EDO: Difference between revisions

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An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[~~2/1|~~octave]] into a whole number of [[equal-step tuning|equal steps]]. A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" (or "''n''-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into ''n'' steps, each with frequency ratio 21/n. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 21/12. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well.

An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[octave]] into a whole number of [[equal-step tuning|equal steps]]. A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" (or "''n''-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into ''n'' steps, each with frequency ratio 21/n. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 21/12. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well.

== History ==

Line 20:

To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is

~~<center><math>\displaystyle~~ s = 1200 \cdot k/n~~</math></center>~~

$$ s = 1200 \cdot k/n $$

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''r'' of the ''k'' steps of ''n''-edo is

~~<center><math>\displaystyle c~~ = 2^{k/n}~~</math></center>~~

$$ r = 2^{k/n} $$

In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.

In particular, when ''k'' is 0, ''r'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''r'' is simply 2, because any number to the 1st power is itself.

== Properties ==

Line 46:

Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[~~the~~ Riemann zeta function ~~and tuning~~ #Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[Riemann zeta function #Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

EDOs with a less pronounced, yet still noteworthy [[~~the~~ Riemann zeta function ~~and tuning~~#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs with a less pronounced, yet still noteworthy [[Riemann zeta function#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of {{w|R. H. M. Bosanquet}}. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

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== Individual pages for EDOs ==

Note: Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki:Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below.

{{Note| Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki: Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below. }}

=== 0…999 ===

Line 1,229:

=== 1000…1999 ===

{{EDOs

| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}

| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1205, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}

=== 2000…9999 ===

Line 1,238:

=== 10000 and up ===

{{EDOs

| 10009, 10600, 10729, 11664, 12276, 12348, 12500~~, 13382~~, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813

| 10009, 10600, 10729, 11664, 12276, 12348, 12500, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813

}}

@@ Line 8: / Line 8: @@
 | ro = DEO
 }}
-An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[2/1|octave]] into a whole number of [[equal-step tuning|equal steps]]. A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" (or "''n''-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into ''n'' steps, each with frequency ratio 2<sup>1/n</sup>. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 2<sup>1/12</sup>. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well.
+An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[octave]] into a whole number of [[equal-step tuning|equal steps]]. A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" (or "''n''-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into ''n'' steps, each with frequency ratio 2<sup>1/n</sup>. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 2<sup>1/12</sup>. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well.
 == History ==
@@ Line 20: / Line 20: @@
 To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is
-<center><math>\displaystyle s = 1200 \cdot k/n</math></center>
+$$ s = 1200 \cdot k/n $$
-To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 2<sup>1/12</sup> (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is
+To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 2<sup>1/12</sup> (≈ 1.059). So the ratio ''r'' of the ''k'' steps of ''n''-edo is
-<center><math>\displaystyle c = 2^{k/n}</math></center>
+$$ r = 2^{k/n} $$
-In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.
+In particular, when ''k'' is 0, ''r'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''r'' is simply 2, because any number to the 1st power is itself.
 == Properties ==
@@ Line 46: / Line 46: @@
 Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.
-If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.
+If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[Riemann zeta function #Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.
-EDOs with a less pronounced, yet still noteworthy [[the Riemann zeta function and tuning#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.
+EDOs with a less pronounced, yet still noteworthy [[Riemann zeta function#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.
 EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of {{w|R. H. M. Bosanquet}}. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:
@@ Line 84: / Line 84: @@
 == Individual pages for EDOs ==
-Note: Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki:Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below.
+{{Note| Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki: Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below. }}
 === 0…999 ===
@@ Line 1,229: / Line 1,229: @@
 === 1000…1999 ===
 {{EDOs
-| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}
+| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1205, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}
 === 2000…9999 ===
@@ Line 1,238: / Line 1,238: @@
 === 10000 and up ===
 {{EDOs
-| 10009, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813
+| 10009, 10600, 10729, 11664, 12276, 12348, 12500, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813
 }}