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The Natural Scale

The teK Project is promoting the introduction and use of scales based on the Natural Harmonic Sequence, also known as the Chord of Nature. The Sequence is listed in Table 6 at the end of this document and forms the basis of all musical scales, but has never been used in its entirety by any musical tradition known to history. Table 1 shows its relation to other important scales, including the modern equal-tempered (ET) scale. The numbers in the Cents column (¢) give the ratio between each note and the lowest or keynote (G), and are explained in another document on this site.

Table 1   THE NATURAL & TRADITIONAL MUSICAL SCALES
Natural¢PentatonicPythagorean   Just   ¢ET
1612001200120012001200G'
151088-110910881100F#
14969906906---
13841--884900E
12702702702702700D
11471-(498)498500C
10386408408386400B
 9204204204204200A
 800000G

Most early scales arose from the practical considerations of tuning and tempering. The following discussion relies on an understanding of the material in the document on tempering on this site. The natural harmonics of a string or pipe include pure thirds and fifths, and a sequence of Perfect Fifths forms the basis of common tunings and the Pentatonic and Pythagorean scales.

Table2   Frequency ratios in violin tuning
StringNoteRatio to previousRatio to lowestSingle octave
HighestE3:227:827:16
3A3:29:49:8
2D3:23:23:2
LowestG---

Table 2 shows that a violin is tuned using consecutive Perfect Fifths. If this sequence is continued (G D A E B F# C#), seven notes of a scale are obtained, the seventh degree (F#) being tolerably dissonant, and the fourth degree (C#) intolerably so. The most common solution to this perennial problem is to move back down the sequence (C G D A E B F#) to obtain a Perfect Fourth (C) that is an inversion of a Perfect Fifth, yielding an extended Pythagorean scale:

Table 3   Notes of the Pythagorean Scale
DegreeNoteRatio to previousRatio to lowest¢
8G-2:11200
7F#9:8243:1281109
6E9:827:16906
5D9:83:2702
4C-(4:3)(498)
3B9:881:64408
2A9:89:8204
1G--0

This scale is still imperfect. The third degree is tolerably sharp (408 vs 386) but the sixth degree (906) lies between two natural notes (841 and 969). A common solution to this was provided by Just Temperament, in which each note was tuned relative to its predecessor rather than to the keynote. By a happy coincidence, a series of alternating Major (9:8) and Minor (10:9) Tones and two Major Semitones (16:15) gives a scale with pure thirds, fifths, seconds and sevenths, a tolerable sixth and a Perfect Fourth:

Table 4   Notes of the Just Scale
DegreeNoteRatio to previous¢
8G16:151200
7F#9:81088
6E10:9884
5D9:8702
4C16:15498
3B10:9386
2A9:8204
1G-0

This scale was most popular with choirs because of the purity of the thirds. Other tempering schemes such as the many varieties of meantone appeared and disappeared across the years, but with the rise of equal temper the whole topic was eventually forgotten. The impurity of modern intervals can be seen from Table 5:

Table 5   MUSICAL INTERVALS
Interval Ratio Cents ExampleET Cents
Unison1:10C - C0
Octave2:11200C - C'1200
Perfect Fifth3:2702C - G700
Perfect Fourth4:3498 C - F500
Major Sixth5:3884C - A900
Major Third5:4386C - E400
Minor Seventh7:4969C - Bb1000
Major Tone9:8204( C - D )200
Minor Tone10:9182( C - D )200
Major Semitone16:15112( C - C# )100
Minor Semitone17:16105( C - C# )100

The new scale being advocated for investigation by the teK Project is, in one sense, the oldest scale in existence, being the notes of the Fourth Octave of the Natural Harmonic Sequence. However, tuning an instrument accurately to this scale was too difficult prior to the advent of modern electronics, and it has never appeared in any known musical tradition. Today, many electronic synthesizers and any Personal Computer can be programmed to generate this scale, but because the whole topic of musical temperament has been all but forgotten, it still remains unknown musical territory. Table 6 below shows that the notes of the Fourth Octave can be used as a diatonic scale, with those of the Fifth Octave supplying a chromatic scale.

The purpose of the teK Project is to encourage exploration of these scales by modern musicians. For suggestions about naming and practising the new scale, see the page on Choirs on this site.

Table 6   THE NATURAL HARMONIC SEQUENCE
OctaveHarmonicRatioNote
6th321:1C
5th3131:16-
3015:8B
2929:16Bb
287:4-
2727:16A
2613:8-
2525:16Ab
243:2G
2323:16F#
2211:8-
21221:16F
205:4E
1919:16Eb
189:8D
1717:16C#
161:1C
4th1515:8B
147:4Bb
1313:8A
123:2G
1111:8F
105:4E
 99:8D
 81:1C
3rd 77:4Bb
 63:2G
 55:4E
 41:1C
2nd 33:2G
 21:1C
1st 11:1C