source file: m1468.txt Date: Tue, 7 Jul 1998 17:16:14 -0400 Subject: decatonic scales in Partchian terms From: "Paul H. Erlich" This message is in MIME format. Since your mail reader does not understand this format, some or all of this message may not be legible. ------ =_NextPart_000_01BDA9EC.7A5BD3BA Content-Type: text/plain Carl Lumma wrote me, privately, regarding my 22-tET paper: >I recommend cutting the indian scale bit, and instead write >about the just versions of the decatonic scales using standard Partchian terms. Did you mean something like this? Let's start with the familiar diatonic (heptatonic) scale in JI. 1. 1/1 2. 10/9 or 9/8 3. 5/4 4. 4/3 5. 3/2 6. 5/3 7. 15/8 Mode 1 is major, mode 6 is minor. The tuning of note 2 depends on the chordal context. The following chords are complete through the 5-limit: 1,3,5 = 1/1:5/4:3/2 (1/1-O) 2,4,6 = 10/9:4/3:5/3 (5/3-U) 3,5,7 = 5/4:3/2:15/8 (15/8-U) 4,6,1 = 4/3:5/3:1/1 (4/3-O) 5,7,2 = 3/2:15/8:9/8 (3/2-O) 6,1,3 = 5/3:1/1:5/4 (5/4-U) 7,2,4 = X A meantone tuning, including 12-, 19-, or 31-tET, allows a single pitch to be used for note 2. Now the pentachordal (so named because it is built of pentachords instead of tetrachords) decatonic scale in JI: 1. 1/1 2. 21/20 or 15/14 3. 9/8 4. 5/4 5. 21/16 6. 10/7 7. 3/2 8. 5/3 9. 7/4 or 50/28 0. 15/8 Mode 1 is standard major, mode 4 is standard minor, mode 6 is alternate major, mode 9 is alternate minor. The following chords are complete through the 7-limit: 1,4,7,9 = 1/1:5/4:3/2:7/4 (1/1-O) 2,5,8,0 = X 3,6,9,1 = X 4,7,0,2 = 5/4:3/2:15/8:15/14 (15/8-U) 5,8,1,3 = X 6,9,2,4 = 10/7:50/28:15/14:5/4 (10/7-O) 7,0,3,5 = 3/2:15/8:9/8:21/16 (3/2-O) 8,1,4,6 = 5/3:1/1:5/4:10/7 (5/4-U) 9,2,5,7 = 7/4:21/20:21/16:3/2 (21/16-U) 0,3,6,8 = X The symmetrical decatonic scale has at least two equally economical representations in JI: REPRESENTATION 1: 1. 1/1 or 63/64 2. 21/20 or 15/14 3. 9/8 4. 5/4 5. 21/16 or 75/56 6. 45/32 or 10/7 7. 3/2 8. 63/40 or 45/28 9. 7/4 or 50/28 0. 15/8 REPRESENTATION 2: 1. 1/1 or 63/64 2. 21/20 or 15/14 3. 9/8 4. 49/40 or 5/4 5. 21/16 6. 7/5 or 45/32 7. 3/2 8. 63/40 or 45/28 9. 7/4 0. 147/80 or 15/8 Modes 1 and 6 are static major, modes 2 and 7 are dynamic major, modes 5 and 0 are static minor, and modes 4 and 9 are dynamic minor. The following chords are complete through the 7-limit: REPRESENTATION 1: 1,4,7,9 = 1/1:5/4:3/2:7/4 (1/1-O) 2,5,8,0 = 15/14:75/56:45/28:15/8 (15/14-O) 3,6,9,1 = X 4,7,0,2 = 5/4:3/2:15/8:15/14 (15/8-U) 5,8,1,3 = 21/16:63/40:63/64:9/8 (63/64-U) 6,9,2,4 = 10/7:5/4:15/14:5/4 (10/7-O) 7,0,3,5 = 3/2:15/8:9/8:21/16 (3/2-O) 8,1,4,6 = X 9,2,5,7 = 7/4:21/20:21/16:3/2 (21/16-U) 0,3,6,8 = 15/8:9/8:45/32:45/28 (45/32-U) REPRESENTATION 2: 1,4,7,9 = 1/1:5/4:3/2:7/4 (1/1-O) 2,5,8,0 = 21/20:21/16:63/40:147/80 (21/20-O) 3,6,9,1 = X 4,7,0,2 = 5/4:3/2:15/8:15/14 (15/8-U) 5,8,1,3 = 21/16:63/40:63/64:9/8 (63/64-U) 6,9,2,4 = 7/5:7/4:21/20:49/40 (7/5-O) 7,0,3,5 = 3/2:15/8:9/8:21/16 (3/2-O) 8,1,4,6 = X 9,2,5,7 = 7/4:21/20:21/16:3/2 (21/16-U) 0,3,6,8 = 15/8:9/8:45/32:45/28 (45/32-U) 22-tET allows a single pitch to be used for each note in the decatonic scales. ------ =_NextPart_000_01BDA9EC.7A5BD3BA Content-Type: application/ms-tnef Content-Transfer-Encoding: base64 eJ8+IhIVAQaQCAAEAAAAAAABAAEAAQeQBgAIAAAA5AQAAAAAAADoAAEIgAcAGAAAAElQTS5NaWNy b3NvZnQgTWFpbC5Ob3RlADEIAQWAAwAOAAAAzgcHAAcAEQAQAA4AAgAUAQEggAMADgAAAM4HBwAH ABEAEAAQAAIAFgEBCYABACEAAABEMzg5MzdDRUYwMTREMjExQUNEMjAwODA1RkJFM0MwNQAsBwEE gAEAJAAAAGRlY2F0b25pYyBzY2FsZXMgaW4gUGFydGNoaWFuIHRlcm1zAEENAQ2ABAACAAAAAgAC AAEDkAYAPAsAAC4AAAADADYAAAAAAAMABIAIIAYAAAAAAMAAAAAAAABGAAAAAFKFAAC3DQAAHgAF gAggBgAAAAAAwAAAAAAAAEYAAAAAVIUAAAEAAAAEAAAAOC4wAAMABoAIIAYAAAAAAMAAAAAAAABG AAAAAAGFAAAAAAAACwAAgAggBgAAAAAAwAAAAAAAAEYAAAAAA4UAAAAAAAALAA+ACCAGAAAAAADA AAAAAAAARgAAAAAOhQAAAAAAAAMAEIAIIAYAAAAAAMAAAAAAAABGAAAAABCFAAAAAAAAAwARgAgg BgAAAAAAwAAAAAAAAEYAAAAAEYUAAAAAAAADABSACCAGAAAAAADAAAAAAAAARgAAAAAYhQAAAAAA AB4AJIAIIAYAAAAAAMAAAAAAAABGAAAAADaFAAABAAAAAQAAAAAAAAAeACWACCAGAAAAAADAAAAA AAAARgAAAAA3hQAAAQAAAAEAAAAAAAAAHgAmgAggBgAAAAAAwAAAAAAAAEYAAAAAOIUAAAEAAAAB AAAAAAAAAAIBCRABAAAAGwYAABcGAAC0DgAATFpGdTh03jADAAoAcmNwZzEyNdIyAPszNgHoIAKk A+SfBxMCgwBQA9QCAGNoCsDQc2V0MBC2fQqACMhUIDsJbzAT72UOMDWzAoAKgXVjAFALA2MAQUUL YG4OEDAzMwumIEZDCsADIEx1bQDAICJ3A2B0ZSAHgCwgknAFEHZhGXBseRnALRQAZwsRC4BnGZB5 IEAyMi10RVQZ0GE6cASQOgqiCoQKgD5JDxqBBaAZAAnwZCBjdfMCQBrydGgZgAuAGuADkccE8AdA GYBiaXQZwABw6x2gC4BzGXBhHaAZQB9gPmUclQGgCGAFQB5CanV5IAAgdgSQAJACIAQgb/5mHjMF gRogAiAN4B7kBCCfIdAa8iAAH6ELESBQCsAOdBIAHrIZcHJtcy7pHDpEaR2geQhgGZEewo8DcBJQ JWAbAWxpaxmA+Sfhcz8cOhcxDlAAQBjQfRJQJwQgJIEAIBkwH2Bo8R4zZmFtAxAHMAXAHqHdI1Qo HlAFMCNEKR7lC4AIIEpJJgsxLiAxFi8W0wqAMi7xMC85oyKQBcA5LzgcNDMu8Cg1LzQcNDQu8DQv yxfwHEM1LvAzLw5QHEP2NjDyMfU3LvExEDB1HDS+TQRxLwAecAQgAMBqBbDzGcAEYiA2NXMLgAWw LvDyVB5RdHUDABsBIqE24P0ZcTIjARvwHZAigSWRHlE/EgAFsCTAAyAFoAIwZXj2dDcUAhBsCQAD 8BsBOXP/BCAKwDlRA3ALUBJQKHIDYGR1ZysUNS0oQCuQdAEcKzEsMyw1ID31LwI6MRE6MpIBkSyA LxGILU8pL0UsNCw2cPs/AS/hOjHRP1EX8D/jM1HMLVVAhT7BLDc+8T9lvjo0Ez/kNCFCx0ERLCoA /z8AQcVEQC8hP+Mx0UBnQ3HuLDhgPwBEFjowUj/jMpF/QGdF4T6wQ6NGoz9SQjY0r0LHSDFBAD7x