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In discussion of musical tuning there is often talk about “equal divisions” of an octave or other interval. There is potential here for confusion as there are two different ways of equally dividing an octave.

Wait, two ways? You’d think there was only one way to divide an octave equally. You make all the notes the same size! But I’m being serious. You can equally divide an interval either **arithmetically** or **logarithmically**.

Arithmetically equal = each step is the same Hz difference from the next

Logarithmically equal = each step is the same ratio difference from the next

Let’s say you have two notes, A 440Hz and A’ 880Hz. The interval between these two notes is an octave. If I want to divide this octave into arithmetically equal notes then we simply split it in a way that the difference in frequencies are equal.

For example we could split it into 8 intervals. We get the difference between the two notes by subtracting the frequencies. 880 – 440 = 440. Then divide this into 8. 440 / 8 = 55. So the base step size is 55Hz.

Constructing the scale is done by starting from 440Hz and then adding 55Hz each time until we reach the octave at 880Hz:

440Hz | 495Hz | 550Hz | 605Hz | 660Hz | 715Hz | 770Hz | 825Hz | 880Hz | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

55Hz | 55Hz | 55Hz | 55Hz | 55Hz | 55Hz | 55Hz | 55Hz | ||||||||||

9/8 | 10/9 | 11/10 | 12/11 | 13/12 | 14/13 | 15/14 | 16/15 |

Each note is 55 Hz apart from the next. You can move up and down the scale by adding/subtracting 55 Hz. The result is the harmonic series segment 8:9:10:11:12:13:14:15:16 with a fundamental frequency of 55 Hz. This scale sounds nothing at all like the equal tempered scale, in fact it sounds similar to the scales you get by playing harmonics on a guitar string.

To hear what it sounds like, you can follow this Scale Workshop link – it will open a page where you can press qwerty keys to play in this tuning.

Let’s take the octave from A 440Hz to A’ 880Hz and look at it as a ratio. 880/440. This can be simplified to 2/1.

To divide 2/1 into 12 logarithmically equal steps we need to find the step size. 2/1^{1/12} = 1.05946309436 (approx).

Constructing the scale is done by starting from 440 Hz and then multiplying this value by the step size 1.05946309436 twelve times until we reach 880 Hz:

440Hz | 466.16Hz | 493.88Hz | 523.25Hz | 554.37Hz | 587.33Hz | 622.25Hz | 659.25Hz | 698.46Hz | 739.99Hz | 783.99Hz | 830.61Hz | 880Hz | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

26.16Hz | 27.72Hz | 29.37Hz | 31.12Hz | 32.96Hz | 34.92Hz | 37Hz | 39.21Hz | 41.53Hz | 44Hz | 46.62Hz | 49.39Hz | ||||||||||||||

1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 | 1.059 |

These steps don’t look equal in terms of frequency – the Hz values get larger with every step. Maybe by now you have noticed one way in which the steps *are* equal… Each step is an equal ratio difference from the next.

To hear what it sounds like (I mean it’s just 12edo at A440Hz), you can follow this Scale Workshop link – it will open a page where you can press qwerty keys to play in this tuning.

In an equal temperament, you can modulate between keys and every key will sound equally in-tune. Whereas arithmetic divisions will only give you perfectly-tuned harmonics of a single fundamental with no concept of tonal modulation.

Which of these two methods *sound* equal to the ear?

It’s the logarithmic version – also known as equal temperament or EDO (equal division of the octave).

When tuning theory people mention “equal” scales, safely assume that they’re probably talking about this method.

Equal temperaments sound equal because our perception of pitch is logarithmic itself. The hearing system isn’t listening out for equal difference in Hz, it’s listening out for equal difference in step size ratio. This isn’t immediately obvious until you’ve compared the two for yourself.

So this was just a quick and simple post to explain something that has caused a little confusion in the past. Hope someone will find it useful.

I started a small project where I take existing Sevish tracks and adapt them to different musical tuning systems. It is called Re-Tuned and is available from my Bandcamp page to stream or as a free download. In this project, you’ll hear many tracks with altered tunings. Many of these are originally written as microtonal pieces that are re-tuned to standard 12 equal!

Why have I done this‽ The project originally started out as trolling youtube videos which rendered familiar microtonal tunes in 12edo. It turned out to be an effective demonstration that 12edo is just another tuning with its own character, and that this kind of retuning subtly or dramatically altered the feeling of the music. I then started to produce microtonal retunings, for example Desert Island Rain (originally in 313edo) is rendered in 24edo, 19edo, 14edo and 9edo.

My musical philosophy is that music is a fun craft to be involved with and I’m glad it exists. When expressing yourself through music there are various tools in the toolbox – for example dynamics, timbre, tempo, tuning – though tuning is often forgotten about. I hope Re-Tuned makes the case that deliberate use of tuning is one more powerfully expressive and aesthetic parameter that musicians can use to hone their craft further.

