• A generating set is a finite collection of pitches (ratios, cents, etc.) from which the full tuning is derived.
• In instruments and software, this set is mapped across the audible range, whether the system is periodic or not.
• This is distinct from the abstract group‑theory definition of a generating set.

Finite vs. Infinite Chroma Sets

• Octave‑periodic tunings (period = 2ᵏ/1) → finite chroma set.
• Non‑octave‑periodic tunings → infinite chroma set, because no finite set of chromas repeats exactly under octave equivalence.

Example: Bohlen–Pierce (13‑ED3)

• Often described as having “13 classes” (13 equal divisions of the tritave).
• On a standard 6‑string guitar tuned in 13‑ED3 (each string a 4th fret above the previous), you actually get 28 unique chromas across the fretboard, far more than the nominal 13.
• A Bohlen–Pierce piano might have 80 keys, many of which are chromas unavailable to the guitarist.

Practical Implications

• In octave‑based systems, musicians can play in any register and still match pitch classes (e.g., C major chord in multiple octaves).
• In non‑octave systems, playing in different “periods” can disrupt harmonic function unless all players know the tuning’s full chroma content and relationships.
• This makes non‑octave tunings powerful for solo exploration or specialized ensembles, but challenging for traditional collaborative workflows.

6.2 Why Frets Matter in Microtonality

Something a lot of people miss when they imagine “just go fretless” as the universal microtonal solution.

On paper, fretless seems like the ultimate freedom: no fixed grid, infinite pitch choice, no compromise to a temperament. But in practice, the physics of strings and the psychology of pitch perception pull you toward certain “gravitational wells”, the harmonic series, familiar consonances, and the natural resonances of the instrument. That’s why bowed strings like violin, viola, cello tend to orbit tonal centers even in free improvisation: the instrument wants to sing in tune with its own overtones.

Frets, paradoxically, are what let you escape that gravity. They impose a structure that might feel “wrong” at first, but that’s exactly what allows you to:

• Accumulate subtle interval hacks, 5‑EDO example: stacking 480¢ “sub‑fourths” to land exactly on a 2400¢ double octave, versus 12‑EDO’s 500¢ fourths overshooting to 2500¢. Without frets, hitting those micro‑targets consistently is cognitively exhausting.
• Exploit non‑intuitive cycles, you can train your hands to navigate a temperament’s quirks without your ear constantly pulling you back to “natural” intervals.
• Play complex chords in JI or odd lattices, on fretless, the intonation demands for multi‑note chords are brutal; with frets, the geometry locks them in.

So frets aren’t just a fetishized relic of guitar design, they’re a cognitive prosthetic. They let you live inside a tuning system’s logic long enough to internalize it, instead of constantly being dragged back to harmonic gravity. Once it’s in your muscle memory, sure, you can take it fretless and keep the system alive in your hands. But without that scaffold, most players never get past the pull of “natural” intonation.

It’s almost poetic:

• Fretless = the ocean, infinite but with strong currents.
• Frets = a map of an invented continent, letting you explore terrain that doesn’t exist in nature.

7. Limitations & Workarounds

7.1 Regarding Fretboard Customization

The application presents certain inherent limitations regarding the direct creation of highly customized fretboard configurations, specifically:

1. Unique Individual Frets: It is not possible to directly add a single, unique fret at a specific position on only one string without incorporating that interval into the underlying pitch set definition for the entire instrument.
2. Fret-Skipping Patterns: Creating patterns where entire frets are omitted across all strings cannot be achieved through a direct fret-removal function.

These limitations are not arbitrary but stem from the application's core design philosophy, which prioritizes the generation and analysis of scales and patterns based on consistent, underlying pitch sets. Allowing arbitrary fret additions or removals would undermine this analytical framework. However, practical workarounds exist, grounded in established music theory principles. Understanding these limitations and their justifications clarifies the application's logic, which shares conceptual roots with historical approaches to instrument notation.

7.2 Historical Precedent and Theoretical Foundation

The methodology adopted finds a parallel in historical solutions developed during the Middle Ages for fretted instruments like the cittern. As instrument makers experimented with different fretting patterns (including microtonal variations, just intonation, and skipped frets), tablatures became inconsistent and difficult to interpret across instruments. A crucial development, particularly coinciding with the standardization towards 12-tone temperament, was to number frets according to their position within the complete theoretical chromatic scale, even if the physical fret was absent. (See Duodecimability)

For instance, if a string only possessed a single fret precisely at its midpoint (the octave), it was labeled "fret 12," not "fret 1." This indicated its position within the 12-tone system. This elegant solution not only standardized tablature but also facilitated theoretical understanding. It implicitly treated transposition algebraically (akin to coset shifts in modern terminology), ensuring that musical patterns like chord shapes remained consistent conceptually, even when physical frets varied. Attempting to transpose patterns based solely on naive fret counting on irregularly fretted instruments would quickly lead to inconsistencies.

