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10_o major_n  _fw-fr to_o 1_o 17_o major_n 4_o to_o 3_o four_o  _fw-fr  _fw-fr to_o 2_o 8_o  _fw-fr  _fw-fr to_o 1_o 15_o  _fw-fr 3_o to_z 2_o 5_o  _fw-fr  _fw-fr to_o 1_o 12_o  _fw-fr 2_o to_z 1_o 8_o  _fw-fr 6_o to_z 5_o 3d_o minor_n 5_o to_o 4_o 3d_o major_n 4_o to_o 3_o four_o  _fw-fr 3_o to_z 2_o 5_o  _fw-fr 2_o to_z 1_o 8_o  _fw-fr all_o that_o be_v concord_n to_o the_o unison_n be_v also_o concord_n to_o the_o octave_n and_o all_o that_o be_v discord_n to_o the_o unison_n be_v discord_n to_o the_o octave_n and_o some_o of_o the_o intermediate_n concord_n be_v concord_n one_o to_o another_o as_o the_o two_o three_o to_o the_o five_o and_o the_o four_o to_o the_o two_o sixth_n so_o that_o the_o unison_n three_o fifth_z and_o octave_n or_o the_o unison_n four_o six_o and_o octave_n may_v be_v sound_v together_o to_o make_v a_o complete_a close_o of_o harmony_n i_o do_v not_o mean_v a_o close_a to_o conclude_v with_o for_o the_o plagal_n be_v not_o such_o but_o a_o complete_a close_n as_o it_o include_v all_o concord_n within_o the_o compass_n of_o diapason_n a_o scheme_n of_o which_o i_o have_v set_v down_o at_o the_o end_n of_o the_o forego_n staff_n of_o five_o line_n which_o contain_v the_o note_n by_o which_o the_o aforesaid_a concord_n be_v express_v the_o former_a two_o which_o ascend_v from_o the_o unison_n gamut_n by_o three_o major_n or_o minor_n and_o fifth_z up_o to_o the_o octave_n be_v usual_o call_v authentic_a as_o relate_v principal_o to_o the_o unison_n and_o best_a satisfy_v the_o ear_n to_o rest_v upon_o the_o other_o two_o which_o ascend_v by_o the_o fonrth_n and_o sixth_n minor_n or_o major_a up_o to_o the_o same_o octave_n be_v call_v plagal_n as_o more_o combine_v with_o the_o octave_n seem_v to_o require_v a_o more_o proper_a base_a note_n uzi._n a_o eight_o below_o the_o four_o and_o therefore_o not_o make_v a_o good_a conclude_v close_o and_o on_o the_o continual_a shift_v these_o or_o often_o change_v they_o depend_v the_o variety_n of_o harmony_n as_o far_o as_o consonancy_n reach_v which_o be_v but_o as_o the_o body_n of_o music_n in_o all_o contrapunct_a chief_o but_o indeed_o in_o all_o kind_n of_o composition_n i_o do_v not_o exclude_v a_o sprinkle_v of_o discord_n nor_o here_o meddle_v with_o air_n measure_n and_o rhythmus_n which_o be_v the_o soul_n and_o spirit_n of_o music_n and_o give_v it_o so_o great_a a_o command_a power_n the_o plagal_n mood_n descend_v by_o the_o same_o interval_n by_o which_o the_o authentic_a ascend_v which_o be_v by_o three_o and_o fifths_o and_o the_o authentic_a descend_v the_o same_o by_o which_o the_o plagal_n ascend_v viz._n by_o fourths_o and_o sixths_n one_o chief_o relate_v to_o the_o unison_n the_o other_o to_o the_o octave_n but_o that_o for_o which_o i_o describe_v these_o full_a close_n be_v chief_o to_o give_v as_o i_o promise_v a_o large_a account_n of_o the_o beforementioned_a combination_n of_o concord_n which_o increase_v the_o consonancy_n of_o each_o note_n and_o make_v a_o wonderful_a variegation_n and_o delightfulness_n of_o the_o harmony_n cast_v your_o eye_n upon_o the_o first_o of_o they_o in_o the_o authentic_a scale_n you_o will_v see_v that_o b_o mi_fw-mi have_v 3_o relation_n of_o consonnacy_n viz._n to_o the_o