IFDO

An IFDO (inverse-arithmetic frequency division of the octave), or UDO (utonal division of the octave) is a periodic tuning system which divides the octave according to the inverse-arithmetic progression of frequency.

The inverse-arithmetic progression is known in general mathematics as the harmonic progression, but it would have been confusing to name this tuning a "harmonic division of the octave" because this mathematical sense of harmonic conflicts with the relevant musical sense of harmonic: divisions according to the harmonic mean correspond to subharmonic sequences, which are the opposite of harmonic sequences. And so "inverse-arithmetic progression" was coined to avoid this conflict, as well as to point to its relationship with the arithmetic progression.

For example, in 12ifdo the first degree is 24/23, the second is 24/22 (12/11), and so on. For an IFDO system, the difference between inverse interval ratios is equal (they form an inverse-arithmetic progression), rather than their difference between interval ratios being equal as in AFDO systems (an arithmetic progression). All IFDOs are subsets of just intonation, and up to transposition, any IFDO is a superset of a smaller IFDO and a subset of a larger IFDO (i.e. n-ifdo is a superset of (n - 1)-ifdo and a subset of (n + 1)-ifdo for any integer n > 1).

When treated as a scale, the IFDO is equivalent to the undertone scale, also known as an aliquot scale. An IFDO is equivalent to a UDO (utonal division of the octave). It may also be called an n-ELDO (equal length division of the octave) since it includes the pitches found by dividing the length of a string or resonating chamber into n equal parts; 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 n-ifdo, the frequency ratio c of the k-th step is

[math]\displaystyle c = (2n)/(2n - k)[/math]

Equal divisions of length

The equal division of length (EDL) is equivalent to the IFDO. However, n-edl corresponds to (n/2)-ifdo for any even number n. Therefore, EDL cannot be used to represent an odd-numbered IFDO.

n-edl divides a string length to n equal divisions, so we have n/2 divisions per octave. If the first division is l₁ and the last, l_n, we have:

l₁ = l₂ = l₃ = … = l_n

So the sum of divisions is l or the string length. Note that the number of divisions in octave is half of the string length. By dividing a string length of l to n divisions we have:

n:(n - 1):(n - 2):(n - 3):…:(n - m):…:1

where n - m is n/2.

For example, by dividing string length to 12 equal divisions we have a series as:

12:11:10:9:8:7:6:5:4:3:2:1

which shows 12-edl:

12:12 means 12 from 12 divisions, 12:11 means 11 from 12 divisions and so on. Ratios as 12:11 shows active string length for each degree, which is vibrating. EDL system shows ascending trend of divisions sizes due to its inner structure and if compared with EDO:

Relation to superparticular ratios

An IFDO has step sizes of superparticular ratios with decreasing numerators. For example, 5ifdo has step sizes 10/9, 9/8, 8/7, 7/6, and 6/5.

Relation to utonality and subharmonic series

We can consider an IFDO system as a utonal system. Utonality is a term introduced by Harry Partch to describe chords whose notes are the undertones (divisors) of a given fixed tone. Considering IFDO, a utonality is a collection of pitches which can be expressed in ratios that have the same numerators. For example, 7/4, 7/5, 7/6 form an utonality in which 7 as the numerator is called a "Numerary nexus".

Individual pages for IFDOs

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

By prime family

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

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

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

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

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

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

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

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

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

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

Under-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

Prime powers (without primes): 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

See also

• Through other Pythagorean means:
  • AFDO – arithmetic frequency division of the octave
  • EDO – equal division of the octave
• UD, or utonal division

External links

• 96~EDO | Equal divisions of length