User:BudjarnLambeth/Generalising equal divisions of the octave - Revision history

BudjarnLambeth: Marked as editable user page

2024-03-05T04:01:07Z

Marked as editable user page

← Older revision		Revision as of 04:01, 5 March 2024
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	I have found myself confused trying to wrap my head around ~~AFDOs~~, ~~IFDOs~~, and ~~all the similar acronyms~~. 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]].		{{Editable user page}}

			= 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.		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.
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	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.		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) ==		== EDO (equal divisions of the octave) ==

BudjarnLambeth at 05:15, 29 January 2024

2024-01-29T05:15:16Z

← Older revision		Revision as of 05:15, 29 January 2024
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	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.		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 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.		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.

BudjarnLambeth at 03:49, 27 January 2024

2024-01-27T03:49:26Z

← Older revision		Revision as of 03:49, 27 January 2024
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	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.		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.


	== EDO (equal divisions of the octave) ==		== EDO (equal divisions of the octave) ==
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	Collectively, every EDn falls under the umbrella of '''[[equal-step tuning]]s'''.		Collectively, every EDn falls under the umbrella of '''[[equal-step tuning]]s'''.


	== EPDn (equal pitch divisions of n) ==		== 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.		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'''.

BudjarnLambeth: /* EDn (equal divisions of n) */

2024-01-27T03:48:27Z

EDn (equal divisions of n)

← Older revision		Revision as of 03:48, 27 January 2024
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	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.		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 ~~tritave~~ (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 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.		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.

BudjarnLambeth: /* EDn (equal divisions of n) */

2024-01-27T03:47:01Z

EDn (equal divisions of n)

← Older revision		Revision as of 03:47, 27 January 2024
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	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.		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 tuning]]s.		Collectively, every EDn falls under the umbrella of '''[[equal-step tuning]]s'''.


	== EPDn (equal pitch divisions of n) ==		== EPDn (equal pitch divisions of n) ==

BudjarnLambeth: Created draft of page

2024-01-27T03:46:29Z

Created draft of page

New page

I have found myself confused trying to wrap my head around AFDOs, IFDOs, and all the similar acronyms. 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.

== 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 tritave (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 tuning]]s.

== 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 ([[EFD]]s)'''.

=== 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 ([[ELD]]s)'''.

=== 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 tuning]]s'''. 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 tuning]]s'''.

== 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}}.

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.