User:BudjarnLambeth/Generalising equal divisions of the octave

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Author note

I have found myself confused trying to wrap my head around IFDOs, arithmetic tunings, harmonotonic tunings and other related concepts. This is my attempt to summarise what I have understood from what I have read on the wiki. Especially what I have read in the articles harmonotonic tuning, arithmetic tuning, equal-step tuning, AFDO and IFDO.

Either here or in the discussion section, please correct any errors I have made. Please also mention any more types of equal-ish tuning that I haven’t yet mentioned here but really should be mentioned.

Once all errors have been corrected and deficiencies ironed out, I plan to release this into main space as a beginner page version of harmonotonic tuning and arithmetic tuning. In the same way that comma basis is a beginner page version of dual list.

I think that I am an ideal person to do this, because I am a beginner, so if I can write a page that even I can understand, then I think anyone will be able to understand it.

The page itself

EDO (equal divisions of the octave)

The interval between a note, and another note with double its frequency, is called an octave, or 2/1.

If you divide the pitch space within an octave into equal parts, you get a tuning called an equal division of the octave (EDO).

For example, if you divide the octave into 12 equally spaced pitches, you get 12edo. If you divide it into 19 equally spaced pitches, you get 19edo.


EDn (equal divisions of n)

Remember that the octave is also known as 2/1, because if one note has a frequency of 1 unit, then the note an octave above has a frequency of 2 units. (The one above, 4 units, then 8, 16 and so on.)

This means that an EDO could also be called an ED2/1 or an ED2. So other names for 12edo include 12ed2/1 or 12ed2.

You could use some other interval instead of the octave, though. For example you could divide the perfect twelfth (3/1) to make an ED3/1, also known as ED3 or EDT. Or you could divide the perfect fifth (3/2) to make an ED3/2, also known as EDF.

You can use any number you want here. You can have ED7, ED5/4, EDπ, EDϕ, etc. All of these represent some interval you can divide into equal pitch slices.

Collectively, every EDn falls under the umbrella of equal-step tunings.


EPDn (equal pitch divisions of n)

So far we’ve been dividing the pitch space of our interval into equal slices. So another way you could say 12EDO or 12ED2/1, is 12EPDO or 12EPD2/1. Another way you could say 10ED5/4 is 10EPD5/4.


EFDn (equal frequency divisions of n)

But what if we divide the frequency space into equal slices instead? If we divide the octave into equal frequency slices, we get equal frequency divisions of the octave (EFDO or EFD2/1). If we divide 5/4 into equal frequency slices, we get EFD5/4.

These types of tunings are equal frequency divisions (EFDs).

AFDO

Equal frequency divisions of the octave (EFDO) can also be called arithmetic frequency divisions of the octave (AFDO), because they will divide up into just intonation intervals. However, equal frequency divisions of anything else beside the octave cannot be called arithmetic, because the intervals within them are not just.

To convert back and forth:

  • 1 EFDO = 1 AFDO
  • 2 EFDO = 2 AFDO
  • 3 EFDO = 3 AFDO
  • 4 EFDO = 4 AFDO

and so on.


ELDn (equal length divisions of n)

What if we find our interval on a string of an instrument, and then divide that interval into string segments of equal length? If we divide the octave into equal string length slices, we get equal length divisions of the octave (ELDO or ELD2/1). If we divide 5/4 into equal string length slices, we get ELD5/4.

These types of tunings are equal length divisions (ELDs).

IFDO

Equal length divisions of the octave (ELDO) can also be called inverse-arithmetic frequency divisions of the octave (IFDO), because they will divide up into just intonation intervals. However, equal frequency divisions of anything else beside the octave cannot be called arithmetic, because the intervals within them are not just.

To convert back and forth:

  • 1 ELDO = 2 IFDO
  • 2 ELDO = 4 IFDO
  • 3 ELDO = 6 IFDO
  • 4 ELDO = 8 IFDO

and so on.


Arithmetic tuning

All of the tunings we have described so far fall under the umbrella of arithmetic tunings. Arithmetic tunings include all tunings with equal step sizes of any kind of quantity: frequency, pitch, or length.

It may seem there is no further way to extend the concept of equal tuning beyond this point: after all how can there be an equal tuning that does not have equal steps. However there is still one final way to extend it.


Harmonotonic tuning

If you take an EFDn, and then multiply every one of its pitches by some number m, then you will end up with a stretched EFDn or a compressed EFDn.

If you do the same to an ELDn, you will end up with a stretched ELDn, or a compressed ELDn.

These no longer have equal steps, but they do still have steps that either grow or shrink by a consistent proportion. So while they are unequal, they still have a kind of ‘monotone’ overarching structure.

All tunings discussed here so far, including these ones, belong to the category of harmonotonic tunings.


Equal division? Why not equal multiplication?

So far, we have discussed taking some interval, like the octave, and dividing it into equal, or stretched equal, slices. But what about taking an interval and multiplying it?

Well, if you multiply an interval by 1, that is the same as dividing it by 1. For example, 1 equal division of the octave (1EDO) is equal to 1 equal multiplication of the octave (1EMO)⁠ ⁠[idiosyncratic term ].

If you multiply an interval by 2, that is the same as dividing it by 1/2. So 2EMO is the same as 1/2EDO. 3EMO is the same as 1/3EDO. 37ELMπ/2 is the same as 1/37ELDπ/2.

So, equal multiplications are actually already included within the categories we’ve discussed.

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