Ptolemy's intense diatonic scale

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Diatonic scale on C, equal tempered Play and Ptolemy's intense or just Play. Diatonic scale on C.png
Diatonic scale on C, equal tempered Play and Ptolemy's intense or just Play .

Ptolemy's intense diatonic scale, also known as the Ptolemaic sequence, [1] justly tuned major scale, [2] [3] [4] Ptolemy's tense diatonic scale, or the syntonous (or syntonic) diatonic scale, is a tuning for the diatonic scale proposed by Ptolemy, [5] and corresponding with modern 5-limit just intonation. [6] While Ptolemy is famous for this version of just intonation, it is important to realize this was only one of several genera of just, diatonic intonations he describes. He also describes 7-limit "soft" diatonics and an 11-limit "even" diatonic.

Contents

This tuning was declared by Zarlino to be the only tuning that could be reasonably sung, it was also supported by Giuseppe Tartini, [7] and is equivalent to Indian Gandhar tuning which features exactly the same intervals.

It is produced through a tetrachord consisting of a greater tone (9:8), lesser tone (10:9), and just diatonic semitone (16:15). [6] This is called Ptolemy's intense diatonic tetrachord (or "tense"), as opposed to Ptolemy's soft diatonic tetrachord (or "relaxed"), which is formed by 21:20, 10:9 and 8:7 intervals. [8]

Structure

The structure of the intense diatonic scale is shown in the tables below, where T is for greater tone, t is for lesser tone and s is for semitone:

NoteNameCDEFGABC
SolfegeDoReMiFaSolLaTiDo
Ratio from C 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1
Harmonic 24 27 30 32 36 40 45 48
Cents020438649870288410881200
StepName TtsTtTs
Ratio9:810:916:159:810:99:816:15
Cents204182112204182204112
NoteNameABCDEFGA
Ratio from A1:19:86:54:33:28:59:52:1
Harmonic of Fundamental B120135144160180192216240
Cents020431649870281410181200
StepName TstTsTt
Ratio9:816:1510:99:816:159:810:9
Cents204112182204112204182

Comparison with other diatonic scales

Ptolemy's intense diatonic scale can be constructed by lowering the pitches of Pythagorean tuning's 3rd, 6th, and 7th degrees (in C, the notes E, A, and B) by the syntonic comma, 81:80. This scale may also be considered as derived from the just major chord (ratios 4:5:6, so a major third of 5:4 and fifth of 3:2), and the major chords a fifth below and a fifth above it: FAC–CEG–GBD. This perspective emphasizes the central role of the tonic, dominant, and subdominant in the diatonic scale.

In comparison to Pythagorean tuning, which only uses 3:2 perfect fifths (and fourths), the Ptolemaic provides just thirds (and sixths), both major and minor (5:4 and 6:5; sixths 8:5 and 5:3), which are smoother and more easily tuned than Pythagorean thirds (81:64 and 32:27) and Pythagorean sixths (27:16 and 128/81), [9] with one minor third (and one major sixth) left at the Pythagorean interval, at the cost of replacing one fifth (and one fourth) with a wolf interval.

Intervals between notes (wolf intervals bolded):

CDEFGABC′D′E′F′G′A′B′C″
C1:19:85:44:33:25:315:82:19:45:28:33:110:315:44:1
D8:91:110:932:274:340:275:316:92:120:964:278:380:2710:332:9
E4:59:101:116:156:54:33:28:59:52:132:1512:58:33:116:5
F3:427:3215:161:19:85:445:323:227:1615:82:19:45:245:163:1
G2:33:45:68:91:110:95:44:33:25:316:92:120:95:28:3
A3:527:403:44:59:101:19:86:527:203:28:59:52:19:412:5
B8:159:152:332:454:58:91:116:156:54:364:458:516:92:132:15
C′1:29:165:82:33:45:615:161:19:85:44:33:25:315:82:1
Pythagorean diatonic scale on C Play. Johnston's notation; + indicates the syntonic comma. Pythagorean diatonic scale on C.png
Pythagorean diatonic scale on C Play . Johnston's notation; + indicates the syntonic comma.

