Syntonic comma: Difference between revisions

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A small musical interval in just intonation

Syntonic comma

The syntonic comma is a small musical interval that is significant in the study of just intonation and musical temperament. It is the difference between a justly intonated major third and a Pythagorean third. This interval is crucial in understanding the tuning systems used in Western music.

Definition

In terms of frequency ratios, the syntonic comma is the difference between the ratio of a justly tuned major third (5:4) and a Pythagorean major third (81:64). Mathematically, it can be expressed as:

\[ \text{Syntonic comma} = \frac{81}{80} \]

This ratio, 81:80, represents a very small interval, approximately 21.51 cents, which is slightly larger than a Pythagorean comma.

Historical Context

The concept of the syntonic comma has been known since ancient times and was extensively studied by Pythagoras and later by Ptolemy. It became particularly important during the Renaissance when musicians and theorists sought to reconcile the mathematical purity of just intonation with the practical needs of musical performance.

Musical Implications

The presence of the syntonic comma in tuning systems means that it is impossible to perfectly tune all intervals in a scale using just intonation. This led to the development of various temperament systems, such as meantone temperament, which temper the syntonic comma to achieve a more usable tuning across all keys.

Meantone Temperament

In meantone temperament, the syntonic comma is distributed across several intervals, allowing for more consonant thirds at the expense of slightly less pure fifths. This system was widely used during the Baroque period and is still of interest to musicians and musicologists today.

Mathematical Properties

The syntonic comma is a prime example of how small discrepancies in tuning can accumulate to create noticeable differences in musical performance. It is one of several commas that arise in the study of musical acoustics and tuning theory.

Visual Representation

File:Just vs Pythagorean.svg

Comparison of just and Pythagorean tuning. The syntonic comma is the small difference between the two tuning systems.

The diagram above illustrates the difference between just and Pythagorean tuning, highlighting the syntonic comma as the small interval that separates them.