7.3 Application Context: Timbre and Octave Equivalence

It is pertinent to note that while the application offers various synthesized guitar timbres, all are fundamentally harmonic. This guarantees the perceptual validity of the octave as the primary cycle of pitch repetition. Consequently, the application's chroma-based color-coding system is fixed to octave equivalence, reinforcing the focus on cyclical pitch structures.

7.4 Workaround: Simulating Unique Fret Additions

Consider the scenario of adding a single extra fret to a standard 12-tone guitar – for example, a quarter-tone fret between the 3rd and 4th traditional frets on one string. This might represent a neutral third, a specific microtonal inflection, or serve another modal/contextual purpose.

Directly modeling this single fret addition is not supported. The required workaround involves redefining the instrument's fundamental pitch set to include the desired new interval. This means the interval will be replicated at octave-equivalent positions across the entire fretboard. While a single physical fret might be intended as ornamental, incorporating its pitch interval categorically into the underlying set maintains theoretical consistency.

This approach offers significant advantages:

• It preserves a coherent, cyclical pitch structure, which is easier to implement computationally.
• It results in a more intuitive playing and analysis experience, as patterns remain consistent.
• Chord progressions and scale analyses remain valid across the fretboard. Integrating the new note reveals its systematic relationship within the established structures.

To implement this, you would define the new pitch set using cents notation, incorporating the new interval. For instance, adding a quarter tone (50 cents) near the beginning of the scale: {50, 100, 200, 300, ... , 1200}. Tuning shifts can then be used to position this interval relative to the open string as desired.

7.5 Workaround: Simulating Fret-Skipping

Fret-skipping, common in constructing instruments based on subsets of larger microtonal systems (e.g., selecting specific pitches from 31-EDO), presents a different challenge. It's crucial to distinguish between:

• Removing specific notes from a system: This is how regular temperaments are often realized (e.g., a 7-note diatonic scale derived from 12-EDO). On a guitar, this might appear as skipped frets, but the same set of notes remains available across the entire fretboard, just spaced differently.
• Skipping entire frets across all strings: This creates a complex scenario where the availability of specific pitches becomes inconsistent across different positions and octaves.

While it is technically possible to pre-calculate the exact pitches resulting from a fret-skipping pattern and input them as a fixed, non-repeating scale into the application, this is highly impractical for analysis. It essentially defines a unique, non-standard instrument configuration, limiting the applicability of standard analytical tools and pattern recognition.

The recommended and more practical approach mirrors the historical solution: analyze the instrument based on the full, underlying pitch set (e.g., the complete 31-EDO scale if the physical instrument omits some 31-EDO frets). Although diagrams might show chords or scales utilizing frets physically absent on the specific instrument, this method allows:

• Understanding the underlying theoretical structure and harmony.
• Identifying alternative voicings or inversions of desired chords/patterns that are physically playable.
• Avoiding the creation of highly specific notation tied exclusively to one instrument's unique fretting.

7.6 Conclusion

The application's limitations on direct, arbitrary fret manipulation encourage users to work within consistent theoretical frameworks. By defining instruments based on complete, cyclical pitch sets (standard or custom), the application facilitates robust musical analysis, pattern recognition, and understanding of transposition and harmony, echoing effective principles developed historically for managing complex fretting systems. The focus remains on the underlying musical structure rather than idiosyncratic physical layouts.

8. Appendix

Mathematical Formalism of Chroma

Using logarithmic and modular notation makes chroma computation explicit and practical for coding. For example, in Python:

chroma = 2 ** (math.log2(x) % 1)

This method bypasses manual interval reduction (e.g., repeated division by 2 for values > 2 or multiplication for values < 1), offering a direct way to compute a pitch’s position within the octave cycle.

Equivalence Relation and Group Structure

Chroma can be formalized via an equivalence relation on positive real numbers:
\[
x \sim y \iff x = 2^n \cdot y \quad \text{for some } n \in \mathbb{Z}
\]
This defines a multiplicative equivalence class under octave scaling. The quotient space \( (0, \infty)/\sim \) maps naturally to the unit circle via logarithmic and exponential functions:
\[
\frac{(0, \infty)}{\sim} \xrightarrow{\log_2(\bullet)} \mathbb{R}/\mathbb{Z} \xrightarrow{\exp(2\pi i \bullet)} \mathbb{S}^1 \subseteq \mathbb{C}
\]
Or more compactly:
\[
[x] \mapsto \log_2(x) + \mathbb{Z} \mapsto e^{2\pi i \log_2(x)}
\]
This mapping preserves structure and enables the construction of pitch class diagrams, such as the circle of fifths, and even spectral representations like hue wheels, where pitch chroma is visualized on the unit circle in the complex plane.

Musical Implications

This formalism reveals that melodies and chords rely on more than octave equivalence alone. The fractional parts of the log₂ scale, the “colors” of pitch, are essential to musical identity. This becomes especially relevant in tuning systems that deviate from the traditional octave structure, where chroma must be redefined relative to alternative cycles (e.g., tritave, π-periodic systems).

9. References & Further Reading

Interval Matrix, Tonal Constancy, Interval Space Randomness, Spiral Harp, Group Theory In Tunings