unison_n or_o give_v note_n g_o to_o the_o five_o and_o to_o the_o octave_n to_o the_o unison_n as_o a_o three_o minor_n to_o the_o five_o as_o a_o three_o major_n to_o the_o octave_n a_o six_o major_n so_o that_o its_o motion_n join_v after_o every_o five_o recourse_n i._n e._n at_o every_o six_o course_n with_o the_o unison_n every_o five_o with_o the_o diapente_fw-la or_o five_o every_o six_o course_n with_o the_o octave_n then_o consider_v the_o diapente_fw-la d_o sol_fw-mi re_fw-mi as_o a_o five_o to_o the_o unison_n it_o join_v with_o it_o every_o three_o course_n and_o as_o a_o four_o to_o the_o octave_n they_o join_v every_o four_o course_n then_o the_o octave_n with_o the_o unison_n join_v after_o every_o second_o vibration_n i._n e._n at_o every_o course_n now_o take_v a_o review_n of_o the_o variety_n of_o consonancy_n in_o these_o four_o note_n here_o be_v mix_v together_o in_o one_o consort_n the_o ration_n of_o 2_o to_o 1_o 3_o to_z 2_o 4_o to_z 3_o 5_o to_z 4_o 6_o to_z 5_o 5_o to_o 3._o and_o just_o so_o it_o be_v in_o the_o other_o close_v only_o change_v alternate_o the_o sixth_n you_o may_v see_v here_o within_o the_o space_n of_o three_o interval_n from_o the_o unison_n viz._n 3d_o 5_o and_o 8_o what_o a_o concourse_n there_o be_v of_o consonant_a ration_n to_o variegate_n and_o give_v as_o it_o be_v a_o
divide_v as_o far_o as_o you_o can_v find_v a_o common_a divisor_n to_o they_o both_o which_o be_v here_o do_v by_o 3_o and_o that_o bring_v they_o to_o the_o ration_n of_o 8_o to_o 5._o by_o this_o you_o see_v that_o a_o three_o minor_n 6_o to_o 5_o add_v to_o a_o four_o 4_o to_o 3_o make_v a_o six_o minor_n 8_o to_o 5._o if_o more_o ration_n be_v to_o be_v add_v set_v they_o all_o under_o each_o other_o and_o multiply_v the_o first_o antecedent_n by_o the_o second_o and_o that_o product_v by_o the_o three_o and_o again_o that_o product_v by_o the_o four_o and_o so_o on_o and_o in_o like_a manner_n the_o consequent_n substraction_n of_o one_o ration_n from_o another_o great_a be_v perform_v in_o like_a manner_n by_o multiply_v the_o term_n but_o this_o be_v do_v not_o lateral_o as_o in_o addition_n but_o crosswise_o by_o multiply_v the_o antecedent_n of_o the_o former_a i._n e._n of_o the_o great_a by_o the_o consequent_a of_o the_o latter_a which_o produce_v a_o new_a antecedent_n and_o the_o consequent_a of_o the_o former_a by_o the_o antecedent_n of_o the_o latter_a which_o give_v a_o new_a consequent_a and_o therefore_o it_o be_v usual_o do_v by_o a_o oblique_a decussation_n of_o the_o line_n for_o example_n if_o you_o will_v take_v 6_o to_o 5_o out_o of_o 4_o to_o 3_o you_o may_v set_v they_o down_o thus_o then_o 4_o multiply_v by_o 5_o make_v 20_o and_o 3_o by_o 6_o give_v 18._o so_o 20_o to_o 18_o i_o e._n 10_o to_o 9_o 9_o 4_o 3_o 6_o 5_o 20._o 28._o 10._o 9_o be_v the_o remainder_n that_o be_v subtract_v a_o three_o minor_a out_o of_o a_o four_o and_o there_o will_v remain_v a_o tone_n minor_n multiplication_n of_o ration_n be_v the_o same_o with_o their_o addition_n only_o it_o be_v not_o wont_a to_o be_v of_o divers_a ration_n but_o of_o the_o same_o be_v take_v twice_o thrice_o or_o often_o as_o you_o please_v and_o as_o before_o in_o addition_n you_o add_v divers_a ration_n by_o multiply_v they_o so_o here_o in_o multiplication_n