EDO: Difference between revisions

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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 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, 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 }}

| 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, 1460, 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, 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, 12655, 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, 72569, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813

}}

@@ 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 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, 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 }}
+| 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, 1460, 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, 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, 12655, 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, 72569, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813
 }}