Natural Pure Tuning consists of the Pitch Reference Frequency and of pure intervals, whose frequency ratios are the quotients of natural numbers.  All is based on the natural overtone series. Natural Pure Tuning begins with 1 Hz and includes 432 HZ as well as 256 and 440 HZ.

The Natural Pure Tuning System consists of
Pitch Reference Frequencies
Intervals in Just Intonation
whose frequency ratios are the
quotients of natural numbers.

Just intonation or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of natural numbers.

Any interval tuned in this way is called a pure or just interval.

Scientific Tuning builds on Scientific Pitch (Verdi Tuning)
which is based on middle C (C4) = 256 Hz.

256 divided by 2 = 128
divided by 2 = 64
divided by 2 = 32
divided by 2 = 16
divided by 2 = 8
divided by 2 = 4
divided by 2 = 2
divided by 2 = 1 Hz (1 cycle per second)

1 Hz is the basic reference frequency of Natural Pure Tuning System.

256 Hz is the eighth octave of 1 Hz.

An octave means the doubling of an frequency.

Row of octaves:
0-1-2-4-8-16-32-64-128-256-512-1024-2048-4096-8192-16384-32768 .........

The Pitch Reference Frequencies are always

odd numbers, integral numbers, natural numbers

and bear reference to the traditonal 12 keynotes of western scales.

The second reference frequency could be 2 Hz.

Because 2 Hz belongs to the octavation of 1 Hz,

3 Hz is the next reference frequency for a pure scale.

4 Hz belongs to the octavation of 1 Hz. Next is 5 Hz and so on.....

1 3 5 7 9 11 13 15 17 19 27 45
C G E Bb D F Ab B Db Eb A Gb
Various builds of just intonation
Cosmo Welfare uses Zarlino by default

Just Intonation

Reference Frequencies

Natural Rure Tuning builds on scientific pitch, first proposed in 1713 by Joseph Sauveur. It is an absolute concert pitch standard which is based on middle C (C4) being set to 256 Hz rather than 261.62 Hz as in the common A440 pitch standard. I have expanded scientific pitch to the Scientific tuning system.


C is a symbol, which stands for the first note of a musical scale. C has no information about the real pitch. The frequenz of C is the important thing. We say: C has a frequency reference of 1 HZ. That is precise! The Concert pitch (A4) is 426,67 Hz. Another example: A is the first note. A has a frequency reference of 27 HZ. The Concert pitch (A4) is 432 Hz.

Concert pitch

Concert pitch refers to the pitch reference to which a group of musical instruments are tuned for a performance. Concert pitch may vary from ensemble to ensemble, and has varied widely over musical history. In Scientific Tuning the concert pitch varies according to the reference frequency.

Why does Just Intonation sound so good?


There are several reasons for this. First of all our auditory systems find it easier to process simple harmonic information than complex harmonic information. Just Intonation intervals are much easier for our ears to "digest". For example, a pure major third has the two tones in the ratio of 4:5, whereas a major third in 12 Tone Equal Temperament has them in a ratio which is approximately 504:635.

Our brains tell us that the first interval is more consonant, simply because it requires less effort to process than the second one. Of course, in the real world, things are rarely absolutely perfect and two tones in the ratio of 40001:50001 will be heard as "near enough" to 4:5.

However, the further an interval strays from a nice simple ratio, the harsher it will tend to sound. Intervals that do not form perfect small number ratios with each other produce beats.


  • Scientific Pitch
  • Octave
  • Hertz
  • Scale
  • Just Intonation
  • Interval
  • Chromatic
  • Natural Number
  • Odd Number
  • Integer Number
  • Overtone