you_o add_v the_o same_o ration_n to_o itself_o after_o the_o same_o manner_n viz._n by_o multiply_v the_o term_n of_o the_o same_o ratio_fw-la by_o themselves_o i_o e._n the_o antecedent_n by_o itself_o and_o the_o consequent_a by_o itself_o which_o in_o other_o word_n be_v to_o multiply_v the_o same_o by_o 2_o and_o will_v in_o the_o operation_n be_v to_o square_a the_o ration_n first_o propound_v or_o give_v the_o second_o ordinal_n power_n the_o ration_n first_o give_v be_v the_o first_o power_n or_o side_n and_o to_o this_o product_n if_o the_o simple_a ration_n shall_v again_o be_v add_v after_o the_o same_o manner_n as_o before_o the_o aggregate_v will_v be_v the_o triple_a of_o the_o ration_n first_o give_v or_o the_o product_n of_o that_o ration_n multiply_v by_o 3_o viz._n the_o cube_n or_o three_o ordinal_n power_n it_o be_v biquadrate_n or_o four_o power_n proceed_v from_o multiply_v it_o by_o 4_o and_o so_o successive_o in_o order_n as_o far_o as_o you_o please_v you_o may_v advance_v the_o power_n for_o instance_n the_o duple_n ration_n 2_o to_o 1_o be_v add_v to_o itself_o duple_v or_o multiply_v by_o 2_o produce_v 4_o to_o 1_o the_o ration_n quadruple_a and_o if_o to_o this_o the_o first_o again_o be_v add_v which_o be_v equivalent_a to_o multiply_v that_o say_v first_o by_o 3_o there_o will_v arise_v the_o ration_n octuple_a or_o 8_o to_o 1._o whence_o the_o ration_n 2_o to_o 1_o be_v take_v for_o a_o root_n its_o duple_n 4_o to_o 1_o will_v be_v the_o square_a its_o triple_a 8_o to_o 1_o the_o cube_n thereof_o etc._n etc._n as_o have_v be_v say_v above_o and_o to_o use_v another_o instance_n to_o duple_v the_o ration_n of_o 3_o to_o 2_o it_o must_v be_v thus_o square_v 3_o by_o 3_o gives_z 9_o 2_o by_o 2_o give_v 4._o so_o the_o duple_n or_o square_a of_o 3_o to_o 2_o be_v 9_o to_o 4._o again_o 9_o by_o 3_o be_v 27_o and_o 4_o by_o 2_o be_v 8_o so_o the_o cubic_n ration_n of_o 3_o to_o 2_o be_v 27_o to_o 8._o again_o to_o find_v the_o four_o power_n or_o biquadrate_n i._n e._n squared_a square_n 27_o by_o 3_o be_v 81_o 8_o by_o 2_o be_v 16_o so_o 81_o to_o 16_o be_v the_o ration_n of_o 3_o to_o 2_o quadruple_v as_o it_o be_v duple_v by_o the_o square_n tripled_a by_o by_o the_o cube_n etc._n etc._n to_o apply_v this_o instance_n to_o our_o present_a purpose_n 3_o to_o 2_o be_v the_o ration_n of_o diapente_fw-la or_o a_o five_o in_o harmony_n 9_o to_o 4_o be_v the_o ration_n of_o twice_o diapente_fw-la or_o a_o nine_o viz._n diapason_n with_o tone_n major_n 27_o to_o 8_o be_v the_o ration_n of_o thrice_o diapente_fw-la or_o three_o fifths_o which_o be_v diapason_n with_o six_o major_n viz._n 13_o major_n the_o ration_n of_o 81_o to_o 16_o make_v four_o fifths_o i._o e._n dis-diapason_n with_o two_o tone_n major_a i._n e._n a_o seventeen_o major_n and_o a_o comma_n of_o 81_o to_o 80._o to_o divide_v any_o ration_n you_o must_v take_v the_o contrary_a way_n and_o by_o extract_v of_o these_o root_n respective_o division_n by_o their_o indices_fw-la will_v be_v perform_v e._n gr_n to_o divide_v it_o by_o 2_o be_v to_o take_v the_o square_a root_n of_o it_o by_o 3_o the_o cubic_n root_n by_o 4_o the_o biquadratick_a etc._n etc._n thus_o to_o divide_v 9_o to_o 4_o by_z 2_o the_o square_a root_n of_o 9_o be_v 3_o the_o square_a root_n of_o 