Cover artwork by Juha Penttinen.

A new original Sevish album is coming out next month…

People working with musical scales have found various ways to categorise them. Recently I’ve heard of a new way of sorting scales depending on how big the difference is between the large and small step. This idea, the “step ratio spectrum” applies to *moment of symmetry* scales.

Reminder about moment of symmetry (MOS) scales: MOS scales have have exactly two step sizes: large and small, usually notated as L and s. One example is the major scale (LLsLLLs).

On the “soft” end of the spectrum, the large and small steps are almost equal in size. The melodic quality of these scales seems smoother to me.

As for the “hard” end, large steps are very exaggerated in how large they are in comparison to the razor-thin small steps. Melodically this sounds quite dramatic to me because of the obvious difference in the step sizes.

In the below video, you’ll hear one piece of music which explores various points on the hard side of this spectrum. The scale used is LsLsLsLsL, tuned to various equal temperaments:

EDO | L:s ratio | L (cents) | s (cents) | TAMNAMS name |
---|---|---|---|---|

313 | 53:12 | 203.2 | 46.0 | Superhard |

24 | 4:1 | 200 | 50 | Superhard |

19 | 3:1 | 189.5 | 63.2 | Hard |

14 | 2:1 | 171.4 | 85.7 | Basic |

9 | 1:1 | 133.3 | 133.3 | Equalized |

It’s because I only recently heard about scale hardness and found it an interesting perspective that I’m posting about the topic here. Maybe this could be a jumping off point for further reading and study about interesting tuning thingies.

The Xenharmonic Wiki is an online knowledge base relating to microtonal music and tuning theory. For a few years a bunch of us from the community have used the Xen Wiki to maintain a list of software plugins that you can use to make microtonal music in the DAW.

This week the list has been updated because of wonderful developments happening in the music technology world that will allow composers to more easily make microtonal music with a wide variety of synths and virtual instruments. That development is the widespread adoption of MIDI Polyphonic Expression (MPE). Widespread MPE support means that new tuning tools can be developed which systematically manage the tunings of various instruments at the same time. And indeed, such tools are already coming out this year, for example Oddsound MTS-ESP Suite and Infinitone DMT. A new section of the list has been added to catalogue these tools. An additional section about MPE synths in general was also added.

Two new tools have just appeared that will interest people working with microtonal scales and tunings: Leimma and Apotome. These tools were launched as part of CTM Festival 2021 and were created by Khyam Allami and Counterpoint.

Leimma is a browser-based tool for exploring, creating, hearing, and playing microtonal tuning systems.

Apotome is a browser-based generative music environment based on octave-repeating microtonal tuning systems and their subsets (scales/modes).

Eight mostly loop-based compositions for a Yamaha DX7ii FD synthesizer that was re-tuned to a microtonal scale. Released February 5, 2021.

Marcus Satellite’s new album uses tunings based on Erv Wilson’s work.

Terra Octava is a collaboration album from the collective at STAFFcirc.

Sounds of digital fusion, chiptune, jazz and electronic music in a variety of equal tempered tunings.

Featuring Cryptovolans, Reuben Gingrich, Jaq, Chimeratio, petet, manfish, STC_1001, STC_1002, STC_1003, STC_1004, Vince Kaichan, Hunter Van Brocklin, Tancla, Emelia K., Abd al-Mahdi, Themnotyou, Sintel, 0x70457465, limeboiler, clown core, b-knox, amimifafa, ordinate and Sevish.

The microtonal tunings include 34edo, 22edo, 19edo, 16edo, 10edo and others – but also the tiny ones like 6edo, 4edo, 3edo, 2edo, 1edo, 0edo!

My track Fuschiamarine in 7edo is on there.

I’m actually blown away by what everybody was able to achieve here. They’re all deep into their musical craft – and some of them are pros at tuning already – some are trying microtones for the first time. Hope you’ll enjoy the listen!

- Approximate scale by harmonics of an arbitrary denominator
- Approximate scale by subharmonics of an arbitrary numerator
- Approximate scale to equal divisions

- ‘Stretch/compress’ now works as it should
- ‘Tempo-sync beating’ now works as it should

- ‘Clear scale’ function now moved into ‘New’ menu
- ‘Mode’ renamed to ‘Subset’
- Various updates to the user guide

- Scale Workshop now works with Harmor and Sytrus by Image-Line, via .fnv tuning file (thanks to Azorlogh).
- The ‘Approximate by ratios’ option walks you through each step of your scale and proposes ratios that are approximately close.
- Generate scales from an ‘Enumerate chord’ e.g. 4:5:6:7:8

Here is some new microtonal music I heard recently.