AFDO

An AFDO (arithmetic frequency division of the octave) or ODO (otonal division of the octave) is a periodic tuning system which divides the octave according to an arithmetic progression of frequency.

For example, in 12afdo the first degree is 13/12, the second is 14/12 (7/6), and so on. For an AFDO system, the difference between interval ratios is equal (they form an arithmetic progression), rather than their ratios between interval ratios being equal as in EDO systems (a geometric progression). All AFDOs are subsets of just intonation, and up to transposition, any AFDO is a superset of a smaller AFDO and a subset of a larger AFDO (i.e. n-afdo is a superset of (n - 1)-afdo and a subset of (n + 1)-afdo for any integer n > 1).

When treated as a scale, the AFDO is equivalent to the overtone scale. An AFDO is equivalent to an ODO (otonal division of the octave). It may also be called an EFDO (equal frequency division of the octave), however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

Formula

Within each period of any n-afdo system, the frequency ratio r of the m-th degree is

[math]\displaystyle{ \displaystyle r = (n + m)/n }[/math]

Alternatively, with common frequency difference d = 1/n, we have:

[math]\displaystyle{ r = 1 + md }[/math]

In particular, when m = 0, r = 1, and when m = n, r = 2.

Relation to string lengths

If the first division has ratio of r₁ and length of l₁ and the last, r_n and l_n , we have: l_n = 1/r_n and if r_n > … > r₃ > r₂ > r₁, then l₁ > l₂ > l₃ > … > l_n

These lengths are related to the inverse of ratios in the system. The above picture shows the differences between divisions of length in 12ado system. On the contrary, we have equal divisions of length in EDL systems (→ EDL system):

Relation to superparticular ratios

An AFDO has step sizes of superparticular ratios with increasing numerators. For example, 5ado has step sizes of 6/5, 7/6, 8/7, and 9/8.

Relation to otonality & harmonic series

We can consider an AFDO system as an otonal system. Otonality is a term introduced by Harry Partch to describe chords whose notes are the overtones (multiples) of a given fixed tone. Considering AFDO, an otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an otonality because they can be written as 4/4, 5/4, 6/4. Every otonality is therefore part of the harmonic series. An otonality corresponds to an arithmetic series of frequencies or a harmonic series of wavelengths or distances on a string instrument.

History

In the earliest materials, the AFDO was known as the ADO, for arithmetic division of the octave. The term was proposed by Shaahin Mohajeri in 2006, along with the term EDL (equal division of length). Previously, the set of pitch materials equivalent to n-ado's had been known as "mode n of the harmonic series", "over-n scales", and n-edl's had been known as "aliquot-n" scales, as the distinction between tunings and scales were not made. Neither of Shaahin's two new concepts were systematic extensions of the term EDO (equal division of the octave), and no one else used these two terms besides Shaahin himself.

In 2021, a team consisting of Douglas Blumeyer, Billy Stiltner, and Paul Erlich developed the first systematic extension of EDO from equal divisions of pitch to equal divisions of frequency and length, including special terms for divisions of rational intervals such as the octave; under this system, an n-ADO would be an n-ODO.

In 2023, Flora Canou revived the old term, reinterpreting the word "arithmetic" as a reference to the arithmetic mean in addition to arithmetic progressions, then extended it through the other Pythagorean means, and later through all power means. As it was shown that arithmetic alone was insufficient to define the object since frequency could not be assumed, the term was eventually changed to AFDO, showing both the type of power mean and the sampled resource.

Individual pages for AFDOs

By size

0…99

0	1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99

200 and above

• 3746579

By prime family

Over-2: 2, 4, 8, 16, 32, 64, 128, 256, 512

Over-3: 3, 6, 12, 24, 48, 96, 192, 384

Over-5: 5, 10, 20, 40, 80, 160, 320

Over-7: 7, 14, 28, 56, 112, 224, 448

Over-11: 11, 22, 44, 88, 176, 352

Over-13: 13, 26, 52, 104, 208, 416

Over-17: 17, 34, 68, 136, 272, 544

Over-19: 19, 38, 76, 152, 304

Over-23: 23, 46, 92, 184, 368

Over-29: 29, 58, 116, 232, 464

Over-31: 31, 62, 124, 248, 496

By other properties

Prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Semiprime: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187

Odd squarefree semiprime: 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303

Nonprime prime power: 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048

Using AFDOs to make EDO interval tables

To make a quick and dirty table of intervals in an edo, you can go to Scale Workshop and do the following:

1. New scale > equal temperament > the edo you want
2. Paste the result into the first column in Excel
3. Go back to Scale Workshop
4. Modify > approximate by harmonics > an afdo selected according to the table below (or choose your own)
5. Modify > reduce
6. Paste the result into the second column in Excel
7. Title the first column "Edo step" and the second column "JI ratios"

Your table is now complete!

Do note that this is not a replacement for a manually curated table of intervals, nor is it even a replacement for a more cleverly computed table of intervals such as those generated by Template:Interval table or Template:Huge table. This method is very crude.

But it is very computationally light, so it can be helpful for really, really big edos that are difficult to document any other way.

Here is the table recommending what afdo to choose (i.e. what number to put for "approximate by harmonics"). The afdos recommended are the superior highly composite numbers (OEIS A002201):

EDO size	Ideal AFDO to map to	Included denominators	Excluded denominators
1edo to 20edo	60afdo	1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30	7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32
21edo to 40edo	120afdo	1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30	7, 9, 11, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32
41edo to 120edo	360afdo	1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30	7, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32
121edo to 840edo	2520afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30	11, 13, 16, 17, 19, 22, 23, 25, 26, 27, 29, 31, 32
841edo to 1680edo	5040afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30	11, 13, 17, 19, 22, 23, 25, 26, 27, 29, 31, 32
1681edo to 18480edo	55440afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30	13, 17, 19, 23, 25, 26, 27, 29, 31, 32
18481edo to 240240edo	720720afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30	17, 19, 23, 25, 27, 29, 31, 32
240241edo to 480480edo	1441440afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32	17, 19, 23, 29, 31
480480edo to 1,441,440edo	4324320afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32	17, 19, 23, 29, 31
1,441,441edo to 7,207,200edo	21621600afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32	17, 19, 23, 29, 31
7,207,201edo to 122,522,400edo	367567200afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32	19, 23, 29, 31
122,522,401edo to 2,327,925,600edo	6983776800afdo	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32	23, 29, 31

See also

• AFS (arithmetic frequency sequence)
• IFDO (inverse-arithmetic frequency division of the octave)
• Frequency temperament
• 5- to 10-tone scales from the modes of the harmonic series

External links

• Approximate EDO and AFDO systems with each other (Excel sheet)^{[dead link]}
• Fret position calculator (Excel sheet)
• Magic of Tone and the Art of Music by the late Dane Rhudyar