Related Pages

@@ Line 1: / Line 1: @@
-'''Syntonic Comma''' (also known as the '''Didymus comma''' or '''meantone comma''') is a small musical interval, specifically a type of [[comma (music)|comma]], that is mathematically represented as the frequency ratio 81:80. This interval, approximately equal to 21.51 cents, plays a significant role in the theory and practice of [[tuning systems]], particularly in the context of [[Just Intonation]] and [[Meantone temperament]].
+{{Short description|A small musical interval in just intonation}}
+==Syntonic comma==
+The '''syntonic comma''' is a small musical interval that is significant in the study of [[just intonation]] and [[musical temperament]]. It is the difference between a [[justly intonated]] [[major third]] and a [[Pythagorean third]]. This interval is crucial in understanding the tuning systems used in Western music.
 ==Definition==
-The syntonic comma arises in musical tuning systems when comparing the sizes of major thirds. In [[Pythagorean tuning]], which is based on a stack of perfect fifths, the major third (from C to E, for example) is slightly larger than the sum of two whole tones (from C to D and D to E). This discrepancy is due to the fact that the Pythagorean major third is derived from four perfect fifths (C-G-D-A-E), resulting in a ratio of 81:64, whereas the just major third, more consonant to the human ear, has a simpler ratio of 5:4 (80:64). The difference between these two intervals is the syntonic comma.
+In terms of frequency ratios, the syntonic comma is the difference between the ratio of a justly tuned major third (5:4) and a Pythagorean major third (81:64). Mathematically, it can be expressed as:
+\[
+\text{Syntonic comma} = \frac{81}{80}
+\]
+This ratio, 81:80, represents a very small interval, approximately 21.51 cents, which is slightly larger than a [[Pythagorean comma]].
 ==Historical Context==
-The syntonic comma has been a subject of interest since the ancient Greek theorists, but it was particularly noted by Didymus and Ptolemy in their explorations of musical tuning. The awareness of this interval influenced the development of various tuning systems throughout history, including meantone temperament, which attempts to temper the syntonic comma in order to achieve more consonant thirds.
+The concept of the syntonic comma has been known since ancient times and was extensively studied by [[Pythagoras]] and later by [[Ptolemy]]. It became particularly important during the [[Renaissance]] when musicians and theorists sought to reconcile the mathematical purity of just intonation with the practical needs of musical performance.
+==Musical Implications==
+The presence of the syntonic comma in tuning systems means that it is impossible to perfectly tune all intervals in a scale using just intonation. This led to the development of various [[temperament]] systems, such as [[meantone temperament]], which temper the syntonic comma to achieve a more usable tuning across all keys.
+===Meantone Temperament===
+In [[meantone temperament]], the syntonic comma is distributed across several intervals, allowing for more consonant thirds at the expense of slightly less pure fifths. This system was widely used during the [[Baroque]] period and is still of interest to musicians and musicologists today.
-==Impact on Tuning Systems==
+==Mathematical Properties==
-In the context of [[Meantone temperament]], the syntonic comma is tempered, meaning it is distributed among several intervals to reduce its dissonance. This is achieved by narrowing the fifths slightly, which in turn makes the major thirds closer to the pure ratio of 5:4. This compromise allows for more consonant harmonies in a wider range of keys than Pythagorean tuning, though it introduces slight impurities in other intervals.
+The syntonic comma is a prime example of how small discrepancies in tuning can accumulate to create noticeable differences in musical performance. It is one of several commas that arise in the study of [[musical acoustics]] and [[tuning theory]].
-[[Just Intonation]] also deals with the syntonic comma, but rather than tempering it, this system embraces the pure intervals, including the pure major third. This results in a tuning system that can be highly consonant in certain keys but may lead to significant dissonances in others, due to the accumulation of syntonic commas.
+==Visual Representation==
+[[File:Just_vs_Pythagorean.svg|thumb|right|Comparison of just and Pythagorean tuning. The syntonic comma is the small difference between the two tuning systems.]]
-==Mathematical Description==
+The diagram above illustrates the difference between just and Pythagorean tuning, highlighting the syntonic comma as the small interval that separates them.
-Mathematically, the syntonic comma can be described as the difference between two kinds of major thirds: the Pythagorean major third (81:64) and the just major third (5:4). This difference is calculated as the ratio 81:80. When expressed in cents, a logarithmic measure of intervals, the syntonic comma is approximately 21.51 cents.
-==Conclusion==
+==Related Pages==
-The syntonic comma is a fundamental concept in the study of musical tuning, illustrating the compromises and choices made in the development of musical scales and temperaments. Its existence highlights the complexity of achieving consonance within a musical system and the intricate balance between mathematical ratios and perceptual qualities of sound.
+* [[Just intonation]]
+* [[Pythagorean tuning]]
+* [[Musical temperament]]
+* [[Meantone temperament]]
+* [[Comma (music)]]
+[[Category:Musical tuning]]
 [[Category:Music theory]]
-[[Category:Musical tuning]]
-[[Category:Musical intervals]]
-{{music-stub}}
-<gallery>
-File:Syntonic_comma_on_C.mid|Syntonic_comma
-File:Syntonic_comma_on_C_HE_notation.png|Syntonic_comma
-File:Syntonic_comma_on_C.png|Syntonic_comma
-File:Just_perfect_fifth_on_D.png|Syntonic_comma
-File:Just_perfect_fifth_on_D.mid|Syntonic_comma
-File:Major_second_on_C.svg|Syntonic_comma
-File:Major_tone_on_C.mid|Syntonic_comma
-File:Minor_tone_on_C.png|Syntonic_comma
-File:Minor_tone_on_C.mid|Syntonic_comma
-File:Syntonic_comma_minor_third_Cuisenaire_rods_just.png|Syntonic_comma
-File:Syntonic_comma_major_third_Cuisenaire_rods_ET.png|Syntonic_comma
-File:Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png|Syntonic_comma
-</gallery>