WBw6QSdD7yNRN1YZwAuAYwpAGuMOILItGcAxOVABBbEzQFB/G5EfgTriO7Ee4BrxHyFw9x9gEgAe MG8fQBmAIdAJgO86sQXAOBQmC047AB4zOKH7AZA5dignoDgAK4BTEVLAnx8AUvEecAVANYFidQMQ fwVAIqFVOAQgH+YiohJQcr9YVS1AIx0ttD3cLvwyLxD3GGEFsTQRMTE1MOEwVzGhZzEXMmFdYTE2 Mscv0TerM4cylzgzOTku8DcxINUwEjUv4DIwdTAz/zUJ/ySHNbpNAGZLNtI2CgdAJbH/VkAZcjXJ MABpjDbYOs873/085jc9fz6CQRBIMDAAPws+OmOBDIRyREAeQ3A4LO8ScE0nPsBF4DlF801FcLL+ MEhDQ9pEQl4hcnhFCnPi/0qUTUV08kzkYTI/UGQCd0S/S2hhMkBndgI+xEiaOmBU/3J3Sbp5MkEU SvlEQGFCcnf/TAp6QkN0Y4F+kgHQfpQ/mG9gU0LHfVJF4DhNHjcyc/8GwCfBBRAfAVpPEhBtoQVA Fx8gieAhYXdSoGVxdf9RMRtABZE24CuQiJIUABngHweQVUId8CJiW18KUkUCUI4wU0VOVEFUeElP Ti8AW68vIDASNv0ykDYxNV0/Xk9fX2BiMBLqNzEQNWCYNDNROGCR4vdhT2JVkKE0kcOV4WQmY1// ZG+N347iRDCPT5BfkW+Sf/8xojBQmDSTvzLHmXA+4JhkX5YQlr+Xz5jbmjg0mXA4/59WZQ8EICoA H6I2cG3CJIH/HfAjkDW5BCA4YB+iQ5BtwvxkeVZBqr4+4B+iEnCqOv9otB+iq1RNAB+iMACsO2wf /20vbj9vT5uPjw9wr3G/cs//c9d7dJUDQbCYokRMn9BAZ/9033Xvdv94D3kYhJSlg8NC/55gSRie M0LHei6BgnuKfA//fR9+L38/gElNRYNfhG+Ff/+GhsmXleO8JEb1lfJCx7V//5x/t5+4r7l/c33N ysNUp7T/zrcB0L2Pvp+/r8C5wS/CP//DT9dnxK9M1aNxuPLNtaFD/9dYo3HIL8k/yknKv8vPzN// ze/Ov8/P0N/XZ9I/L3IbdP9RP1JPBbEgIPRROBMtwSLfFyPhJgUTAQD4MAADACYAAAAAAAMALgAA AAAACwACAAEAAAAeAHAAAQAAABwAAABTdGFuZGFyZCBQZW50YWNob3JkYWwgbWFqb3IAAgFxAAEA AAAbAAAAAb2kW9Hq1bvcHg9UEdKs0gCAX748BQFfviPgAEAAOQDws1B57Km9AQMA8T8JBAAAHgAx QAEAAAAGAAAAUEFVTEUAAAADABpAAAAAAB4AMEABAAAABgAAAFBBVUxFAAAAAwAZQAAAAAADAP0/ 5AQAAAMAgBD/////AgFHAAEAAAA0AAAAYz1VUzthPSA7cD1BY2FkaWFuLUFzc2V0O2w9TUFSUy05 ODA3MDcyMTE2MTRaLTE0MzAzAAIB+T8BAAAASwAAAAAAAADcp0DIwEIQGrS5CAArL+GCAQAAAAAA AAAvTz1BQ0FESUFOLUFTU0VUL09VPUFBTS9DTj1SRUNJUElFTlRTL0NOPVBBVUxFAAAeAPg/AQAA AA8AAABQYXVsIEguIEVybGljaAAAHgA4QAEAAAAGAAAAUEFVTEUAAAACAfs/AQAAAEsAAAAAAAAA 3KdAyMBCEBq0uQgAKy/hggEAAAAAAAAAL089QUNBRElBTi1BU1NFVC9PVT1BQU0vQ049UkVDSVBJ RU5UUy9DTj1QQVVMRQAAHgD6PwEAAAAPAAAAUGF1bCBILiBFcmxpY2gAAB4AOUABAAAABgAAAFBB VUxFAAAAQAAHMIY5ghDbqb0BQAAIMLrTW3rsqb0BHgA9AAEAAAABAAAAAAAAAB4AHQ4BAAAAJAAA AGRlY2F0b25pYyBzY2FsZXMgaW4gUGFydGNoaWFuIHRlcm1zAB4ANRABAAAALgAAADxCRTFCRkQw NzM5RDhEMTExQUNDODAwODA1RkJFM0MwNTE0OTYxNEBNQVJTPgAAAAsAKQAAAAAACwAjAAAAAAAD AAYQgTZ4nAMABxDvCAAAAwAQEAAAAAADABEQAQAAAB4ACBABAAAAZQAAAENBUkxMVU1NQVdST1RF TUUsUFJJVkFURUxZLFJFR0FSRElOR01ZMjItVEVUUEFQRVI6SVJFQ09NTUVORENVVFRJTkdUSEVJ TkRJQU5TQ0FMRUJJVCxBTkRJTlNURUFEV1JJVEUAAAAAAgF/AAEAAAAuAAAAPEJFMUJGRDA3MzlE OEQxMTFBQ0M4MDA4MDVGQkUzQzA1MTQ5NjE0QE1BUlM+AAAAKBk= ------ =_NextPart_000_01BDA9EC.7A5BD3BA-- ------------------------------ End of TUNING Digest 1468 *************************