Rhythmic numerical systems

Rhythmic numerical systems,^[1] represent a branch of the science of musical numerical systems^[2] attributed to researcher Masih Aslani. This interdisciplinary field investigates numerical patterns, systems, and relationships associated with rhythm, meter, and tempo in music, focusing on their connections to psychological states and the four temperaments of humans.^[3] Musical numerical systems aim to provide a framework for analyzing the effects and psychological states induced by musical elements and for revising music theories through computational methods.^[3][2] This area intersects musicology, mathematics, music psychology, and music therapy.

Historical background

The mathematical perspective on music has a long history. Pythagoras and his followers explored the relationship between numerical ratios and musical intervals (harmony), considering music a branch of the mathematical sciences.^[4] In Islamic civilization, scholars such as Al-Farabi in his Kitab al-Musiqa al-Kabir (The Great Book of Music) and Avicenna (Ibn Sina) in the music section of his Kitab al-Shifa (The Book of Healing) engaged in mathematical analyses of musical dimensions.^[5] Avicenna, in particular, believed in the therapeutic effects of specific melodic modes based on the theory of the four humors (four temperaments).^[6]

In contemporary times, numerical approaches to rhythm have continued in fields such as computer music (e.g., Euclidean rhythms) and ethnomusicology (classification of rhythmic patterns across cultures).^[7]

Contemporary approaches

In recent decades, research has sought to quantify the relationship between musical parameters (such as tempo, meter) and a listener's emotional or physiological responses. This work often falls within the domains of music psychology and music therapy.^[8] One line of inquiry examines the role of tempo in emotional arousal, with faster tempos generally associated with higher excitement and slower tempos with calmness or sadness.^[9]

Models based on the four temperaments theory

Some contemporary research has attempted to propose systematic models for classifying the effects of rhythm and tempo, drawing inspiration from historical concepts such as the four temperaments theory found in traditional medicine. In these models, each musical quality (such as a specific time signature or tempo range) may be assigned to one of the four temperaments: "warm and dry", "warm and moist", "cold and dry", or "cold and moist".^[3] The stated aim of such models is to create a predictive framework for selecting music with specific therapeutic effects in music therapy.

One example of such a model was proposed by Masih Aslani (2025), in which specific mathematical operations are performed on the numerator and denominator of a time signature to derive a number; based on the remainder when this number is divided by 4, a temperament is assigned to the meter.^[3] This model also considers tempo as an "acquired temperament" that interacts with the "innate temperament" of the time signature.^[3]

Despite its conceptual foundations, there is a significant lack of comprehensive scholarly research on this topic, both from a historical and a contemporary perspective. As one of the first dedicated explorations of the subject, this article examines these issues and establishes a foundational understanding of rhythmic numerical systems.^[3]

Overview

Rhythm constitutes a fundamental element of music, serving as its temporal foundation. It encompasses the organization of sound and silence into structured patterns over time, essentially governing the flow and duration of musical events.^[10]

The new science of musical numerical systems is founded on reasoning and numerical logic to understand music's effects on living beings. A key aspect within music therapy is identifying the temperamental qualities of rhythm and tempo. This novel approach offers promising avenues for research in medical engineering.

The analytical approaches demonstrate strong consistency across different time signature types. The agreement between methods for simple, compound, and complex time signatures provides mutual verification, validating the methodology and reinforcing the reliability of the identified rhythmic temperament classifications.^[3] The system provides distinct calculation methods for different types of time signatures: simple, compound, and irregular.

Calculation Methods

The methodology involves numerical operations on the upper and lower numbers of a time signature. The final result is reduced modulo 4, with the remainder (0, 1, 2, or 3) corresponding to one of the four elemental temperaments: Earth =0 (cold and dry), Fire =1 (warm and dry), Wind =2 (warm and moist), and Water =3 (cold and moist).^[3]

Calculation Notation

The analytical framework employs specific notations for its calculations, as defined below:^[3]

Table 1: Calculation Notations

Notation	Explanation
${\displaystyle Re}$	Number of the element (Remainder)
${\displaystyle a}$	Upper number (numerator) of the time signature
${\displaystyle b}$	Lower number (denominator) of the time signature
${\displaystyle c}$	Sum of the upper and lower numbers (${\displaystyle c=a+b}$)
${\displaystyle e}$	Extracted number (Constant value used in compound time calculations)
${\displaystyle a_{1},a_{2},...}$	Intermediate results for irregular time signatures
${\displaystyle TC}$	Total calculation result (Final value before modulo operation)

The core operation involves calculating a final value (${\displaystyle TC}$) from the time signature components and then determining the elemental temperament by finding the remainder when ${\displaystyle TC}$ is divided by 4:

${\displaystyle TC\equiv Re{\pmod {4}}}$

The remainder ${\displaystyle Re}$ corresponds to one of the four classical elements and its associated temperament.

