This week I stumbled upon what’s called the Divine 9 tuning, which should be an alternative to Equal Temperament. The Divine 9 website claims that this temperament is based on the order principles of nature itself, and looks more like a new age site than a music site. It keeps mentioning the importance of the number 9 and how this is derived straight from spirals using the Fibonacci sequence (including Egyptian gods, Chinese dragons and ‘Cosmic creation principles’). The Divine 9 tuning is supposed to use this number to give it it’s specific properties.

There’s nothing wrong with trying out different tunings, but the Divine 9 tuning somehow annoys me. More because of its presentation than because of the tuning itself. First of all I’m not sure whether it’s really an alternative to Equal Temperament. Secondly, the mathematics behind the importance of the number 9 and its relationship with nature is flawed. Finally, the way the number nine is part of the Divine 9 tuning is very artificial, and has hardly any influence on the tuning itself. From what I’ve heard of music played in Divine 9 tuning it sounds pleasant, but so does for example Pythagorean tuning (yes, my digital piano can play Pythagorean). I haven’t checked in which keys the Divine 9 example are, but I doubt it sounds that good in all 12 keys (which is exactly why we use equal temperament on a piano).

## Equal Temperament

Musical intervals sound pretty if their ratio is a fairly simple fraction. The simplest ones are 2:1 (octave), 3:2 (perfect fifth) and 4:3 (perfect fourth). So if you choose A = 440 Hz, you would like E to be 660 Hz (440 * 3/2). The problem is that if you stack 12 perfect fifth, you would multiply by (3/2)^12 = 129.75. However, 12 perfect fifth should be 7 octaves, or 2^7 = 128. So using only perfect ratios goes wrong in the end. We have to make small adjustments to end up at an octave in the end. Pythagorean tuning tries to use as many nice intervals as possible, but if you change keys everything goes wrong. So a piano tuned in Pythagorean C sounds great in C, but horrible in other keys. Equal temperament distributes the adjustments fairly over all notes, so all of the 12 keys have the same ‘error’. If you move away from equal temperament, some keys will sound better, but others get worse. There is a Divine 9 tuned piano, and the examples on youtube have a nice sound, but I haven’t heard a demonstration in all 12 keys. In some keys Divine 9 sounds better than equal temperament, but the same holds for e.g. Pythagorean tuning. A piano should sound good in all keys, and I doubt whether this is true for Divine 9 tuning.

## Divine 9 tuning

The website doesn’t really explain the details of the tuning, it just gives the frequencies of the notes. Also, it’s not clear what the ‘important’ number 9 has to do with it. The only reference I find is that adding up to 12 decimals of the frequency results in a multiple of 9. The weird thing is that the Divine 9 tuning isn’t based on intervals (as any other temperament that I’m aware of), but on absolute frequencies. Since Hertz is based on the pretty arbitrary definition of seconds, the number 9 showing up in absolute frequencies is also arbitrary. Anyway, here are the frequencies as far as I could reconstruct them:

A | 213.5742188 | 3^7 * 5^2 / 2^8 |
---|---|---|

A# | 227.8125 | 3^6 * 5^1 / 2^4 |

B | 240.3259277 | 3^2 * 5^6 * 7^1 / 2^12 |

C | 253.125 | 3^4 * 5^2 / 2^3 |

C# | 270 | 2^1 * 3^3 * 5^1 / 2^0 |

D | 284.765625 | 3^6 * 5^2 / 2^6 |

D# | 303.75 | 3^5 * 5^1 / 2^2 |

E | 321.865081787109375 | 3^3 * 5^8 / 2^15 |

F | 341.71875 | 3^7 * 5^1 / 2^5 |

F# | 360 | 2^3 * 3^2 * 5^1 / 2^0 |

G | 379.6875 | 3^5 * 5^2 / 2^4 |

G# | 405 | 3^4 * 5^1 / 2^0 |

A | 427.1484375 | 3^7 * 5^2 / 2^7 |

Note that A is tuned down from the usual 440 Hz all the way to 427.15 Hz. The Divine 9 site notes that the sum of the digits of the frequencies are a multiple of 9, which is obvious since all frequencies have a factor 9 in the nominator. Unfortunately this is only true for the arbitrary unit of Hertz, using more than 10 completely non-significant decimals. Anyway, music depends on intervals, not that much on absolute frequencies. The author claims that the number 9 shows up in any interval, but some of the perfect fifth for example are really perfect, that is they have a ratio of 3:2. That doesn’t look like number 9 to me.

Of course choosing your frequencies as fairly simple ratios, the intervals are automatically also integer ratios. For the sound of the temperament however, only the interval ratios matter. Divine 9 would sound pretty much the same if you multiply all frequencies by a constant close to 1, messing up the factor 9. All intervals would stay the same. So why not define the much more important interval ratios instead of absolute frequencies?

