﻿

# A new musical scale

proposed by Carsten Peust (© 2015)

On this page, I propose a new kind of musical scale which has significant advantages over the traditional Western scale, but which has never been used before (to my knowledge, at least).

F1
F2
F3
F4
F5
F6
F7
F1
(F8)
F2
(F9)
F3
(F10)
F4
(F11)
F5
(F12)
Last interval:
Just hit one of the function keys on your keyboard and hear what happens! (If you hear nothing, you may need to click on the page first.)

The traditional Western scale has three essential properties:
• The octave, the interval between two sounds whose frequencies have the ratio 2 : 1, is the basic unit and the only interval that can be played exactly.
• The octave is partitioned into 12 segments.
• The partition is equally spaced on a logarithmic scale (so-called equal temperament), so that the quotient of the frequencies of any two subsequent tones is constant. This facilitates the construction of musical instruments, but at the cost that almost all intended intervals can only be played in approximations.

The key requirement to any musical scale is making it possible to play good intervals. An interval between two tones will sound "good" if their frequencies have a simple rational quotient. The best, or cleanest, interval is the octave with a 2 : 1 relation. Other very harmonic relations are 3 : 2 (= 1.5 : 1) or 4 : 3 (= 1.333 : 1). By contrast, if the quotient is not (close to) a simple rational fraction, the interval will sound dissonantic.

To state this more precisely, the perceived quality of the interval depends on both the numerator and the denominator of the fraction (after reducing the fraction): they should both be low. I therefore define the quality of an interval as the maximum of its numerator and its denominator (after reducing). The transition between good and bad intervals is somewhat smooth, but Western tradition has set the limit at quality level 8 as defined here. The minor sixth, which approximates the fraction 8 : 5 (making up a quality level 8), is still counted as a canonical interval, whereas the major second, which (with 22/12 = 1.122) approximates the fraction 9 : 8 (quality level 9), is not usually regarded as harmonic. My perception may have been influenced by Western musical traditions, but I would confirm after some acoustical experiments that setting the limit at quality level 8 seems reasonable.

It turns out that a slight frequency deviation from the theoretical value is tolerable and the interval will still sound OK, though somewhat deteriorated. In the traditional Western scale, almost all intervals need to be approximated, which is the price to pay for fulfilling the above-mentioned equal temperament condition. Acoustical experiments show that deviations up to ≈ 1% sound acceptable.

If we agree on the quality limit at level 8, the following is an exhaustive list of all harmonic intervals. It becomes clear that four of them cannot be produced at all on the traditional Western scale. It is deplorable that most human beings have never heard these intervals even though they sound just as harmonic as the traditional ones.

Fraction Approximation on
level
8 : 7 = 1.143 (missing) (missing) 8
7 : 6 = 1.167 (missing) (missing) 7
6 : 5 = 1.200 23/12 = 1.189 Minor third 6
5 : 4 = 1.250 24/12 : 1 = 1.260 Major third 5
4 : 3 = 1.333 25/12 = 1.335 Perfect fourth 4
7 : 5 = 1.400 (missing) (missing) 7
3 : 2 = 1.500 27/12 = 1.498 Perfect fifth 3
8 : 5 = 1.600 28/12 = 1.587 Minor sixth 8
5 : 3 = 1.667 29/12 = 1.682 Major sixth 5
7 : 4 = 1.750 (missing) (missing) 7
2 : 1 = 2.000 212/12 = 2.000 Octave 2

On the other hand, five of the twelve intervals that can be played on the traditional scale are entirely unusable, namely 21/12 : 1 = 1.059, 22/12 : 1 = 1.122, 26/12 : 1 = 1.414, 210/12 : 1 = 1.782 and 211/12 : 1 = 1.888. The probability for a random interval (the interval between two randomly chosen tones) to be harmonic is 7/12 ≈ 58%, the remainder being wasted.

## A new musical scale

Let me now explain the innovative scale that I am suggesting here. While I accept the advantage of a scale based on the octave, I drop the requirement of equal spacing that was only posed by the mechanics of traditional musical instruments, but no longer makes any sense in the computer age. I therefore propose a scale that divides the octave into segments of unequal sizes. Numerous such scales could be conceived, but the scale implemented here partitions the octave into seven segments as follows.[1] The proposed names of the tones are given on the right; for reasons that will become clear below, the names are composed of "F" and a digit:

Fraction Name
1 : 1 = 1.000 F1
7 : 6 = 1.167 F2
5 : 4 = 1.250 F3
4 : 3 = 1.333 F4
35 : 24 = 1.458 F5
5 : 3 = 1.667 F6
7 : 4 = 1.750 F7
2 : 1 = 2.000 F1

Properties of this scale:
• All of the intervals defined above as harmonic can be produced on this scale.
• All these intervals will be exact rather than approximated.
• For any two radomly hit keys, the probability for the interval to be harmonic is ≈ 67%.

The following table is ordered by the intervals and shows how to play them:

Interval How to play
8 : 7 = 1.143 F2-F4, F5-F6, F7-F1
7 : 6 = 1.167 F1-F2, F3-F5
6 : 5 = 1.200 F5-F7, F6-F1
5 : 4 = 1.250 F1-F3, F2-F5, F4-F6
4 : 3 = 1.333 F1-F4, F3-F6, F7-F2
7 : 5 = 1.400 F3-F7, F6-F2
3 : 2 = 1.500 F2-F7, F4-F1, F6-F3
8 : 5 = 1.600 F3-F1, F5-F2, F6-F4
5 : 3 = 1.667 F1-F6, F7-F5
7 : 4 = 1.750 F1-F7, F4-F2, F6-F5

Several of these intervals have been unheard of to the present day, such as 7 : 5 = 1.4. Just hit the keys F3 and F7 to be one of the first humans ever to have heard this beautiful interval!

## How to play the scale

Playing the scale in a browser requires the Web Audio API which is part of HTML5 but not yet supported by all browsers (as of 2015). I tested this page on Google Chrome version 40.0 and Mozilla Firefox version 31.5; other browsers may not work. JavaScript must be active (which is usually true for most browsers).

The tones are played with the Function keys (F1-F12) that are available on most keyboards. I decided to start with 312 Hertz for F1, by which all frequencies come out as integer numbers. The highest tone that you can hear on F12 comes at 910 Hertz (being the octave of F5). The range covers approximately 1½ octaves.