Simple Time Signatures

For simple time signatures, the sum of the upper (numerator) and lower (denominator) numbers is calculated. This sum is then divided by 4, and the remainder determines the temperament.^[3]

${\displaystyle TC=a+b,\quad TC\equiv Re{\pmod {4}}}$

Example:

• Time signature ${\displaystyle {\frac {3}{4}}}$: ${\displaystyle TC=3+4=7,\quad 7\equiv 3{\pmod {4}}}$. Remainder 3 corresponds to Cold and Moist (Water).

The computational method for simple time signatures is specifically designed for a lower number (denominator) of 4 or its multiples, because the system is fundamentally based on four temperaments. Time signatures with lower numbers, like 1 or 2, are analyzed by conceptually adapting them to an equivalent form with a denominator of 4 before calculation.,^[3]

Compound Time Signatures

Two methods are described. In Method A, a constant value (e=5, derived from 3+2, reflecting the 3/2 ratio in compound meters) is subtracted from the sum of the time signature's numbers before the modulo operation.^[3]

${\displaystyle c=a+b,\quad TC=c-e,\quad TC\equiv Re{\pmod {4}}}$

Example:

• Time signature ${\displaystyle {\frac {9}{8}}}$: ${\displaystyle c=9+8=17,\quad TC=17-5=12,\quad 12\equiv 0{\pmod {4}}}$. Remainder 0 corresponds to Cold and Dry (Earth).

Method B for Compound Time Signatures

Method B involves subtracting the sum of the upper and lower numbers from the sequence 1, 2, 3, and 4 to reach a single digit. If the result is between 1 and 4, it is the final temperament number; if between 5 and 9, it is divided by 4 and the remainder is used.^[3]

Example for 9/8:

The calculation is as follows:

• Sum of numerator and denominator: ${\displaystyle c=9+8=17}$
• ${\displaystyle TC=17-(2+3+4)=8}$
• ${\displaystyle 8\equiv 0{\pmod {4}}}$ (remainder 0 corresponds to Cold and Dry temperament)

Notes:^[3]

1. The final result must be a single digit.
2. Two-digit numbers starting with 1 are excluded from subtraction for the number 1.
3. Two-digit numbers starting with 2 are excluded from subtraction for the number 2.

Irregular Time Signatures

Irregular time signatures consist of two or more unequal beat groups. The calculation method differs depending on whether the underlying beat structure is simple or compound.^[3]

With Simple Beat Structure

For irregular signatures with simple beats (e.g., 5/8, 7/8), the sum of numerator and denominator is calculated for each beat group. These sums are then added together, and the total is divided by 4. The remainder determines the temperament.^[3]

Example for 7/8 (3+4):

• Beat group 1 (3/8): ${\displaystyle a_{1}=3+8=11}$
• Beat group 2 (4/8): ${\displaystyle a_{2}=4+8=12}$
• Total calculation: ${\displaystyle TC=11+12=23}$
• ${\displaystyle 23\equiv 3{\pmod {4}}}$ (remainder 3 = Cold and Moist)

With Compound Beat Structure

For irregular signatures with compound beats, methods similar to compound time signatures (Method A or B) are applied to each beat group. The results are summed and then divided by 4.^[3]

Example 1 - Method A for ${\displaystyle {\frac {15}{8}}^{(3+2)}}$ (9+6):

• Beat group 1 (9/8): ${\displaystyle 17-5=12}$
• Beat group 2 (6/8): ${\displaystyle 14-5=9}$
• ${\displaystyle TC=12+9=21}$
• ${\displaystyle 21\equiv 1{\pmod {4}}}$ (remainder 1 = Warm and Dry)

Method B for Irregular Time Signatures with Compound Beats

For irregular time signatures with a compound beat structure, Method B applies sequential subtraction from the numbers 1, 2, 3, and 4 to the sum of each beat group's numbers. The results for each group are then summed, and the final value is reduced modulo 4.^[3]

Example for ${\displaystyle {\frac {15}{8}}^{(3+2)}}$:

• Beat group 1 (3 beats = 9/8 equivalent):

• • ${\displaystyle a_{1}=9+8=17}$
  • ${\displaystyle 17-(2+3+4)=8}$

• Beat group 2 (2 beats = 6/8 equivalent):

• • ${\displaystyle a_{2}=6+8=14}$
  • ${\displaystyle 14-(2+3+4)=5}$

• Total calculation: ${\displaystyle TC=8+5=13}$
• Temperament determination: ${\displaystyle 13\equiv 1{\pmod {4}}}$
  • Result: Warm and Dry temperament (Fire element)