## Spirals?

Where it gets really weird is the reason why this number 9 should be so important. The Divine 9 websites mentions all kinds of spirals in nature, like sunflower seeds and nautilus shells. Many books have been written about the way Fibonacci sequences are related to spirals in nature, as well as to the golden ratio. The author Gert Kramer claims that the number 9 is really special in the Fibonacci sequence. Well, if you keep looking for 9 you’ll find it everywhere, as long as you ignore all the other numbers showing up. Numerologists seem to have a preference for computations modulo 9, since this is simply a matter of adding up the individual digits. That is, as long as you use the (arbitrary) base 10 number system. The strange thing is that Kramer mentions the importance of a base 9 number system, based on results you only get from using a base 10 number system. For the importance of the number 9, Kramer refers to his webpage ‘Bereken the orde’ (compute the order). He shows that if you compute the Fibonacci sequence modula 9, you get:

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 (with ‘9’ written for ‘0’)

Now if you take the first 12 numbersĀ (why not 9?), together with the next 12 (starting at 8 8), you get:

1 1 2 3 5 8 4 3 7 1 8 9

8 8 7 6 4 1 5 6 2 8 1 9

Add the corresponding numbers, and you end up with 9 each time. Of course this is nothing special. You’ll find an 8-8 sequence somewhere, and summing 1+8 twice results in 0 (modula 9). Once you’ve got a sequence of two zeroes, adding them (following the Fibonacci rule) doesn’t get anything but zeroes. According to Kramer this explains why Chinese dragons have 117 scales. It’s even the title of one of his books. (Actually Chinese dragons have 81 yang scales and 36 yin scales, both squares. And 117 is not only a multiple of 9 but also of 13. Again, choose what you like and ignore the obvious rest.)

Suppose we only had 6 fingers and we would thus be using a base 6 number system. Adding the digits of a base 6 number results in its value modulo 5. Now compute the Fibonacci sequence modulo 5, again using the number ‘5’ instead of ‘0’ to obscure things a little more:

1 1 2 3 5 3 3 1 4 5 4 4 3 2 5 2 2 4 1 5

Again, split the sequence at the part starting with 4-4:

1 1 2 3 5 3 3 1 4 5

4 4 3 2 5 2 2 4 1 5

Add up corresponding values again, and we get a multiple of 5 each time. Should we invent a Divine 5 tuning now? The whole idea of the importance of the number 9 in nature via spirals and Fibonacci sequences, is based on a simple mathematical trick that works for lots of different numbers as well. The only advantage of 9 is that computations modula 9 are easier due to our base 10 number system. If the reason dragons have 117 scales is that we have 10 fingers, so be it. But I don’t see how people having 10 fingers should influence a musical tuning.

## Conclusion

Apart from the pseudo-scientific ‘bs’ on the Divine 9 site, the tuning itself could be interesting. Every tuning has its advantages and disadvantages. Divine 9 sounds sweet (in selected keys), and the different sound can stimulate experimentation (just as e.g. Pythagorean tuning does, or for example dadgad on a guitar). However, it would be nice if it had been presented as a collection of nice (simple fraction) intervals, instead of a set of absolute frequencies which happen to be multiples of 9 in a specific unit of measurement. The suggestion that Divine 9 is somehow divine because it is derived from spirals in nature is absolute nonsense.

Hello Simon,

I agree with your vieuws about the divinity of the number 9 etc.

I myself dont argue normally with this type of reasoning.

The Divine 9 tuning doent seem to have very much to do with it anyway .Nice reconstruction of the calculation of the frequencies.

I would like to add something to the discussion.

In Equal Temperament Tuning the adaptation required to the frequencies is as you mentioned equally distributed , leading to an error of about 0.5 percent both in the terts and the quint interval , which makes all chords slighty, but equally , different from the ideal, sounds still good though.

The divine tuning gives errors of up to 1.5 % but not always in both the terts and quint at the same time, and the errors are different for different major chords (C,D etc, a small Excel table will do)

As you know from your Ukelele that nearly all music is made up of a melody and accompanying notes , mostly connected to different chords.

This is true for simple melodies ( f.i. silent night, only three chords) but also for most classical music. Most guitar players will realize this with the chords given in the annotation , players of Chopin waltzes dont normally realize that the left hand consists mainly of chords, normally quite a number of different ones.

In Divine 9 a change in key will make the music sound different , because the chords are going to be different , some which have a perfect terts ( say E C ratio) and imperfect quint( say G C ratio) can have the reverse when the piece is playedin a different key.

As you say the sound can be quite pleasurable.

It would be very interesting to hear a Chopin walz in Divine 9 tuning in a transposition to some other keys. It could change the mood completely.

By the way you cant have divine 9 on your Ukelele because the frets are for equal temperament.

Greetings,

Ru

Thank you for your comments. Of course even orchestral music consists mainly of chords, with the individual notes distributed over different instruments. As you mentioned, the difference with equal temperament depends on the key. It’s clear from the interval fractions, that some keys (like C) probably sound much better than others. It’s not that difficult to try out the Divine 9 tuning. I guess most digital piano’s can be retuned (at least my clavinova can). It is a bit of trouble though. Another possibility is to get one of the many midi recordings of e.g. Chopin and play it (potentially transposed) through a retuned midi synth. I haven’t tried, since I was more interested in the strange reasoning behind Divine 9, than the tuning itself.

Hi Simon,

I have an usual question for you. Are you able to create a scala file from the devine9 tuning? Please check out this site:

http://www.huygens-fokker.org/scala/

Kramer doesn’t seem to respond to my question.

I haven’t used scala for ages, but I guess it’s easy to create one. The frequency ratios are in my article (as products of primes). For example A=54675/256, A#=3645/16, B=984375/4096 etc. The scala file is a simple text file with the relative ratios, so starting at A the first line of the data is (3645/16)/(54675/256) = (3645*256)/(54675*16) = 933120/874800. The next line would be B/A = (984375/4096)/(54675/256) = (984375*256)/(54675*4096) = 252000000/223948800. Of course you could simplify these fractions. Scala understands fractions, so you could just enter 12 lines of fractions. I don’t know how exactly scala works, but you might need to create a separate file for each key, and in the case of A tune the A down to 427.15 Hz.