Note that D–F is a Pythagorean minor third or semiditone (32:27), its inversion F–D is a Pythagorean major sixth (27:16); D–A is a wolf fifth (40:27), and its inversion A–D is a wolf fourth (27:20). All of these differ from their just counterparts by a syntonic comma (81:80). More concisely, the triad built on the 2nd degree (D) is out-of-tune.

F-B is the tritone (more precisely, an augmented fourth), here 45:32, while B-F is a diminished fifth, here 64:45.

Related Research Articles

In music theory, a diatonic scale is any heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.

<span class="mw-page-title-main">Equal temperament</span> Musical tuning system with constant ratios between notes

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.

<span class="mw-page-title-main">Just intonation</span> Musical tuning based on pure intervals

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.

<span class="mw-page-title-main">Pythagorean tuning</span> Method of tuning a musical instrument

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.

<span class="mw-page-title-main">Meantone temperament</span> Musical tuning system

Meantone temperaments are musical temperaments; that is, a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are temperaments in that the fifths are not pure.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

<span class="mw-page-title-main">Chromatic scale</span> Musical scale set of twelve pitches

The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.

<span class="mw-page-title-main">Syntonic comma</span> Musical interval

In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125). Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.

<span class="mw-page-title-main">Wolf interval</span> Dissonant musical interval

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.

In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B, or D and C. It is equal to the frequency ratio (1.5)1227 = 531441524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often "temper" is the Pythagorean comma.

<span class="mw-page-title-main">Major second</span> Musical interval

In Western music theory, a major second is a second spanning two semitones. A second is a musical interval encompassing two adjacent staff positions. For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff positions. Diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones.

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees (of a major scale are called major.

<span class="mw-page-title-main">Semitone</span> Musical interval

A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C; the interval between them is a semitone.

In the musical system of ancient Greece, genus is a term used to describe certain classes of intonations of the two movable notes within a tetrachord. The tetrachordal system was inherited by the Latin medieval theory of scales and by the modal theory of Byzantine music; it may have been one source of the later theory of the jins of Arabic music. In addition, Aristoxenus calls some patterns of rhythm "genera".

<span class="mw-page-title-main">Ditone</span> Interval in music

In music, a ditone is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also called a comma-redundant major third; the smallest is the interval with a ratio of 100:81, also called a comma-deficient major third.

<span class="mw-page-title-main">Comma (music)</span> Very small interval arising from discrepancies in tuning

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Quarter-comma meantone, or  1 / 4 -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning ; the result is  3 / 2 × [ 80 / 81 ] 1 / 4 = 45 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

<span class="mw-page-title-main">53 equal temperament</span> Musical tuning system with 53 pitches equally-spaced on a logarithmic scale

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 2153, or 22.6415 cents, an interval sometimes called the Holdrian comma.

<span class="mw-page-title-main">Pythagorean interval</span> Musical interval

In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/ 21) and the perfect fourth with ratio 4/3 (equivalent to 22/ 31) are Pythagorean intervals.

<span class="mw-page-title-main">Regular diatonic tuning</span>

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

<span class="mw-page-title-main">Five-limit tuning</span>

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.

References

  1. Partch, Harry (1979). Genesis of a Music , pp. 165, 173. ISBN   978-0-306-80106-8.
  2. Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, pp. 172–73. ISBN   978-0-19-816505-7.
  3. Wright, David (2009). Mathematics and Music, pp. 140–41. ISBN   978-0-8218-4873-9.
  4. Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p. 78. ISBN   978-0-252-03098-7.
  5. see Wallis, John (1699). Opera Mathematica, Vol. III. Oxford. p. 39. (Contains Harmonics by Claudius Ptolemy.)
  6. 1 2 Chisholm, Hugh (1911). The Encyclopædia Britannica , Vol.28, p. 961. The Encyclopædia Britannica Company.
  7. Dr. Crotch (October 1, 1861). "On the Derivation of the Scale, Tuning, Temperament, the Monochord, etc.", The Musical Times, p. 115.
  8. Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN   0-945996-04-7 Chapter 2, Page 9
  9. Johnston, Ben; Gilmore, Bob (2006). 'Maximum Clarity' and Other Writings on Music. p. 100. ISBN   978-0-252-03098-7.