Special Notes:^[3]

• Identical numbers in beat groups are eliminated pairwise
• For single-beat compound structures, subtract 1 from the final result
• The calculations for simple and compound beat structures provide mutual verification

Tempo Temperament Model

The system also incorporates tempo, measured in beats per minute (BPM), as an "acquired temperament" that interacts with the "innate temperament" of the time signature. The tempo range is divided into four segments, each assigned to a temperament quality (e.g., cold and dry, warm and moist), with thresholds varying based on the number of beats per measure (duple, triple, quadruple time).^[3]

Discussion and Applications

Identifying the temperament of rhythm and tempo could have applications in music therapy, possibly helping to select music that produces specific psychological effects or balances a listener's mood.^[3] For instance, a piece with a "warm and moist" rhythm might be used to counteract a "cold and dry" state. The internal consistency between the different calculation methods for simple, compound, and irregular time signatures is presented as evidence supporting the model's validity.^[3]

Interaction of Innate and Acquired Temperament

The model distinguishes between the innate temperament of a time signature (derived from its numerical structure) and the acquired temperament imposed by performance tempo. For example, the time signature 2/4 has an innate warm and moist temperament. However, when performed at different tempos, it exhibits acquired temperaments:^[3]

• At tempos of 40 BPM and below: acquires a **cold and dry** temperament
• At tempos of 41-61 BPM: acquires a **cold and moist** temperament
• At tempos of 103 BPM and above: acquires a **warm and dry** temperament
• At tempos of 62-102 BPM: the innate **warm and moist** temperament is reinforced

This explains why a 4/4 time signature (innately cold and dry) can evoke qualities similar to 2/4 when performed within the warm and moist tempo range (102-182 BPM).^[3]

The psychological effects attributed to 2/4 at different tempos are described as follows:^[3]

Table 2: Character traits associated with 2/4 time signature at different tempos

Tempo Temperament	Personality Traits
Cold and Dry	Enhances patience, endurance, and to some extent, thoughtful and cautious courage
Cold and Moist	Strengthens introversion and behavioral calmness along with willpower
Warm and Moist	Greatly strengthens strong will and reasonable courage
Warm and Dry	Sharpens intelligence and quick reactions, enhancing willpower and courageous boldness

Theoretical Basis

The classification of rhythm speed based on four temperaments draws parallels with elemental molecular movement in traditional medicine. Slower tempos, analogous to earth element stability, are posited to have longer-lasting effects on memory, while faster tempos, comparable to fire element movement, create more transient but excited states.^[3]

See also

• Music therapy
• Time signature
• Four temperaments

References

1. Aslani, M. (2025). "An Innovative Numerical System Model for Identify Effects and Psychological States of Musical Rhythm and Tempo through Four Temperaments" (PDF). Mathematics and Computational Sciences. 6 (1): 147–159. doi:10.30511/mcs.2025.2044621.1252.
2. Aslani, M. (2025). "Innovative 17-note Number System and its Computational Method to Identify the Effects and Psychological States of Music Scale" (PDF). Transactions on Combinatorics (TOC). doi:10.22108/toc.2025.145560.2290.
3. Aslani, M. (2025). "An Innovative Numerical System Model for Identify Effects and Psychological States of Musical Rhythm and Tempo through Four Temperaments". Mathematics and Computational Sciences. 6 (1): 147–159. doi:10.30511/mcs.2025.2044621.1252.
4. Huffman, Carl. "Pythagoreanism". Stanford Encyclopedia of Philosophy. Retrieved 2025-03-20.
5. Farmer, Henry George (1936). The Science of Music in the Mamluk Kingdom: With Special Reference to the Works of Ibn Sina and al-Farabi. London: Luzac & Co. OCLC 1018076456.
6. Ameli, S. M. R. (2004). "Music Therapy in the View of Avicenna" (PDF). Journal of the International Society for the History of Islamic Medicine. 3 (5): 38–43.
7. Toussaint, Godfried (2005). "The Euclidean Algorithm Generates Traditional Musical Rhythms" (PDF). Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science. Banff: 47–56.
8. Juslin, Patrik N.; Sloboda, John A. (2010). Handbook of Music and Emotion: Theory, Research, Applications. Oxford: Oxford University Press. ISBN 978-0199230143.
9. Husain, G.; Thompson, W. F.; Schellenberg, E. G. (2002). "Effects of Musical Tempo and Mode on Arousal, Mood, and Spatial Abilities". Music Perception. 20 (2): 151–171. doi:10.1525/mp.2002.20.2.151.
10. Kamien, Roger (2018). Music: An Appreciation (12th ed.). McGraw-Hill Education. ISBN 978-1259892707.

Category:Music theory