4_o be_v 2_o then_o 3_o to_o 2_o be_v a_o ration_n just_o half_o so_o much_o as_o 9_o to_o 4._o from_o hence_o it_o will_v be_v obvious_a to_o any_o to_o make_v this_o inference_n that_o addition_n and_o multiplication_n of_o ration_n be_v in_o this_o case_n one_o and_o the_o same_o thing_n and_o these_o hint_n will_v be_v sufficient_a to_o such_o as_o bend_v their_o thought_n to_o these_o kind_n of_o speculation_n and_o no_o great_a trespass_n upon_o those_o that_o do_v not_o the_o advantage_n of_o proceed_v by_o the_o ordinal_n power_n square_n cube_n etc._n etc._n as_o be_v before_o mention_v may_v be_v very_o useful_a where_o there_o be_v occasion_n of_o large_a progression_n as_o to_o find_v for_o example_n how_o many_o comma_n be_v contain_v in_o a_o tone_n major_n or_o other_o interval_n let_v it_o be_v how_o many_o be_v in_o diapason_n which_o must_v be_v do_v by_o multiply_v comma_n i_o e._n add_v they_o till_o you_o arrive_v at_o a_o ration_n equal_a to_o octave_n if_o that_o be_v seek_v viz._n duple_n or_o else_o by_o divide_v the_o ration_n of_o diapason_n by_o that_o of_o a_o comma_n and_o find_v the_o quotient_a which_o may_v be_v do_v by_o logarithm_n and_o herein_o i_o meet_v with_o some_o difference_n of_o calculation_n mersennus_n find_v by_o his_o calculation_n 58_o 1_o â—_o comma_n and_o somewhat_o more_o in_o a_o octave_n but_o the_o late_a nicholas_n mercator_n a_o modest_a person_n and_o a_o learned_a and_o judicious_a mathematician_n in_o a_o manuscript_n of_o he_o of_o which_o i_o have_v have_v a_o sight_n make_v this_o remark_n upon_o it_o in_o solvendo_fw-la hoc_fw-la problemate_fw-la aberrat_fw-la mersennus_n and_o he_o work_v by_o the_o logarithm_n find_v out_o but_o 55_o and_o a_o little_a more_o and_o from_o thence_o have_v deduce_v a_o ingenious_a invention_n of_o find_v and_o apply_v a_o least_o common_a measure_n to_o all_o harmonic_a interval_n not_o precise_o perfect_a but_o very_o near_o it_o suppose_v a_o comma_n to_o be_v 1_o 53_o part_n of_o diapason_n for_o better_a accommodation_n rather_o than_o according_a to_o the_o true_a partition_n 1_o 55_o which_o 1_o 53_o he_o call_v a_o artificial_a comma_n not_o exact_a but_o differ_v from_o the_o true_a natural_a comma_n about_o 1_o 20_o part_n of_o a_o comma_n and_o 1_o 1000_o of_o diapason_n which_o be_v a_o difference_n imperceptible_a then_o the_o interval_n within_o diapason_n will_v be_v measure_v by_o comma_n according_a to_o the_o follow_a table_n which_o you_o may_v prove_v by_o add_v two_o or_o three_o or_o more_o of_o these_o number_n of_o comma_n to_o see_v how_o they_o agree_v to_o constitute_v those_o interval_n which_o they_o ought_v to_o make_v and_o the_o like_a by_o substract_v intervals_n â—_o 53_o comma_n 1_o diesis_fw-la 2_o semit_fw-la minus_n 3_o semit_n medium_fw-la 4_o semit_fw-la majus_n 5_o semit_n max._n 6_o tone_n minor_n 8_o tone_n major_a 9_o 3d_o minor_a 14_o 3d_o major_n 17_o four_o 22_o tritone_n 26_o semidiapente_v 27_o 5_o 31_o 6_o minor_n 36_o 6_o major_n 39_o seven_o minor_n 45_o seven_o major_n 48_o octave_n 53_o this_o i_o think_v fit_a on_o this_o occasion_n to_o impart_v to_o the_o reader_n have_v leave_n so_o to_o do_v from_o mr._n mercator_n friend_n to_o who_o he_o present_v the_o say_a manuscript_n here_o i_o may_v advertise_v